Portfolio Protection Strategies - DiVA portal1229553/FULLTEXT01.pdf · portfolio’s equity...

IN DEGREE PROJECT MATHEMATICS,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2018

Portfolio Protection StrategiesA study on the protective put and its extensions

GUSTAV ALPSTEN

SERCAN SAMANCI

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ENGINEERING SCIENCES

Portfolio Protection Strategies A study on the protective put and its extensions GUSTAV ALPSTGEN SERCAN SAMANCI Degree Projects in Financial Mathematics (30 ECTS credits) Degree Programme in Applied and Computational Mathematics KTH Royal Institute of Technology year 2018 Supervisor at KTH: Anja Janssen Examiner at KTH: Anja Janssen

TRITA-SCI-GRU 2018:303 MAT-E 2018:70 Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

Acknowledgements

We would like to express our gratitude to Claes Wachtmeister for introducing us

to the subject and his supervising of this thesis. We would also like to thank Anja

Janssen for her support and valuable remarks towards our report.

1

Abstract

�e need among investors to manage volatility has made itself painfully clear over

the past century, particularly during sudden crashes and prolonged drawdowns in

the global equity markets. �is has given rise to a liquid portfolio insurance mar-

ket in the form of options, as well as a�racted the a�ention of many researchers.

Previous literature has, in particular, studied the e�ectiveness of the widely known

protective put strategy, which serially buys a put option to protect a long position

in the underlying asset. �e results are o�en uninspiring, pointing towards few, if

any, protective bene�ts with high option premiums as a main concern. �is raises

the question if there are ways to improve the protective put strategy or if there

are any cost-e�cient alternatives that provide a relatively be�er protection. �is

study extends the previous literature by investigating potential improvements and

alternatives to the protective put strategy. In particular, three alternative put spread

strategies and one collar strategy are constructed. In addition, a modi�ed protective

put is introduced to mitigate the path dependency in a rolling protection strategy.

�e results show that no option-based protection strategy can dominate the other

in all market situations. Although reducing the equity position is generally more

e�ective than buying options, we report that a collar strategy that buys 5% OTM

put options and sells 5% OTM call options has an a�ractive risk-reward pro�le and

protection against drawdowns. We also show that the protective put becomes more

e�ective, both in terms of risk-adjusted return and tail protection, for longer matu-

rities.

2

Abstrakt

Hantering av volatilitet i �nansiella marknader har under de senaste decennierna

visat sig vara nodvandigt for investerare, framfor allt i samband med krascher och

langdragna nedgangar i de globala aktiemarknaderna. De�a har ge� upphov till

en likvid derivatmarknad i form av optioner samt vackt e� intresse for forskning i

omradet. Tidigare studier har i synnerhet undersokt e�ektiviteten i den valkanda

protective put-strategin som kombinerar en lang position i underliggande aktie med

en put-option. Resultaten ar o�a inte tilltalande och visar fa fordelar med strategin,

dar dess hoga kostnader ly�s upp som e� stort problem. Saledes vacks fragan om

protective put-strategin kan forba�ras eller om det mojligtvis �nns nagra kostnad-

se�ektiva alternativ med relativt ba�re sakerhet mot eventuella nedgangar i under-

liggande. Denna studie utvidgar tidigare forskning i omradet genom a� undersoka

forba�ringsmojligheter for och alternativ till protective put-strategin. Sarskilt stud-

eras tre olika put spread-strategier och en collar-strategi, samt en modi�erad ver-

sion av protective put som amnar a� minska pa vagberoendet i en lopande option-

sstrategi.

Resultatet fran denna studie pekar pa a� ingen optionsbaserad strategi ar universellt

bast. Generellt se� ger en avy�ring av delar av aktieinnehavet e� mer e�ektivt

skydd, men vi visar a� det �nns situationer da en collar-strategi som koper 5 % OTM

put-optioner och saljer 5% OTM call-optioner har en a�raktiv risk-justerad pro�l och

sakerhet mot nedgangar. Vi visar vidare a� protective put-strategin blir mer e�ektiv,

bade i termer av en risk-justerad avkastning och som sakerhet mot svansrisker, for

langre forfallodatum pa optionerna.

3

Contents

Introduction 7

�eoretical Background 12

2.1 Option contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 Mathematical de�nition of European options . . . . . . . . . . . . . . . 13

2.1.2 ATM, ITM and OTM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Black-Scholes-Merton Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 Black-Scholes-Merton model with dividends . . . . . . . . . . . . . . . . 17

2.2.2 Black-Scholes-Merton model for a stock index . . . . . . . . . . . . . . . 18

2.2.3 Merton’s Jump Di�usion Model . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.4 Convexity of options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.1 Realized versus implied volatility . . . . . . . . . . . . . . . . . . . . . . 22

2.3.2 Volatility risk premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.3 Volatility surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Protection strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.1 Protective Put Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.2 Bear Put Spread Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4.3 Collar Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Methodology 31

3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 Assumptions and delimitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Option-based protection strategies and portfolios . . . . . . . . . . . . . . . . . 33

3.3.1 Protective Put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3.2 Divested Equity Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4

3.3.3 Low-cost Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3.4 Fractional Protective Put . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3.5 Fractional Put Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4 Evaluation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.5 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.6 Backtesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.6.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.6.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Results 44


4.2 Backtesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2.1 Whole period: December 2002 - Mars 2018 . . . . . . . . . . . . . . . . . 54

4.2.2 Financial crisis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Discussion 66


5.1.1 Risk-adjusted performance . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.1.2 Peak-to-trough drawdown characteristics . . . . . . . . . . . . . . . . . 71

5.2 Backtesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2.1 Risk-adjusted performance . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2.2 Peak-to-trough drawdown characteristics . . . . . . . . . . . . . . . . . 76

5.2.3 Financial crisis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Appendix 86

6.1 Modelling examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.1.1 Protective Put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.1.2 Divested Equity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5

6.2 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.2.1 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.2.2 Backtesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.3 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.3.1 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.3.2 Backtesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.3.3 Total cost of strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6

Introduction

�e watchword among traders of �nancial instruments has been, and is today, volatility. It is

a measure that captures the degree of uncertainty in the price movements of �nancial assets.

Following a sequence of unprecedented global economic events, such as the �nancial crisis in

2008, and the rise of derivatives, it has become increasingly important to understand the causes

and implications of volatility. More importantly, it has become crucial for traders to manage the

presence of volatility in their assets, either by risk-reducing measures or by making a bet on its

direction. �e need among investors to address the volatility has made itself painfully clear over

the past century, in particular during sudden crashes and prolonged drawdowns in the global

equity markets.

�ere are today a whole variety of measures and instruments that allow market participants to

protect their equity investments against market drawdowns. Two widely used hedging meth-

ods are short-selling and the use of derivatives. Unlike long positions, short-selling equity can

become very costly due to margin and collateral requirements, and there are major risks that

can lead to unexpected losses given its speculative character. However, the short-selling strategy

is rather limited, since not all stocks can be sold short. More commonly, investors use deriva-

tives to hedge their equity investments. Derivatives are contracts that can be used to reduce the

portfolio’s equity exposure by providing downside protection during market drawdowns. �e

most common instruments used for hedging are futures and options. Both are traded on liq-

uid and transparent markets. �e distinctive di�erence between them, from a hedging point of

view, resides in the investment �exibility and liability a�ributed to the contract parties. Sophisti-

cated hedgers o�en seek �exibility to construct suitable pay-o� pro�les, which are best achieved

through option contracts. Futures provide li�le �exibility in this ma�er and is generally inferior

to options when it comes to hedging equity investments. Also, the �nancial liability associated

with futures are larger than those for options making them less a�ractive. �e popularity of

options as hedging instruments has led to a multitude of protection strategies (Fodor, Doran,

Carson and Kirch, 2010).

7

�e concept of hedging equity investments against market drawdowns, while allowing upside

participation, is o�en referred to as portfolio insurance (PI). PI strategies have, over the past

decades, become an important tool for investors as well as a popular subject for research. Leland

and Rubinstein (1976) introduced the Option Based Portfolio Insurance (OBPI), which consists

of a long position in a risky asset (usually equity) and long a position in a put option with a

strike price equal to the insured amount. Another strategy, called Constant Proportion Portfolio

Insurance (CPPI), was introduced by Perold (1986) and Black and Jones (1987) as an alternative to

the OBPI. CPPI ensures a prede�ned �oor, or a cushion, by dynamically rebalancing allocations

between the risky and risk-less asset (Bertrand and Prigent, 2005).

�e e�ectiveness of the OBPI and CPPI strategies have been thoroughly studied and compared in

the literature. �e aggregated results from previous studies, Zhu and Kavee (1988), El Karoui et

al. (2005), Bertrand and Prigent (2005), Zagst and Kraus (2011) and Bertrand and Prigent (2011),

conclude that there is no dominant strategy. Perold and Sharpe (1988) and Black and Rouhani

(1989) show that the relative performance of the strategies depend on the market trend and the

level of the volatility. Annaert et al. (2009), similarly concludes that no strategy can dominate the

other in all market situations. �e appropriate strategy must be chosen based on the investor’s

expectations on the market situation.

In more recent studies, Figlewski et al. (2013) and Israelov (2017) examined the performance of

the protective put strategy, which is a development of the OBPI method. Figlewski et al. (2013)

conducted a simulation study to examine the performance of the protective put strategy using

three di�erent types of strike methods: a �xed strike, a �xed percentage strike and a combination

of both. Figlewski et al. (2013) argue that the �xed percentage strike method, which resets the

strike price at a �xed percentage of the stock’s current price at the time of a rollover, is a more

accurate description of the protective put strategy used by actual investors. It is found that

the �xed percentage strike is much less protective than the �xed strike method in a prolonged

bear market and provides very limited protection when out-of-the-money (OTM) puts are used.

�e �xed strike method is more costly during low volatility periods and resembles more a long

stock position over longer investment horizons, as the stock price tends to dri� away from the

�xed strike price. �e results from the study, in terms of combined limited downside risk and

a�ractive mean return, favors a �xed percentage strategy using in-the-money (ITM) or at-the-

8

money (ATM) puts. Figlewski et al. (2013) conclude that the protective put is more approriate

when the true expected return on the stock is higher than the risk-free rate, but the stock is

expected to underperform.

Israelov (2017) measured the e�ectiveness of the protective put strategy by comparing its peak-

to-trough drawdown characteristics to those of a static risk-reducing strategy. �e risk-reducing

strategy, referred to as the divested equity strategy, manages downside risk by reducing the

exposure to the risky asset and does not involve any use of options. Israelov (2017) split his

study into two parts, a real-world implementation using the CBOE S&P 500 5% Put Protection

Index and an idealized environment through a Monte Carlo simulation. Israelov (2017) �nds that

unless the option purchases and their maturities are timed just right around market drawdowns,

the protective put strategy may o�er li�le downside protection. �e culprits are the high cost

of put options and the path dependency of a rolling option strategy. Buying put options reduces

the portfolio’s beta relative to S&P 500 index and provides negative alpha due to the existence

of volatility risk premium. �e combined e�ect of reduced beta and negative alpha impacts the

return more negatively than not having put options in the portfolio. Israelov (2017) continues to

show that even in an idealized environment, where there is no volatility risk premium (i.e. no

extra costs incurred for the options), portfolios that are protected with put options have worse

peak-to-trough drawdown characteristics per unit of expected return than portfolios that have

instead statically reduced their equity exposure in order to reduce risk. In a real-world scenario,

in which there is non-zero volatility risk premium, the situation was shown to be worse.

Israelov (2017) and Figlewski et al. (2013) both extended the theoretical research done by Zhu

and Kavee (1988), El Karoui et al. (2005), Bertrand and Prigent (2005), Annaert et al. (2009),

Zagst and Kraus (2011) and Bertrand and Prigent (2011) to option-based strategies over longer

investment horizons. However, none of the studies examined alternative protection strategies

and their performance relative to the protective put. It would be interesting to shed light on

potential extensions of, or substitutes to, the protective put strategy in order to �nd out if there

are alternatives that have more a�ractive protective characteristics.

A common way to reduce the cost of the protective put is to �nance the strategy by selling an

OTM put or an OTM call. �e former creates a payo� pro�le referred to as a bear put spread

and sacri�ces some protection in exchange for lower net premium paid. �e la�er is known as

9

a collar and caps the upside of the protective put. As both the downside and upside are limited,

the collar strategy usually has a lower beta than the bear put spread. A sold put or call generates

positive alpha that partially o�sets the negative alpha from the purchased put (Benne�, 2014).

Israelov and Klein (2016) compares the protectiveness of the collar to the divested equity strategy.

�ey �nd, in particular, that investors would be be�er o� simply reducing their equity exposure

rather than investing in a collar strategy. �eir study shows that investing in a collar has provided

lower returns and a lower Sharpe ratio than investing directly in the S&P 500 index. �ese

�ndings point to yet another inferior strategy compared to the divested equity strategy.

�e previous literature on the protective put, as well as the article on collar, do not shed a posi-

tive light on their risk-adjusted performance and protectiveness. �is raises the question if there

are other portfolio insurance constructions that provide more a�ractive risk-adjusted return and

tail protection relative to the previous �ndings. It does indeed raise the question if the divested

equity strategy can be outperformed in both risk-adjusted return and tail protection. Although

the divested equity strategy is simple by construction, it requires static risk-reduction, and it

is not an easy task to manually time each market crash. Options, on the other hand, are con-

vex instruments that automatically reduce equity exposure as markets crash, which is o�en a

more convenient construction for the investor who may not have the time to continuously reset

the equity exposure. �is creates incentives to search for protection strategies that are be�er

alternatives than the divested equity strategy.

�is study aims to draw upon the previous �ndings and add several more protection strategies

to the analysis. �e basic setup of the study is similar to the ones carried out by Israelov (2017)

and Figlewski et al. (2013). It consists of a Monte Carlo simulation and a backtesting on the S&P

500 index in the period 20 December 2002 - 23 March 2018. In contrast to previous studies, we

will employ a stochastic jump model in our Monte Carlo simulation to create occasional crashes

similar to the �nancial crisis and other preceding comparable events. �is study will also include

more advanced options strategies, including a version of the Bear Put Spread and the Collar. In

addition, we present a modi�ed version of the protective put and the put spread strategies in an

a�empt to reduce the negative impact of path dependence, which was particularly emphasized

as detrimental to returns by Israelov (2017). �e strategies are benchmarked against each other

and a divested equity strategy which does not include any use of options.

10

We formulate our research questions as follows:

• Is the protective put strategy really an e�ective way to hedge an equity portfolio?

• Are there any other protection strategies that provide relatively better tail protec-

tion?

• Is it, in particular, possible to improve risk-adjusted returns and tail protection

by:

– Making a protection strategy more cost-e�cient?

– Reducing the path dependency of a rolling option strategy?

�e performance of the strategies are based on their risk-adjusted return measured by the Sharpe

ratio in periods of positive excess return. When excess return is negative, which happens during

crashes and prolonged bear markets, the risk-adjusted return is instead measured by the adjusted

Sharpe ratio as proposed by Israelsen (2009). �ere are various metrics for a strategy’s tail pro-

tection, where one of the most widely used is the Value at Risk (VaR). An alternative, which

is employed in this study, is to measure the portfolio’s peak-to-trough drawdowns over prede-

termined time windows. Peak-to-trough drawdowns provide a quantity on the protectiveness

of the di�erent strategies against periods of falling asset prices that persist over di�erent time

windows.

�e study will be structured as follows: We begin by presenting, in Chapter 2, the preliminaries

of option theory. �is is followed by a presentation, in Chapter 3, of the examined protection

strategies, the evaluation methods employed and the setup of the Monte Carlo simulation and

the backtesting. �e results of their performance are presented in tables and graphs in Chapter

4 and are discussed in detail in Chapter 5.

11

�eoretical Background

2.1 Option contracts

An option contract (also called an option) is a �nancial derivative that gives the holder of the

option the right to take action on an underlying asset, but with no obligation to execute this right.

�ere are two basic types of options. A call option gives the holder of the option the right to buy

an underlying asset by a predetermined date for a predetermined price. A put option, on the other

hand, gives the holder of the option the right to sell an underlying asset by a predetermined date

for a predetermined price. �e predetermined date and the predetermined price are commonly

referred to as the maturity date and the strike price, respectively (Hull, 2012).

�ere are various styles of options, each with di�erent terms specifying under what conditions

they are allowed to be executed. �ese are broadly categorized into vanilla options and exotic

options. Vanilla options are further classi�ed as either European or American.

• European options can be exercised only on the maturity date

• American options can be exercised at any time up to the maturity date

Exotic options have more complex features and may have several triggers relating to their exe-

cution. Options derive their value from the underlying asset, which can be anything from a stock

to a foreign currency, a bond, a commodity, an index or a futures contract (Hull, 2012).

�e remaining parts of this report will focus only on European options on stocks. European

options are generally easier to analyze and are one of the most actively traded option types on

the markets. Hence, any mention of stock options herea�er implicitly refers to European stock

options.

12

2.1.1 Mathematical de�nition of European options

�e holder of a European option will claim the payo� Φ(ST ) at maturity time T , where Φ is the

contract function and ST is the stock price at maturity.

�e claim, or value, of a European call option on a non-dividend-paying stock at maturity is

mathematically represented as:

ΦC(ST ) = max(ST −K,0) = (ST −K)+ (2.1.1)

where K is the strike price. �e call option has a non-zero payo� at maturity when the stock

price ST exceeds the strike price K , otherwise it expires worthless. Figure 2.1 shows the payo�-

diagram for a European call option (Bjork, 2009).

With the same notations, the value of a European put option on a non-dividend-paying stock at

maturity is given by the formula:

ΦP (ST ) = max(K − ST ,0) = (K − ST )+ (2.1.2)

where the strike priceK must exceed the stock price ST at maturity to result in a non-zero payo�

(Bjork, 2009). Figure 2.2 shows the payo�-diagram for a European put option.

�e value calculated in equations 2.1.1 and 2.1.2 is the intrinsic value of the option. For any time

t < T , the intrinsic value of an option is calculated as:

(St −K)+ (for a call option)

(K − St)+ (for a put option)(2.1.3)

(Benne�, 2014).

13

0 10 20 30 40 50 60 70 80 90 100

Stock price, ST

0

5

10

15

20

25

30

35

40

45

50

Payoff

Figure 2.1: Payo�-diagram for a European call option with strike price K = 50.

0 10 20 30 40 50 60 70 80 90 100

Stock price, ST

0

5

10

15

20

25

30

35

40

45

50

Payoff

Figure 2.2: Payo�-diagram for a European put option with strike price K = 50.

For any time t < T it is, however, not obvious how to calculate a ”fair” option price. �e price is

determined by the market, in�uenced by the participants’ a�itude to risk and expectations about

the future stock prices. �ese elements are captured by another value component, namely the

time value. At maturity, the time value of the option decays to zero. In general, the value of an

option can be decomposed into two components:

Value of option = Intrinsic value+Time value

where the time value usually is non-zero for 0 ≤ t < T , i.e. all times t prior to maturity T

(Benne�, 2014).

14

2.1.2 ATM, ITM and OTM

When the current stock price equals the strike price, St = K , the option is said to be at-the-money

(ATM). An ATM option has zero intrinsic value and non-zero time value. �e intrinsic value is

also zero when the current stock price is less than the strike price, St < K , for a call option, and

when the current stock price is greater than the strike price, St > K , for a put option. In both

cases, the option is said to be out-of-the-money (OTM). On the contrary, when the intrinsic value

is non-zero, an option is said to be in-the-money (ITM), (Benne�, 2014).

In general, the time value is greatest for ATM options. OTM options tend to trade cheapest,

whereas ITM options are relatively expensive and hence tend to trade in lesser volumes than

their cheaper OTM counterparts. Whether an option is ATM, ITM or OTM thus has an impact

on its market price (Benne�, 2014).

2.2 Black-Scholes-Merton Model

�ere are various models to calculate the ”fair” price of a stock option. One of the most com-

mon pricing models used by market participants is the Black-Scholes-Merton model. �e Black-

Scholes-Merton model is widely used by option market participants and is perhaps the world’s

most well-known option pricing model. �e model is both arbitrage free and complete, mak-

ing the prices it produces unique. Furthermore, the model is developed based on the following

assumptions (Hull, 2012):

1. �e stock price S(t) dynamics is described by the geometric Brownian motion:

dSt = µStdt + σStdWt (2.2.1)

where µ is the stock’s annual expected rate of return, σ is the annual volatility of the stock

price and Wt is a Wiener process. �e stock price is hence assumed to have a lognormal

distribution:

St = S0e(µ− σ22 )t+σWt

15

where S0 is the initial stock price.

2. Volatility is constant.

3. Short selling is permi�ed.

4. �ere are no transaction costs or taxes.

5. �ere are no dividends during the life of the option.

6. �ere are no arbitrage opportunities.

7. Trading is continuous.

8. �e risk-free rate r is deterministic and the same for all maturities.

Given these assumptions, the price formulas for European call and put options are solutions to

the Black-Scholes-Merton di�erential equation problem:

∂F∂t

(t,St) +12S2t σ

2∂2F

∂S2t(t,St)− rF(t,St) = 0

F(T ,ST ) = Φ(ST )

(2.2.2)

F(t,St) is the price of an option as a function of time and underlying stock price, where time t

extends from the day the contract was wri�en to maturity, 0 ≤ t ≤ T . �e boundary condition

F(T ,ST ) = Φ(ST ) ensures that the price of the option at maturity is equal to the payo� of the

contract at maturity (Hull, 2012).

Solving equation 2.2.2 for the price of a call option and a put option, denoted by c(t,St) and

p(t,St), respectively, with strike price K and time of maturity T yields:

c(t,St) = StN[d1(t,St)

]− e−r(T−t)KN

[d2(t,St)

]p(t,St) = Ke

−r(T−t)N[− d2(t,St)

]− StN

[− d1(t,St)

] (2.2.3)

where N is the cumulative distribution function for the N [0,1] distribution and

d1(t,St) =1

σ√T − t

ln(StK

)+(r +

12σ2

)(T − t)

d2(t,St) = d1(t,St)− σ

√T − t

(2.2.4)

16

(Hull, 2012).

�e presence of an expected rate of return, µ, in equation 2.2.1 introduces a risk preference, which

may vary among investors. �e Black-Scholes-Merton di�erential equation 2.2.2 is independent

of investors’ risk preferences. In deriving the Black-Scholes-Merton formula, one can therefore

assume a risk-neutral world, where the expected rate of return on all stocks is the risk-free rate,

r . Under a risk-neutral measure, the stock price process then becomes:

dSt = rStdt + σStdWt (2.2.5)

where the expected rate of return is equal to the risk-free rate r and Wt is a Wiener process under

the risk-neutral measure.

2.2.1 Black-Scholes-Merton model with dividends

�e Black-Scholes-Merton formulas in 2.2.3 are derived based on non-dividend paying stocks.

In reality, stocks o�en pay dividends and to take this into account the Black-Scholes-Merton

formula must be modi�ed. Dividends are paid out to the holder of the stock on the ex-dividend

dates. On this dates the stock price declines by the amount of the dividend. Stock prices can be

decomposed into two components:

• A riskless component, D , that represents the present value of all the dividends during the

life of the option discounted from the ex-dividend dates to the present at the risk-free rate.

• A risky component, S , that corresponds to the stochastic part of the stock price following

the volatility process σ .

�e current stock price, t, including the present value of all the dividends to be paid out from

today until maturity, is the sum of both these components:

S∗t = St +Dt

17

Given that the Black-Scholes-Merton formula is derived based only on the risky component, the

equations in 2.2.3 can be used if the stock price, including dividends, is reduced by the present

value of all the dividends during the life of the option:

St = S∗t −Dt

In principle, the stock price is adjusted for the anticipated dividends and then the option is valued

as though the stock pays no dividend (Hull, 2012).

In the case of a known dividend yield (continuous dividend), q, the adjusted stock price becomes:

S∗t e−q(T−t)

where t is the current time and T is the time of maturity. �e risk-neutral price process of a stock

with dividend yield q can be wri�en as

dSt = (r − q)Stdt + σStdWt (2.2.6)

where the expected rate of return is reduced by the dividend yield (Hull, 2012).

2.2.2 Black-Scholes-Merton model for a stock index

In valuing stock index options, the underlying index can be treated as a stock paying a known

dividend yield, q. �e theory presented in the previous section 2.2.1 can thus be used to derive

the call and put formulas for stock index options. With the assumptions that the underlying

stock index has the current price, S∗t e−q(T−t) and pays no dividends, the Black-Scholes-Merton

equations for call and put options on a stock index are:

18

c(t,S∗t e−q(T−t)) = S∗t e

−q(T−t)N[d1(t,S

∗t e−q(T−t))

]− e−r(T−t)KN

[d2(t,S

∗t e−q(T−t))

]p(t,S∗t e

−q(T−t)) = Ke−r(T−t)N[− d2(t,S∗t e−q(T−t))

]− S∗t e−q(T−t)N

[− d1(t,S∗t e−q(T−t))

](2.2.7)

d1(t,S∗t e−q(T−t)) =

1

σ√T − t

ln(S∗tK

)+(r − q+ 1

2σ2

)(T − t)

d2(t,S

∗t e−q(T−t)) = d1(t,S

∗t e−q(T−t))− σ

√T − t

(2.2.8)

where q is the anticipated annual dividend yield during the life of the stock index option, S∗t is

the current value of the stock index and σ is the volatility of the stock index (Hull, 2012).

2.2.3 Merton’s Jump Di�usion Model

�e Black-Scholes-Merton model assumes that stock returns have a lognormal distribution de-

scribed by the geometric Brownian motion presented in section 2.2. Empirically, stock returns

tend to have fat tails, i.e. a distribution that assigns higher probability to extreme returns com-

pared to the lognormal distribution. Merton’s Jump Di�usion model was introduced as an al-

ternative to the Black-Scholes-Merton model to address the issue of fat tails. By modifying

the geometric Brownian motion to include an independent Poisson process, dpt , the modelled

stock prices will occasionally experience a jump, more similar to the empirically encountered

behaviour. �e suggested model is dependent on two additional parameters: the average num-

ber of jumps per year, λ, and the average jump size measured as a percentage of the stock price,

k (Hull, 2012).

�e jumps contribute to the growth rate of the stock price with a value equal to−λk. �e modi�ed

risk-neutral process for the stock price is:

dSt = (r − q −λk)Stdt + σStdWt + dpt (2.2.9)

19

where dWt and dpt are assumed to be independent.

�e percentage jump size, X, is assumed to be drawn from a particular probability distribution.

Commonly, the distribution is chosen such that ln(1+X) ∼N (γ,δ2) with mean percentage jump

size k = E(X) = eγ+δ2/2−1. It is assumed that the two sources of randomness, the Poisson process

for when a jump occurs and the lognormal distribution of the jump size, are independent of each

other.

With these modi�cations to the geometric Brownian motion and assumptions for the jumps, one

can show that the price of the European option can be wri�en as:

∞∑n=0

e−λ′T (λ′T )n

n!fn (2.2.10)

where λ′ = λ(1+k) and fn is the Black-Scholes-Merton option price obtained from equation 2.2.7

for an underlying asset with dividend yield q, variance equal to:

σ2 +nδ2

T

and risk-free rate equal to:

r −λk + n ln(1 + k)T

(Hull, 2012).

Merton’s Jump Di�usion model creates the fat tail distribution that is more in line with reality, but

leads to an incomplete market due to the addition of another random source, the Poisson process.

�e price obtained in 2.2.10 is not unique, since there is no unique risk-neutral measure in an

incomplete market. However, by a change of probability measure, and e�ectively by changing

the dri� according to equation 2.2.9, Merton constructed an arbitrage-free model. Yet, due to the

incompleteness, it is not possible to construct a replicating portfolio and no perfect hedge.

20

2.2.4 Convexity of options

�e a�ractiveness of options for hedging revolves around their non-linear (”convex”) payo�

structure. For example, put options provide downside protection while preserving upside po-

tential. �e value of put options rises as the price of the underlying stock falls. �is is captured

by the measure delta, ∆. It is given by the �rst derivative of the price equation for the put option

in 2.2.3 with respect to the underlying stock price:

∆put =∂p

∂S=N (d1)− 1 (2.2.11)

and its value ranges between -1 and 0 (Hull, 2012).

More importantly, a long position in an option has a positive exposure to the second derivative

of the price equations in 2.2.3 with respect to the underlying stock price - or the �rst derivative

of ∆ with respect to the underlying stock price. �e rate of the increase in the price of a put

option increases as the price of the underlying stock falls. Conversely, the rate of the decrease

in the price of a put option decreases as the price of the underlying stock rises. �is measure is

known as the gamma, Γ , and quanti�es the sensitivity of ∆ to changes in the stock price:

Γput =∂∆∂S

=∂2p

∂S2=N ′(d1)

S0σ√T

(2.2.12)

which varies with respect to the underlying stock price, S , as a bell-shaped curve. �e described

relationship between gamma and the underlying stock price is typical for a long position in a put

option. �is asymmetric behavior, which makes a put option more valuable as the price of the

underlying stock falls while preserving upside potential, is what makes it a popular instrument

for portfolio protection (Hull, 2012).

21

2.3 Volatility

2.3.1 Realized versus implied volatility

�e volatility, σ , measures the amount of variability in the stock returns. �ere are two volatility

measures that are of interest to participants in the options market. �e �rst one, realized volatility,

also called historical volatility, is an ex-post estimate of stock return variation. It is de�ned as

the annualized standard deviation of daily stock returns:

σrealized =

√√√252N − 1

N∑t=1

(rt − r)2 (2.3.1)

where N +1 is the number of observations and

rt = lnStSt−1

, r =1N

N∑t=1

rt (2.3.2)

(Hull, 2012).

�e realized volatility implies nothing about the future. To estimate future volatility, option

traders look at a second volatility, the implied volatility, which is derived from the Black-Scholes-

Merton formula by inserting the current market price of the option. It is the volatility parameter

in the geometric Brownian motion process:

dStSt

= µdt + σimplieddWt (2.3.3)

(Hull, 2012).

Understanding the di�erence between realized and implied volatility is fundamental to any in-

vestor engaging in options trading. As can be seen in equations 2.2.3 and 2.2.4, an increase in

the implied volatility, σ , all else being equal, increases the value of a long position in an option,

while the opposite is true for a short position. �e time value of an option is, in practice, heav-

ily dependent on the volatility in the underlying stock. A long position in an option is a long

volatility exposure, i.e. the realized volatility, at the time of maturity, is expected to exceed the

22

implied volatility at which the option was bought (Benne�, 2014).

2.3.2 Volatility risk premium

In a perfect market, there exists only one volatility parameter, σrealized = σimplied = σ . However,

in reality option writers (sellers) tweak the Black-Scholes-Merton formulas in equation 2.2.3 to

compensate for the risk of losses during periods when realized volatility suddenly increases,

also called crash risk. Note in Figure 2.3 how realized volatility spiked higher than the implied

volatility during the �nancial crisis 2007 - 2008. �e implied volatility, however, tends to exceed

the realized volatility on average. �e discrepancy between implied volatility and the subsequent

realized volatility is known as the volatility risk premium. �e volatility risk premium indicates

how expensive an option is and reduces realized returns for the buyer (Israelov, 2017).

�e volatility risk premium over a certain period can, for example, be measured by the di�erence

between the S&P 500 annualized 1-month ATM implied volatility and the S&P 500 annualized

subsequent 1-month realized volatility. �e volatility risk premium at time t is thus the di�erence

between the S&P 500 annualized 1-month ATM implied volatility at time t and the annualized

standard deviation of the S&P 500 measured from time t and one month forward. In Figure 2.3,

the volatility risk premium is plo�ed over the period December 2002 - March 2018.

23

2002 2005 2007 2010 2012 2015 2017 2020-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1A

nn

ua

lize

d v

ola

tilit

yImplied Volatility

Realized Volatility

Volatility Risk Premium

Figure 2.3: �e volatility risk premium measured as the di�erence between the S&P 500 annual-ized 1-month ATM implied volatility and the S&P 500 annualized subsequent 1-month realizedvolatility.

In the period December 2002 - March 2018, the volatility risk premium amounted to 0.8% on

average, with a median of 1.7%. See table 2.1 for more statistics.

24

Table 2.1: Statistics of the volatility in the S&P 500 index in the period December 2002 - March2018.

Percentiles99th 95th 90th 75th Median Mean

Implied volatility 53.3% 31.7% 25.8% 18.9% 14.0% 16.4%Realized volatility 74.2% 32.2% 25.5% 17.7% 13.0% 15.6%Volatility risk premium 11.3% 8.2% 6.4% 4.1% 1.7% 0.8%

2.3.3 Volatility surface

�e implied volatility that is used by traders to price an option depends on its strike price and

time to maturity. A plot of these three dimensions gives a volatility surface, which shows how

implied volatility depends on each of the two parameters. For simplicity, the dependencies are

o�en plo�ed as two separate two-dimensional graphs.

�e plot of implied volatility and strike price is known as the volatility skew, because the curve

is generally skewed. For example, the implied volatility for the S&P 500 on 16 January 2018 is

decreasing with increasing strike price and reaches a minimum before it slightly increases again

(see Figure 2.4). Hence, the volatility used to price an option with low strike price (i.e. a deep

OTM put or a deep ITM call) is signi�cantly higher than that used to price options with higher

strike price (i.e. a deep ITM put or a deep OTM call). By �xing the time to maturity, the volatility

skew can be depicted in a two-dimensional graph (see Figure 2.5), (Hull, 2012).

Plo�ing implied volatility against the time to maturity shows the term structure, which pro-

vides information on how implied volatility varies with increasing time to maturity. �e implied

volatility for the S&P 500 as of 16 January 2018 exhibits a high volatility for the shortest time to

maturity, but approaches quickly a minimum, from where it increases with increasing time to

maturity (see Figure 2.5). �is re�ects an expectation that the volatility will increase over time,

leading to more richly priced long-dated options (Hull, 2012).

25

0

Q2-20

0.2

Q1-20Q4-19

0.4

Q3-19 2Q2-19

Implie

d v

ola

tilit

y

0.6

Q1-191.5Q4-18

Time of maturity

0.8

Q3-18

Strike percentage

Q2-18 1

1

Q1-18Q4-17

0.5Q3-17Q2-17

Q1-17 0

Figure 2.4: A volatility surface for options on the S&P 500 Index. �e time of maturity spansfrom 2 April 2018 to 23 March 2021, where the la�er is the longest dated option available in thedataset. �e strike price is given as a percentage, where 1 represents an ATM option.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Strike percentage

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Imp

lied

vo

latilit

y

Q1-17 Q2-17 Q3-17 Q4-17 Q1-18 Q2-18 Q3-18 Q4-18 Q1-19 Q2-19 Q3-19 Q4-19 Q1-20 Q2-20

Time of maturity

0.14

0.145

0.15

0.155

0.16

0.165

0.17

0.175

0.18

0.185

0.19

Imp

lied

vo

latilit

y

Figure 2.5: Le�: Volatility skew for a �xed time to maturity. Right: Term structure for a �xedstrike price.

�e volatility surface informs traders about how OTM, ATM and ITM puts and calls are priced in

terms of volatility, which in turn relates to the time value of the option. �is information is useful

for building suitable option strategies. For example, a �at (or �a�er) volatility skew curve - OTM

and ITM levels are equal to the ATM level - presents excellent opportunities to put on portfolio

hedges at a cheaper cost than the observed skew in Figure 2.5. �us, the observed volatility skew

in Figure 2.5 informs the trader that the OTM puts are more expensively priced relative to the

ATM level. On the other hand, the skew presents opportunities for option writers who would

like to take advantage of the relatively more expensive OTM puts.

26

2.4 Protection strategies

A myriad of portfolio insurance strategies have been developed to manage volatility and tail

risk exposure. �is section presents some of the most widely employed option-based protection

strategies.

2.4.1 Protective Put Strategy

�e core idea of the protective put strategy is to limit the downside risk of a portfolio consisting

of a stock or a stock index (e.g. S&P 500) held over a certain investment horizon by purchasing

a sequence of shorter-term put options. �e strategy protects the portfolio from losses below a

certain strike level, while allowing unlimited pro�ts as long as the underlying asset’s price rises.

�e total return on the portfolio is reduced by the cost of the put options and also depends on the

path that the underlying asset’s price follows. �e path dependency arises from the fact that the

put options must be rolled over as they mature into new puts at prices that will depend heavily on

the underlying asset’s current price and implied volatility. Figure 2.6 shows the payo� diagram

for a protective put strategy.

0 10 20 30 40 50 60 70 80 90 100

Stock price, ST

-40

-30

-20

-10

0

10

20

30

40

50

Payoff

Figure 2.6: Protective put strategy: A long position in a put option with strike price K = 50 anda long position in an underlying asset.

�e investor primarily needs to consider two parameters in this strategy - the time to maturity

and the strike price of the put options. �e time to maturity is usually shorter than the investment

horizon, e.g. 1 month, 3 months, or 12 months. �e strike price can be one of the following:

27

• A �xed value, K = C, where C is a constant

• A �xed percentage of the underlying asset’s current price, K = p · St , where p is given in

decimal form (e.g. p = 0.95) and St is the underlying asset’s price at the time of purchase

• A combination of both.

Furthermore, it can be chosen such that the option is either ITM (p > 1), ATM (p = 1) or OTM

(p < 1), where the la�er is more common since its the cheapest alternative and is expected to

reduce the portfolio return the least. �e choice of parameters and strike method should therefore

depend on the investor’s expectations on the market.

2.4.2 Bear Put Spread Strategy

A bear put spread involves a long position in a put option on a particular underlying asset, while

simultaneously taking a short position in a put option on the same underlying asset with the

same maturity date, but with a lower strike price. �e strategy takes advantage of the observed

volatility skew in Figure 2.5 by selling a richly priced deeper OTM put and buying a relatively

cheaper OTM put with a higher strike price, which e�ectively reduces the net premium paid.

�e le� graph in Figure 2.7 illustrates a payo� diagram for a bear put spread. �e right graph

shows the payo� diagram when it is used as a hedge on a long position in an underlying asset,

e.g. a stock (Benne�, 2014).

0 10 20 30 40 50 60 70 80 90 100

Stock price, ST

-40

-30

-20

-10

0

10

20

30

40

50

Pa

yo

ff

0 10 20 30 40 50 60 70 80 90 100

Stock price, ST

-40

-30

-20

-10

0

10

20

30

40

50

Pa

yo

ff

Figure 2.7: Le�: A bear put spread consisting of a long position in a put option with strike priceK = 60 and a short position in a put option with strike price K = 30. Right: A bear put spreadand a long position in a stock.

28

A bear put spread as a hedging strategy is more complex compared to its protective put counter-

part. �e maximum protection provided by the bear put spread is equal to the size of the spread

less the net premium paid. �e strategy does not cover any losses beyond the size of the spread,

which requires that the bearish investor is able to proportionately match the spread to the size of

the expected fall in the stock price. However, it lies in the investor’s interest to keep the spread as

small as possible, since the premium gained from the short put is higher the closer it is chosen to

the ATM level. Hence, the strategy becomes a trade-o� between keeping the net premium paid

as low as possible while maximizing the coverage of potential losses. �e right graph in Figure

2.7 illustrates the extent of the hedge obtained by the bear put spread. �e stock is hedged only

down to ST = 30, below which any additional losses are not covered (Benne�, 2014).

2.4.3 Collar Strategy

�e Collar strategy consists of a long position in a put option to cover the downside risk of an

underlying asset and a short position in a call option to reduce (or to completely o�set) the

premium paid for the put. Typically, both the put and the call options are OTM with the same

maturity on the same number of underlying assets. �e net cost of a collar is o�en less than

the net cost of a put spread (see section 2.4.2) and is therefore considered as a low-cost method

for protection. However, to obtain this signi�cant reduction in cost, the strategy gives up some

upside. �e short call position results in a cap on performance, which makes the collar strategy

less exposed to volatility. When the premium received from the short call option completely

o�sets the premium paid for the long put option, the strategy is called a zero cost collar (Benne�,

2014).

�e le� graph in Figure 2.8 illustrates a payo� diagram for the collar strategy. �e right graph

shows the payo� diagram when it is used as a hedge on a long position in an underlying asset,

e.g. a stock (Benne�, 2014).

29

0 10 20 30 40 50 60 70 80 90 100

Stock price, ST

-30

-20

-10

0

10

20

30

40

Pa

yo

ff

0 10 20 30 40 50 60 70 80 90 100

Stock price, ST

-40

-30

-20

-10

0

10

20

30

40

50

Pa

yo

ff

Figure 2.8: Le�: A collar consisting of a long position in a put option with strike price K = 40and a short position in a call option with strike price K = 60. Right: A collar and a long positionin a stock.

�e collar is more exposed to skew compared to the bear put spread because of the parabola-like

shape around the ATM level (see the le� graph in Figure 2.5).

30

Methodology

3.1 Overview

�is study aims to evaluate the protective performance of di�erent protection strategies involv-

ing options. �e performance will be evaluated based on their risk-adjusted return and peak-

to-trough drawdowns. Moreover, each option strategy will be benchmarked against a divested

equity strategy to analyze how it fares against an alternative in which there is no use of options.

�e study is split into two parts. First, a Monte Carlo simulation is performed to evaluate the

di�erent strategies under ideal conditions. �e simulation aggregates the performance of the dif-

ferent strategies over varying stock price developments, including rising markets, bear markets

and extreme crashes. Section 3.5 covers this part in more detail.

�e results from the Monte Carlo simulation cannot be generalized beyond the ideal conditions

it was performed under. �erefore, the strategies will also be backtested against historical data

of the S&P 500 index to assess their performance in a real-world implementation. In reality,

there are signi�cant variations in the implied volatility (as displayed in Figure 2.3) driven by

prevailing market conditions. Implied volatilities tend to move inversely with equity returns,

which may provide positive pressure on a put option’s price during equity losses, improving its

downside hedging properties. At the same time, rolling over into new put options become much

more expensive. It is important to note that the results of a backtesting provide information only

on how the portfolio performed in the speci�c market environment that prevailed during the

backtesting.

In short, the Monte Carlo simulation enables a more extensive analysis, while the backtesting

serves as a reality check.

31

3.2 Assumptions and delimitations

Certain assumptions and delimitations have been made in order to simplify the analysis and to

isolate the protective properties of the di�erent option strategies. In particular, the construction

of the option strategies is based on the following:

1. �e portfolio is fully invested in the underlying asset (S&P 500 index)

2. Rolling purchases and selling of options:

(a) �e �rst option is �nanced by borrowing at the risk-free interest rate and is repaid at

its maturity date, i.e. the cost of the purchased option (including interest expense) is

deducted from the value of the portfolio at the maturity date

(b) �e following options are �nanced by borrowing at the risk-free interest rate if, at

each rollover date, the accumulated payo�s from previous expired options are not

su�cient to cover the cost of buying a new option

(c) Option payo�s are held as cash and grow with the risk-free interest rate between the

rollover dates

(d) No reinvestments of option payo�s

3. Other cash positions also grow with the risk-free interest rate

4. No dividends are collected from the underlying asset (S&P 500 index)

5. No transaction costs

In contrast to a realistic scenario, it is implicitly assumed that the portfolio can never run short

of funds to purchase an option. �e options are either funded by accumulated payo�s or by

borrowing at the risk-free interest rate, where the la�er would reduce returns with an extra

amount equal to the interest expense.

32

3.3 Option-based protection strategies and portfolios

�e core objective of the option-based strategies that are covered in this study is to provide

protection against a long position in the underlying S&P 500 index. �e strategies are intended

to be alternatives to the regular protective put strategy and should therefore be consistent with

its objective. Essentially, the hypothetical investor is bullish on the underlying S&P 500 index,

but seeks protection against potential losses should the price of the S&P 500 index fall below a

tolerated level. Although the hypothetical investor seeks to maximize the risk-adjusted returns,

she does not intend to realize pro�ts from the options. �e protection obtained from the option-

based strategies should therefore be seen as an insurance policy, rather than an investment.

As documented by previous studies, the protective put strategy su�ers from high put option pre-

miums and path dependency. Besides providing protection, the selected option-based strategies

are constructed to address these two problems. In particular, the selected option-based protection

strategies aim to:

1. Reduce the high cost related to put options.

2. Minimize exposure to the path dependency in a rolling put option strategy.

Table 3.1 provides an overview of the selected option-based protection strategies. �e strategies

presented in the table will not only be compared to each other, but also benchmarked against

a risk-reducing strategy that does not involve any options. A divested equity strategy will be

implemented to serve this purpose, which will be further explained in Section 3.3.2.

33

Table 3.1: Overview of the selected option strategies.Strategy Description Protection Maximum lossa,b Rationale

Protective Put Long 5% OTM put Full protection belowstrike 5% + premium Simple

Put Spread 95/80 Long 5% OTM put,short 20% OTM put Protection up to 20% dip 5% + 80% + net premium Low cost

Put Spread 95/85 Long 5% OTM put,short 15% OTM put Protection up to 15% dip 5% + 85% + net premium Low cost

Put Spread 100/90 Long ATM put,short 10% OTM put Protection up to 10% dip 90% + net premium Low cost

Collar 95/105 Long 5% OTM put,short 5% OTM call

Full protection belowstrike, capped upside 5% + net premium Low cost

Fractional Protective Put Long overlapping fof 5% OTM put

Full protection a�era certain time 5% + (1− f )·95% + f ·premium Reduced path

dependency

Fractional Put Spread 95/85 Long/Short overlappingf of 5%/15% OTM put

Full protection a�era certain time

5% + 85% + (1− f )·10% +f ·net premium

Low cost, Reducedpath dependency

a f represents a fraction of one option. For example, the Fractional Protective Put strategy based on 3m options involves buying 1/3 of a 3m option every month. Consequently,the portfolio is fully protected a�er 3 months.b �e maximum loss of the Fractional Protective Put strategy occurs in the �rst month, when the portfolio is only hedged by a fraction f of a put option, e.g. for 3m options:5% +(2/3)·95% + (1/3)·premium

�e option strategies in Table 3.1 will be implemented in four di�erent portfolios. �e portfolios

are presented in Table 3.2 and are designed to show how the di�erent option-based strategies

perform across di�erent maturities of the constituent options.

Table 3.2: Description of the four di�erent portfolios.

Portfolio Descriptiona

1m portfolio Rolling purchases/selling of 1m options every month.3m portfolio Rolling purchases/selling of 3m options every 3 months.6m portfolio Rolling purchases/selling of 6m options every 6 months.12m portfolio Rolling purchases/selling of 12m options every 12 months.”1m”, ”3m”, ”6m” and ”12m” are abbreviations for 1-month, 3-month, 6-month and 12-month.a �e fractional strategies are exceptions. �ey buy fractional amounts of an option every week inthe 1m portfolio and every month in the 3m, 6m and 12m portfolios.

3.3.1 Protective Put

�e protective put strategy buys a 5% OTM put at every rollover date. Documenting its perfor-

mance across the four portfolios (1m, 3m, 6m and 12m) will be important for the analysis, as

the other protection strategies are built to improve its de�ciencies. For example, the low-cost

portfolios are designed to address the negative impact of high premiums on the returns of the

protective put strategy. See Appendix 6.1.1 for a modelling example.

34

3.3.2 Divested Equity Portfolio

Option-based protection strategies are one approach to protect against rising volatility (or down-

side risk). However, volatility can also be dampened via asset allocation. One such passive ap-

proach is the static reduction of equity exposure - in this context, called the divested equity

strategy. It will serve as a benchmark for each option strategy to shed light on how option-based

protection strategies stand against a hedging method that does not use options. �e divested

equity portfolio allocates a portion of its net asset value (NAV) to a long position in the underly-

ing asset, i.e. S&P 500 index, and the remainder to cash. It weighs the investor’s willingness to

assume risk against expected returns through the exposure to the risky asset. See Section 6.1.2

in Appendix for a modelling example.

For the purpose of this study, the allocation given to the S&P 500 is chosen such that the di-

vested equity portfolio yields the same geometric return (see Section 3.4 for the de�nition) as the

option-based strategy it is compared to. �is enables a fair comparison between their drawdown

characteristics. Although no transaction costs are considered, the portfolio will be rebalanced on

a monthly basis to maintain a more realistic approach.

3.3.3 Low-cost Portfolios

�e put spread 95/80, put spread 95/85, put spread 100/90 and collar 95/105 in Table 3.1 are all

low-cost strategies compared to the protective put strategy. �e put spreads are of special interest

because they will shed light on the bene�ts, if there are any, of forgoing some protection for lower

option premiums. �e collar, on the other hand, caps the upside in exchange for lower premiums,

which gives an equity exposure that is di�erent from the protective put, but still interesting

for comparative purposes. Ultimately, the comparison provides a perspective on what is worth

sacri�cing for improved risk-adjusted return and reduced tail risk.

�e size of the spreads were chosen based on S&P 500 index drawdown percentiles in the period

2 January 1990 - 20 December 2002. �e 10-year period was intentionally chosen to precede the

period that this study covers. It represents the information available to the hypothetical investor

at the time the backtesting commences. �e 99th and 95th percentiles of the 20-Day drawdown

35

window are 14.0% and 8.4%, respectively. Over a 63-Day drawdown window, the 99th and 95th

percentiles are 22.6% and 15.8%, respectively. With these numbers in mind, the put spread

95/85 strategy is based on a rounded assumption that the market should fall at maximum 15.0%

from the ATM level during the holding period. �e put spread 95/80 strategy assumes the more

extreme case of a 20% dip, while the put spread 100/90 protects against a moderate 10% dip in

the market from the ATM level.

Indeed, the 99th and 95th percentiles of drawdowns over longer windows (e.g. 250 days) are

larger. If the strategies were designed to match these, then the spreads would be too large and

barely reduce the premiums paid, and therefore fail their purpose.

All four low-cost option-based protection strategies, except for one, buys 5% OTM put options.

�e reason is to keep the costs low, while not giving up too much protection from the ATM level.

In order to capture the e�ect of expensive put options and for comparative purposes, one of the

strategies, namely the put spread 100/90, buys ATM puts.

3.3.4 Fractional Protective Put

As explained in Section 2.4.1, the protective put strategy is exposed to path dependency risk.

When a put option with a �xed percentage strike expires and is rolled over into a new one, the

cost will mainly depend on the implied volatility of the S&P 500 index at that particular time.

If the rollover of the put option is timed with a sudden spike in implied volatility, then this will

result in a higher cost than it would if volatility remained constant. Figure 2.3 shows how implied

volatility spiked during the �nancial crisis, which at the time led to very high premiums for the

put options.

As a measure to address the path dependency risk, a modi�ed protective put strategy has been

constructed. �e fractional protective put strategy aims to reduce the exposure to path depen-

dency risk by diversifying the purchases of put options over overlapping maturity cycles. �e put

options are purchased in fractional amounts to keep the costs on par with the regular protective

put. For example, the 3m portfolio would buy 1/3 of a 3m option every month. Consequently,

full protection is not obtained before the third month.

36

Based on the same rationale, the 6m portfolio buys 1/6 of a 6m option every month and reaches

full protection by the sixth month. Similarly, the 12m portfolio buys 1/12 of a 12m option every

month and reaches full protection by the twel�h month. �e 1m portfolio is slightly di�erent,

but follows the same logic. It buys 1/4 of a 1m option every week and reaches full protection by

the �rst month.

In the �rst months of the strategy, the portfolio is only partially hedged and therefore more

exposed to market downturns. �e longer the maturity of the strategy, the longer it takes for the

portfolio to become fully protected and is thus more exposed to market risk.

3.3.5 Fractional Put Spread

�e fractional put spread 95/85 is designed to combine the bene�ts of lower net cost and reduced

path dependency. Similar to the regular put spread 95/85, it buys 5% OTM puts and sells 15%

OTM puts. Hence, the strategy protects, at maximum, against 15% market dips.

3.4 Evaluation methods

Annualized portfolio return

Both the annualized arithmetic and geometric returns on the portfolio are calculated.

Arithmetic return = 252 ·[R1 +R2 + . . .RN−1

]· 1N−1

Geometric return =[(1 +R1) · (1 +R2) · . . . (1 +RN−1)

] 252N−1 − 1

(3.4.1)

where Ri is the daily return and N is the number of days in the investment horizon.

37

Risk-adjusted performance

�e risk-adjusted performance of a portfolio will be assessed on its excess return per unit of

volatility. �is measure is captured by the Sharpe ratio. It is de�ned as the excess return over the

risk-free rate per unit of volatility:

Sp =rp−rrfσp

(3.4.2)

where rp is the portfolio arithmetic return, rrf is the risk-free interest rate and σp is the portfolio

volatility.

When the market is in a downward trend, the Sharpe ratio can be a misleading tool (Scholz, 2007).

In this case, the risk-free rate o�en outperforms the portfolio returns. A negative excess return

gives a negative Sharpe ratio, which makes the relative ranking between the di�erent portfolios

misleading. In order to address the ranking issue, Israelsen (2009) proposed an adjusted Sharpe

ratio:

Sp,adj =ep

σep/abs(ep )p

(3.4.3)

where ep = rp − rrf is the excess return on the portfolio. �is modi�cation solves the ranking

issue. �e less negative the adjusted Sharpe ratio is, the be�er is the risk-adjusted performance,

whereas larger negative values indicate the opposite.

Peak-to-trough drawdowns

In order to measure the e�ectiveness of a strategy’s tail protection, we will examine the returns

on the portfolio during peak-to-trough drawdowns over rolling overlapping time windows of

sizes 5, 10, 20, 63, 125, and 250 days. �e peak-to-trough drawdowns are measured as the largest

decline in the portfolio value over a given time window. Graphically, it would be seen as a decline

in the portfolio value from its highest peak to its lowest trough over that speci�c time window.

Let VP (ti) and VL(ti) represent the value of the portfolio at its highest peak and lowest trough,

respectively, in the time interval (ti−k , ti), where k is the length of the drawdown window (e.g.

38

5, 10, etc.). �en the peak-to-trough drawdown, D , over overlapping windows is calculated by:

D(ti−k) =VP (ti)−VL(ti)

VP (ti)(3.4.4)

for i = k,k +1, ...,n, where {t0, t1, ..., tN−1} is the set of days within the investment horizon.

�e e�ectiveness of the protection is measured at the 99th, 95th and 50th (median) percentiles

of the peak-to-trough drawdowns over the di�erent windows. �e 99th and 95th percentiles, in

particular, quantify the e�ectiveness of the tail protection of the strategy.

Capital Asset Pricing Model: Beta and Alpha

�e capital asset pricing model (CAPM) describes the relationship between systematic (undiver-

si�able) risk and expected return on an asset, e.g. a stock. �e excess return on the asset over

the risk-free rate is regressed against the excess return on the market over the risk-free rate. �e

formula is given by:

Ra = Rf + β(RM −Rf ) (3.4.5)

where Ra is the expected return on the asset, RM is the return on the market (e.g. an index

such as the S&P 500), Rf is the return on a risk-free investment (e.g. the risk-free rate) and β is

the value of the slope obtained from the linear regression. β captures the sensitivity of returns

from the asset to returns from the market. If the asset is a stock, then the excess return on the

marketRM−Rf is sometimes referred to as the equity risk premium. O�en the linear relationship

in equation 3.4.5 has a non-zero intercept, α, which captures abnormal returns that cannot be

explained by the correlation with the market returns (Hull, 2012).

Investors use β and α as portfolio performance metrics, in which case Ra would be the expected

return on the portfolio. For the purposes of our analysis, a β below 1 means that the portfolio is

less sensitive to changes in the market returns, while a β above 1 implies the opposite. Hence,

a β equal to 1 means that the portfolio excess returns perfectly follow the excess return on the

market. �e β and α will help us to explain the volatility exposure of the di�erent strategies and

the e�ect of volatility risk premium on the returns.

39

3.5 Monte Carlo Simulation

A Monte Carlo simulation is performed to produce di�erent sets of future stock prices following

a distribution based on the Merton’s Jump Di�usion model presented in section 2.2.3. �e simu-

lation enables an idealized environment and captures many di�erent possible outcomes for the

underlying stock. �e di�erent outcomes allow for more generality in the results, but their ap-

plicability to the reality is restricted to the ideal conditions, which o�en are very di�erent from

the actual ones. It is, however, worthwhile to examine how the di�erent protection strategies

compare against each other and how they fare against the divested equity strategy in an ideal

se�ing.

We simulate 1,000 stock paths with a 5 year investment horizon, i.e. T = 5 · 252 = 1,260 days.

�e stock prices are drawn from a lognormal distribution with the addition of a Poisson-driven

jump process in accordance with the risk-neutral process 2.2.9 in the Merton’s Jump Di�usion

model. �e jump size is drawn from a lognormal distribution, N (−0.5,0.052), and with a jump

frequency of 0.1 per year. �ese numbers are based on an assumption that the stock is expected

to fall ca 50% every tenth year similar to a major crash experienced in the real-world equity

markets. �e jump component contributes with an amount of −λk = 0.0393 to the dri� of the

stock price.

Furthermore, without loss of generality, it is assumed that the risk-free interest rate and the

dividend yield are both equal to zero. �e volatility of the stock price is assumed to be 20% based

on an annualized historical volatility of ca 18.8% in the S&P 500 index in the period December

2002 - March 2018. �e volatility risk premium is set to a rounded number of 2.0 percentage

points based on the 1.7% median of the volatility risk premiums in the same period (see Table

2.1). �us, we add 2.0 percentage points on top of the stock price volatility in the pricing formula

for options. Option prices are modeled according to the pricing formula in the Merton’s Jump

Di�usion model, equation 2.2.10.

To summarize, the input parameters are:

40

r = 0% (risk-free interest rate)

S0 = 100 (�e price of the stock at the initial investment)

q = 0% (dividend yield)

σs = 20% (stock volatility)

σp = 22% (stock volatility + volatility risk premium)

dt = 1252 (size time steps)

λ = 0.1 (jump frequency per year)

γ = −0.5 (expected value of the jump size)

δ = 0.05 (standard deviation of the jump size)

(3.5.1)

�e assumption of constant volatility in the Monte Carlo simulation implies no volatility skew

with respect to strike prices and no term structure with respect to time to maturity. �is is an

idealized environment, since OTM put options are cheaper relative to the ATM level compared

to the more realistic scenario where the volatility curve is skewed (see Figure 2.5). Also, the

cost of options is constant across the di�erent maturities. Moreover, the assumption of constant

volatility leads to another unrealistic consequence, namely, the absence of spikes in the volatility

during crashes. We have modelled crashes, but kept the volatility deterministic, which is a super-

idealized environment. �us, as crashes occur in the simulated stock paths, the time value of the

di�erent protection strategies will not be a�ected. In reality, the cost of put options would rise

signi�cantly, making, for example, the protective put more expensive. �is scenario is covered

in the backtesting part of the study.

3.6 Backtesting

3.6.1 Data

�e backtesting of the option-based protection strategies will be performed on the S&P 500 index,

which is a market capitalization weighted index of the 500 largest listed companies in the U.S.

by market value. Options on the S&P 500 index give exposure to the broad-based market with

relatively li�le capital and are actively traded on the Chicago Board Options Exchange (CBOE).

41

�eir popularity, as well as the high data availability, motivates their use in this study.

Daily data is extracted from Bloomberg spanning the period from 20 December 2002 to 23 March

2018 and consists of the constituent parameters of the Black-Scholes-Merton model. Table 3.3

presents the data set in more detail.

Table 3.3: Daily data points for the Black-Scholes-Merton input parameters spanning the period20 December 2002 - 23 March 2018.

Parameter Daily dataUnderlying asset S&P 500 indexStrike pricea Fixed percentages of the S&P 500 index spot level (30%, 35%, …, 200%)Time to maturitya 5 days (1w), 21 days (1m), 63 days (3m), 126 days (6m) and 252 days (12m)Volatility Implied volatility of S&P 500 index across strike price and time to maturityRisk-free interest rate Zero coupon rates for the di�erent time to maturitiesDividend yield Expected S&P 500 dividend yield for the di�erent time to maturitiesa �ese are �xed values and do not vary by days.Source: Bloomberg

3.6.2 Outline

�e option-based protection strategies and portfolios presented in Table 3.1 and Table 3.2 are

modelled based on the assumptions presented in Section 3.2 and are performed on the data set in

Table 3.3. All four portfolios (1m, 3m, 6m and 12m) implement each of the di�erent strategies and

their performance is measured in terms of risk-adjusted return and peak-to-trough drawdowns.

�e backtesting consists of two parts:

�e whole period: 20 December 2002 - 23 March 2018

As depicted in Figure 2.3, the volatility in the S&P 500 index varied signi�cantly during the

whole period. �is part examines the performance of the protection strategies over a long-term

investment horizon and how they fare during shi�ing market environments.

�e �nancial crisis: 14 December 2007 - 5 March 2009

Backtesting the protection strategies on the data from the �nancial crisis show how they fare

during a period of very high volatility. As they are expected to perform during such events, it

is especially interesting to assess how well they live up to these expectations as well as their

42

relative performance.

43

Results


In some cases, the geometric return on an option-based protection strategy could not be matched

with the geometric return on the divested equity strategy. �ose cases (or stock paths) are omi�ed

to enable a fair comparison between them.

In addition to the results presented in this section, Tables 6.1 - 6.4 in Appendix report the draw-

down characteristics in more detail. Figure 6.13 in Appendix visualizes the change in the total

cost as a percentage of the initial investment with respect to option maturity.

44

Tabl

e4.1:

Mea

nva

lueo

fann

ualiz

edre

turn

sand

perfo

rman

cem

easu

resf

orth

esim

ulat

edst

rate

gies

acro

ssth

efou

rpor

tfolio

s.�

eadj

uste

dSh

arpe

ratio

isus

edon

stoc

kpa

thsw

ithne

gativ

ean

nual

ized

arith

met

icre

turn

.Eac

hst

rate

gyis

benc

hmar

ked

agai

nstt

hedi

vest

edeq

uity

.A

rithm

etic

retu

rnGe

omet

ricre

turn

Vola

tility

Shar

pera

tio1m

3m6m

12m

1m3m

6m12

m1m

3m6m

12m

1m3m

6m12

mPr

otec

tive

Put

-0.2%

-0.7%

-1.0%

-1.4%

-1.2%

-1.5%

-1.8%

-2.0%

16.5%

14.7%

13.9%

13.4%

0.17

0.16

0.15

0.15

Div

este

dEq

uity

-0.5%

-0.8%

-1.0%

-1.2%

-1.2%

-1.5%

-1.8%

-2.0%

12.8%

12.2%

12.6%

12.7%

0.27

0.26

0.25

0.25

Di�

+0.3%

+0.1%

0.0%

-0.1%

0.0%

0.0%

0.0%

0.0%

+3.8%

+2.5%

+1.2%

+0.7%

-0.10

-0.10

-0.09

-0.10

PutS

prea

d95

/80

-0.8%

-1.1%

-1.5%

-1.4%

-2.2%

-2.3%

-2.6%

-2.6%

19.1%

17.5%

17.3%

17.8%

0.19

0.18

0.18

0.19

Div

este

dEq

uity

-1.0%

-1.1%

-1.4%

-1.3%

-2.2%

-2.3%

-2.6%

-2.6%

16.2%

15.5%

15.8%

16.5%

0.27

0.25

0.23

0.24

Di�

+0.2%

+0.1%

0.0%

-0.1%

0.0%

0.0%

0.0%

0.0%

+2.9%

+2.0%

+1.4%

+1.3%

-0.07

-0.07

-0.06

-0.05

PutS

prea

d95

/85

-0.2%

-0.8%

-1.0%

-1.0%

-1.7%

-2.2%

-2.4%

-2.4%

19.6%

18.4%

18.4%

19.1%

0.21

0.19

0.19

0.20

Div

este

dEq

uity

-0.5%

-1.0%

-1.1%

-1.0%

-1.7%

-2.2%

-2.4%

-2.4%

15.9%

16.0%

16.5%

17.6%

0.28

0.25

0.23

0.24

Di�

+0.4%

+0.2%

+0.1%

0.0%

0.0%

0.0%

0.0%

0.0%

+3.8%

+2.4%

+1.9%

+1.6%

-0.07

-0.05

-0.04

-0.04

PutS

prea

d10

0/90

-1.8%

-1.3%

-1.1%

-1.0%

-2.7%

-2.3%

-2.4%

-2.4%

15.2%

16.4%

17.6%

18.6%

0.17

0.18

0.19

0.20

Div

este

dEq

uity

-1.8%

-1.3%

-1.2%

-1.0%

-2.7%

-2.3%

-2.4%

-2.4%

13.6%

14.7%

16.1%

17.0%

0.26

0.24

0.24

0.25

Di�

0.0%

0.0%

0.0%

0.0%

0.0%

0.0%

0.0%

0.0%

+1.6%

+1.6%

+1.4%

+1.6%

-0.09

-0.05

-0.04

-0.04

Colla

r95/

105

0.3%

0.2%

0.3%

0.3%

-0.2%

0.0%

0.1%

0.3%

12.1%

7.9%

5.8%

3.9%

0.18

0.19

0.20

0.25

Div

este

dEq

uity

0.3%

0.3%

0.4%

0.5%

-0.2%

0.0%

0.1%

0.3%

10.5%

8.0%

6.6%

5.5%

0.23

0.23

0.22

0.22

Di�

+0.1%

-0.1%

-0.2%

-0.2%

0.0%

0.0%

0.0%

0.0%

+1.6%

-0.1%

-0.8%

-1.6%

-0.05

-0.04

-0.02

+0.02

Frac

tiona

lPro

tect

ive

Put

-0.2%

-0.5%

-0.5%

-0.3%

-1.3%

-1.4%

-1.4%

-1.2%

16.5%

15.0%

14.6%

14.8%

0.17

0.17

0.16

0.17

Div

este

dEq

uity

-0.6%

-0.7%

-0.6%

-0.4%

-1.3%

-1.4%

-1.4%

-1.2%

12.6%

12.6%

12.5%

12.6%

0.27

0.25

0.24

0.25

Di�

+0.4%

+0.1%

+0.1%

+0.1%

0.0%

0.0%

0.0%

0.0%

+3.9%

+2.4%

+2.1%

+2.2%

-0.10

-0.09

-0.08

-0.08

Frac

tiona

lPut

Spre

ad95

/85

-0.7%

-0.8%

-0.6%

-0.6%

-2.2%

-2.2%

-2.2%

-2.3%

19.8%

18.7%

19.1%

+20.0

%0.2

10.1

90.1

90.1

9D

ives

ted

Equi

ty-0

.9%-0

.9%-0

.8%-0

.7%-2

.2%-2

.2%-2

.2%-2

.3%16

.7%16

.7%17

.2%18

.3%0.2

70.2

40.2

30.2

1Di�

+0.2%

+0.1%

+0.2%

+0.2%

0.0%

0.0%

0.0%

0.0%

+3.0%

+2.0%

+1.9%

+1.6%

-0.06

-0.04

-0.03

-0.02

45

Tabl

e4.2:

Mea

nva

lueo

fthe

annu

aliz

edal

pha,

beta

and

tota

lcos

t(as

%of

initi

alin

vest

men

t)fo

rthe

simul

ated

stra

tegi

esac

ross

the

four

portf

olio

s.�

eto

talc

osti

smea

sure

das

thea

ccum

ulat

edco

stov

erth

ewho

lein

vest

men

thor

izon

divi

ded

byth

eini

tiali

nves

tmen

t.�

esiz

eoft

hein

itial

inve

stm

enti

seq

ualf

oral

lstra

tegi

es.

Ann

ualiz

edal

pha

Beta

Tota

lcos

tas%

ofin

itial

inve

stm

ent

1m3m

6m12

m1m

3m6m

12m

1m3m

6m12

mPr

otec

tive

Put

-3.1%

-2.9%

-3.1%

-3.5%

0.63

0.56

0.52

0.48

56.7%

55.1%

47.9%

39.3%

Div

este

dEq

uity

0.0%

0.0%

0.0%

0.0%

0.52

0.50

0.51

0.51

0.0%

0.0%

0.0%

0.0%

Di�

-3.1%

-2.9%

-3.1%

-3.5%

+0.12

+0.06

0.0

-0.03

+56.7%

+55.1%

+47.9%

+39.3%

PutS

prea

d95

/80

-2.3%

-2.0%

-2.0%

-2.0%

0.78

0.71

0.71

0.73

47.2%

44.5%

35.2%

24.8%

Div

este

dEq

uity

0.0%

0.0%

0.0%

0.0%

0.65

0.63

0.65

0.68

0.0%

0.0%

0.0%

0.0%

Di�

-2.3%

-2.0%

-2.0%

-2.0%

+0.13

+0.08

+0.06

+0.05

+47.2%

+44.5%

+35.2%

+24.8%

PutS

prea

d95

/85

-2.0%

-1.6%

-1.6%

-1.6%

0.82

0.76

0.77

0.80

44.3%

38.3%

28.0%

18.5%

Div

este

dEq

uity

0.0%

0.0%

0.0%

0.0%

0.65

0.66

0.69

0.73

0.0%

0.0%

0.0%

0.0%

Di�

-2.0%

-1.6%

-1.5%

-1.5%

+0.17

+0.10

+0.09

+0.08

+44.3%

+38.3%

+28.0%

+18.5%

PutS

prea

d10

0/90

-2.2%

-1.6%

-1.6%

-1.6%

0.60

0.67

0.73

0.79

139.8

%66

.6%39

.3%22

.8%D

ives

ted

Equi

ty0.0

%0.0

%0.0

%0.0

%0.5

60.6

20.6

80.7

10.0

%0.0

%0.0

%0.0

%Di�

-2.2%

-1.6%

-1.6%

-1.6%

+0.03

+0.06

+0.06

+0.07

+139

.8%

+66.6%

+39.3%

+22.8%

Colla

r95/

105

-1.0%

-0.7%

-0.4%

-0.1%

0.49

0.31

0.22

0.14

3.0%

0.1%

-1.0%

-2.1%

Div

este

dEq

uity

0.0%

0.0%

0.0%

0.0%

0.47

0.36

0.30

0.25

0.0%

0.0%

0.0%

0.0%

Di�

-1.0%

-0.7%

-0.4%

-0.1%

+0.02

-0.05

-0.08

-0.11

+3.0%

+0.1%

-1.0%

-2.1%

Frac

tiona

lPro

tect

ive

Put

-3.0%

-2.7%

-2.5%

-2.4%

0.64

0.59

0.58

0.59

56.7%

55.1%

47.9%

39.3%

Div

este

dEq

uity

0.0%

0.0%

0.0%

0.0%

0.51

0.52

0.52

0.52

0.0%

0.0%

0.0%

0.0%

Di�

-3.0%

-2.7%

-2.5%

-2.3%

+0.13

+0.07

+0.06

+0.06

56.7%

55.1%

47.9%

39.3%

Frac

tiona

lPut

Spre

ad95

/85

-1.9%

-1.4%

-0.01

-0.7%

0.82

0.78

0.81

0.85

44.3%

38.3%

28.0%

18.5%

Div

este

dEq

uity

0.0%

0.0%

0.0%

0.0%

0.68

0.69

0.72

0.77

0.0%

0.0%

0.0%

0.0%

Di�

-1.9%

-1.4%

-1.0%

-0.7%

+0.14

+0.09

+0.09

+0.08

44.3%

38.3%

28.0%

18.5%

46

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4

Diff. in Sharpe ratio versus Divesting

0

1

2

3

4

5

6P

robabili

ty D

ensity F

unction

1m Portfolios

Protective Put, (-0.1, 0.14)

Put Spread 95/80, (-0.07, 0.1)

Put Spread 95/85, (-0.07, 0.09)

Put Spread 100/90, (-0.09, 0.14)

Collar 95/105, (-0.05, 0.11)

Frac. Protective Put, (-0.1, 0.14)

Frac. Put Spread 95/85, (-0.06, 0.08)

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6


0

1

2

3

4

5

6

Pro

babili

ty D

ensity F

unction

3m Portfolios


Put Spread 95/80, (-0.07, 0.1)

Put Spread 95/85, (-0.05, 0.08)

Put Spread 100/90, (-0.05, 0.1)

Collar 95/105, (-0.04, 0.14)


Frac. Put Spread 95/85, (-0.04, 0.07)

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6


0

1

2

3

4

5

6

7

8

Pro

babili

ty D

ensity F

unction

6m Portfolios


Put Spread 95/80, (-0.06, 0.09)

Put Spread 95/85, (-0.04, 0.07)

Put Spread 100/90, (-0.04, 0.07)

Collar 95/105, (-0.02, 0.15)


Frac. Put Spread 95/85, (-0.03, 0.05)

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2


0

2

4

6

8

10

12

Pro

ba

bili

ty D

en

sity F

un

ctio

n

12m Portfolios


Put Spread 95/80, (-0.05, 0.08)

Put Spread 95/85, (-0.04, 0.06)

Put Spread 100/90, (-0.04, 0.06)

Collar 95/105, (0.02, 0.19)


Frac. Put Spread 95/85, (-0.02, 0.03)

Figure 4.1: Distribution of the di�erence in Sharpe ratio between the option-based protectionstrategies and the divested equity strategy.

47

0 0.5 1 1.5

Sharpe ratio

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Pro

babili

ty D

ensity F

unction

1m Portfolios

Protective Put, (0.31, 0.25)

Put Spread 95/80, (0.33, 0.27)

Put Spread 95/85, (0.34, 0.28)

Put Spread 100/90, (0.33, 0.26)

Collar 95/105, (0.36, 0.27)

Frac. Protective Put, (0.31, 0.25)

Frac. Put Spread 95/85, (0.34, 0.28)

0 0.5 1 1.5

Sharpe ratio

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Pro

babili

ty D

ensity F

unction

3m Portfolios


Put Spread 95/80, (0.34, 0.27)

Put Spread 95/85, (0.35, 0.28)

Put Spread 100/90, (0.34, 0.28)

Collar 95/105, (0.37, 0.28)


Frac. Put Spread 95/85, (0.36, 0.28)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Sharpe ratio

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Pro

babili

ty D

ensity F

unction

6m Portfolios


Put Spread 95/80, (0.35, 0.28)

Put Spread 95/85, (0.35, 0.28)

Put Spread 100/90, (0.36, 0.28)

Collar 95/105, (0.38, 0.31)


Frac. Put Spread 95/85, (0.36, 0.28)

0 0.5 1 1.5 2 2.5

Sharpe ratio

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Pro

babili

ty D

ensity F

unction

12m Portfolios


Put Spread 95/80, (0.34, 0.28)

Put Spread 95/85, (0.36, 0.29)

Put Spread 100/90, (0.36, 0.29)

Collar 95/105, (0.47, 0.37)


Frac. Put Spread 95/85, (0.37, 0.29)

Figure 4.2: Probability density function of the Sharpe ratios of the di�erent strategies duringperiods of positive return on the portfolios.

48

-0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0

Sharpe ratio

0

10

20

30

40

50

60

70

80

90

100

Pro

babili

ty D

ensity F

unction

1m Portfolios


Put Spread 95/80, (-0.02, 0.01)

Put Spread 95/85, (-0.02, 0.01)

Put Spread 100/90, (-0.01, 0.02)

Collar 95/105, (0, 0)


Frac. Put Spread 95/85, (-0.02, 0.01)

-0.1 -0.08 -0.06 -0.04 -0.02 0

Sharpe ratio

0

10

20

30

40

50

60

70

80

90

100

Pro

babili

ty D

ensity F

unction

3m Portfolios


Put Spread 95/80, (-0.01, 0.01)

Put Spread 95/85, (-0.01, 0.01)

Put Spread 100/90, (-0.01, 0.01)

Collar 95/105, (0, 0)


Frac. Put Spread 95/85, (-0.01, 0.01)

-0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0

Sharpe ratio

0

10

20

30

40

50

60

70

80

90

100

Pro

babili

ty D

ensity F

unction

6m Portfolios


Put Spread 95/80, (-0.01, 0.01)

Put Spread 95/85, (-0.02, 0.01)

Put Spread 100/90, (-0.01, 0.01)

Collar 95/105, (0, 0)

Frac. Protective Put, (-0.01, 0)

Frac. Put Spread 95/85, (-0.01, 0.01)

-0.1 -0.08 -0.06 -0.04 -0.02 0

Sharpe ratio

0

10

20

30

40

50

60

70

80

90

100

Pro

babili

ty D

ensity F

unction

12m Portfolios

Protective Put, (-0.01, 0)

Put Spread 95/80, (-0.01, 0.01)

Put Spread 95/85, (-0.02, 0.01)

Put Spread 100/90, (-0.02, 0.02)

Collar 95/105, (0, 0)


Frac. Put Spread 95/85, (-0.02, 0.02)

Figure 4.3: Probability density function of the adjusted Sharpe ratios of the di�erent strategiesduring periods of negative return on the portfolios.

49

0 2 4 6 8 10 12

Maturity (months)

-1.8%

-1.6%

-1.4%

-1.2%

-1%

-0.8%

-0.6%

-0.4%

-0.2%

0%

Diff.

in

99

th p

erc

en

tile

dra

wd

ow

n v

ers

us D

ive

stin

g

5-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105

Fractional Protective Put

Fractional Put Spread 95/85

0 2 4 6 8 10 12

Maturity (months)

-1%

-0.8%

-0.6%

-0.4%

-0.2%

2.2204e-16%

0.2%

Diff. in 9

9th

perc

entile

dra

wdow

n v

ers

us D

ivesting

10-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-3%

-2%

-1%

0%

1%

2%

3%

4%

5%

6%

Diff.

in

99

th p

erc

en

tile

dra

wd

ow

n v

ers

us D

ive

stin

g

20-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-4%

-3%

-2%

-1%

0%

1%

2%

3%

Diff. in 9

9th

perc

entile

dra

wdow

n v

ers

us D

ivesting

63-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-5%

-4%

-3%

-2%

-1%

0%

1%

2%

3%

Diff. in 9

9th

perc

entile

dra

wdow

n v

ers

us D

ivesting

125-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-6%

-5%

-4%

-3%

-2%

-1%

0%

1%

2%

Diff.

in

99

th p

erc

en

tile

dra

wd

ow

n v

ers

us D

ive

stin

g

250-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



Figure 4.4: Di�erence in the mean 99th percentile peak-to-trough drawdown between the option-based protection strategies and the divested equity strategy versus maturity. �e di�erence ismeasured in percentage points.

50

0 2 4 6 8 10 12

Maturity (months)

-0.5%

-0.4%

-0.3%

-0.2%

-0.1%

0%

0.1%

0.2%

Diff.

in

50

th p

erc

en

tile

dra

wd

ow

n v

ers

us D

ive

stin

g

5-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-0.8%

-0.6%

-0.4%

-0.2%

1.1102e-16%

0.2%

0.4%

Diff. in 5

0th

perc

entile

dra

wdow

n v

ers

us D

ivesting

10-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-1.4%

-1.2%

-1%

-0.8%

-0.6%

-0.4%

-0.2%

0%

0.2%

0.4%

0.6%

Diff.

in

50

th p

erc

en

tile

dra

wd

ow

n v

ers

us D

ive

stin

g

20-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-3%

-2.5%

-2%

-1.5%

-1%

-0.5%

0%

0.5%

1%

Diff.

in

50

th p

erc

en

tile

dra

wd

ow

n v

ers

us D

ive

stin

g

63-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-4%

-3.5%

-3%

-2.5%

-2%

-1.5%

-1%

-0.5%

0%

0.5%

1%

Diff.

in

50

th p

erc

en

tile

dra

wd

ow

n v

ers

us D

ive

stin

g

125-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-5%

-4%

-3%

-2%

-1%

0%

1%

Diff. in 5

0th

perc

entile

dra

wdow

n v

ers

us D

ivesting

250-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



Figure 4.5: Di�erence in the mean 50th percentile peak-to-trough drawdown between the option-based protection strategies and the divested equity strategy versus maturity. �e di�erence ismeasured in percentage points.

51

0 2 4 6 8 10 12

Maturity (months)

-6.5%

-6%

-5.5%

-5%

-4.5%

-4%

-3.5%

-3%

-2.5%

-2%

-1.5%

Mean 9

9th

perc

entile

dra

wdow

n

5-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-12%

-11%

-10%

-9%

-8%

-7%

-6%

-5%

-4%

-3%

-2%

Mean 9

9th

perc

entile

dra

wdow

n

10-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-22%

-20%

-18%

-16%

-14%

-12%

-10%

-8%

-6%

-4%

Mean 9

9th

perc

entile

dra

wdow

n

20-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-30%

-25%

-20%

-15%

-10%

-5%

Mean 9

9th

perc

entile

dra

wdow

n

63-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-35%

-30%

-25%

-20%

-15%

-10%

-5%

Mean 9

9th

perc

entile

dra

wdow

n

125-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-35%

-30%

-25%

-20%

-15%

-10%

-5%

Mean 9

9th

perc

entile

dra

wdow

n

250-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



Figure 4.6: Mean 99th percentile peak-to-trough drawdown for the option-based protectionstrategies versus maturity.

52

0 2 4 6 8 10 12

Maturity (months)

-1.2%

-1%

-0.8%

-0.6%

-0.4%

-0.2%

0%

Mean 5

0th

perc

entile

dra

wdow

n

5-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-2%

-1.8%

-1.6%

-1.4%

-1.2%

-1%

-0.8%

-0.6%

-0.4%

-0.2%

0%

Mean 5

0th

perc

entile

dra

wdow

n

10-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-3%

-2.5%

-2%

-1.5%

-1%

-0.5%

0%

Mean 5

0th

perc

entile

dra

wdow

n

20-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-6%

-5%

-4%

-3%

-2%

-1%

0%

Mean 5

0th

perc

entile

dra

wdow

n

63-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-9%

-8%

-7%

-6%

-5%

-4%

-3%

-2%

-1%

Mean 5

0th

perc

entile

dra

wdow

n

125-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-14%

-12%

-10%

-8%

-6%

-4%

-2%

Mean 5

0th

perc

entile

dra

wdow

n

250-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



Figure 4.7: Mean 50th percentile peak-to-trough drawdown for the option-based protectionstrategies versus maturity.

53

4.2 Backtesting

4.2.1 Whole period: December 2002 - Mars 2018

In addition to the results presented in this section, Tables 6.5 - 6.8 in Appendix report the draw-

down characteristics in more detail. Figures 6.5 - 6.8 in Appendix visualize the value development

of the di�erent option-based protection strategies over the whole investment horizon. Figure 6.14

in Appendix visualizes the change in the total cost as a percentage of the initial investment with

respect to option maturity.

54

Tabl

e4.3:

Sum

mar

yof

annu

aliz

edre

turn

sand

perfo

rman

cem

easu

resf

orth

edi�

eren

tstra

tegi

esac

ross

thef

ourp

ortfo

lios.

Each

stra

tegy

isbe

nchm

arke

dag

ains

tthe

dive

sted

equi

ty.

Arit

hmet

icre

turn

Geom

etric

retu

rnVo

latil

itySh

arpe

ratio

1m3m

6m12

m1m

3m6m

12m

1m3m

6m12

m1m

3m6m

12m

Prot

ectiv

ePu

t5.5

%5.1

%5.5

%5.8

%4.0

%4.3

%4.9

%5.4

%17

.7%13

.3%11

.8%10

.9%0.2

20.2

70.3

30.3

9D

ives

ted

Equi

ty4.2

%4.5

%5.2

%5.8

%4.0

%4.3

%4.9

%5.4

%7.4

%8.2

%9.8

%11

.1%0.3

60.3

60.3

80.3

8Di�

+1.3%

+0.6%

+0.2%

-0.0%

0.0%

0.0%

0.0%

0.0%

+10.3%

+5.1%

+2.0%

-0.3%

-0.14

-0.10

-0.04

+0.01

PutS

prea

d95

/80

6.1%

6.0%

6.8%

7.3%

4.6%

5.0%

5.8%

6.4%

17.6%

15.1%

15.1%

15.0%

0.26

0.30

0.35

0.38

Div

este

dEq

uity

5.0%

5.4%

6.4%

7.2%

4.6%

5.0%

5.8%

6.4%

9.1%

10.1%

12.5%

14.2%

0.37

0.38

0.39

0.40

Di�

+1.1%

+0.6%

+0.4%

+0.1%

0.0%

0.0%

0.0%

0.0%

+8.5%

+4.9%

+2.6%

+0.8%

-0.11

-0.08

-0.04

-0.01

PutS

prea

d95

/85

6.6%

6.7%

7.4%

7.8%

5.1%

5.6%

6.3%

6.7%

18.2%

15.9%

16.0%

16.2%

0.28

0.32

0.37

0.39

Div

este

dEq

uity

5.5%

6.1%

7.1%

7.7%

5.1%

5.6%

6.3%

6.7%

10.3%

11.9%

14.1%

15.5%

0.38

0.39

0.39

0.40

Di�

+1.1%

+0.6%

+0.3%

+0.1%

0.0%

0.0%

0.0%

0.0%

+7.9%

+4.0%

+1.9%

+0.7%

-0.10

-0.06

-0.03

-0.01

PutS

prea

d10

0/90

4.2%

5.5%

6.8%

7.8%

2.9%

4.5%

5.8%

6.8%

16.6%

15.0%

15.4%

15.6%

0.16

0.26

0.34

0.40

Div

este

dEq

uity

2.9%

4.7%

6.4%

7.8%

2.9%

4.5%

5.8%

6.8%

4.4%

8.6%

12.4%

15.6%

0.31

0.37

0.39

0.40

Di�

+1.3%

+0.8%

+0.4%

+0.0%

0.0%

0.0%

0.0%

0.0%

+12.1%

+6.4%

+3.1%

0.0%

-0.15

-0.11

-0.05

-0.00

Colla

r95/

105

5.9%

5.9%

4.6%

2.1%

5.3%

5.7%

4.4%

2.0%

12.5%

8.4%

6.7%

5.5%

0.35

0.52

0.45

0.10

Div

este

dEq

uity

5.8%

6.3%

4.7%

2.0%

5.3%

5.7%

4.4%

2.0%

11.0%

12.2%

8.6%

2.2%

0.38

0.39

0.37

0.19

Di�

+0.2%

-0.4%

-0.1%

+0.1%

0.0%

0.0%

0.0%

0.0%

+1.5%

-3.8%

-1.9%

+3.3%

-0.03

+0.13

+0.08

-0.09

Frac

tiona

lPro

tect

ive

Put

5.6%

5.3%

5.8%

5.9%

4.2%

4.5%

5.2%

5.5%

17.4%

13.2%

11.7%

10.6%

0.23

0.28

0.36

0.41

Div

este

dEq

uity

4.4%

4.8%

5.7%

6.0%

4.2%

4.5%

5.2%

5.5%

7.8%

8.7%

10.7%

11.6%

0.36

0.37

0.38

0.39

Di�

+1.2%

+0.5%

+0.1%

-0.1%

0.0%

0.0%

0.0%

0.0%

+9.5%

+4.5%

+1.0%

-1.0%

-0.13

-0.09

-0.02

+0.03

Frac

tiona

lPut

Spre

ad95

/85

6.7%

6.8%

7.6%

7.9%

5.2%

5.7%

6.5%

6.9%

17.9%

15.9%

16.1%

16.1%

0.29

0.33

0.37

0.40

Div

este

dEq

uity

5.7%

6.3%

7.3%

7.9%

5.2%

5.7%

6.5%

6.9%

10.7%

12.3%

14.6%

15.9%

0.38

0.39

0.40

0.40

Di�

+1.0%

+0.5%

+0.2%

0.0%

0.0%

0.0%

0.0%

0.0%

+7.1%

+3.6%

+1.5%

+0.2%

-0.10

-0.06

-0.02

-0.00

55

Tabl

e4.4:

Sum

mar

yof

annu

aliz

edal

pha,

beta

and

tota

lcos

t(as

%of

initi

alin

vest

men

t)fo

rthe

prot

ectio

nst

rate

gies

acro

ssth

efou

rpor

tfolio

s.�

etot

alco

stis

mea

sure

das

thea

ccum

ulat

edco

stov

erth

ewho

lein

vest

men

thor

izon

divi

ded

byth

eini

tiali

nves

tmen

t.�

esiz

eoft

hein

itial

inve

stm

enti

sequ

alfo

rall

stra

tegi

es.

Ann

ualiz

edal

pha

Beta

Tota

lcos

tas%

ofin

itial

inve

stm

ent

1m3m

6m12

m1m

3m6m

12m

1m3m

6m12

mPr

otec

tive

Put

-2.3%

-0.8%

0.3%

1.0%

0.83

0.59

0.51

0.46

+182

.0%17

7.1%

162.4

%14

8.2%

Div

este

dEq

uity

-0.1%

-0.1%

0.0%

0.0%

0.39

0.43

0.52

0.59

0.0%

0.0%

0.0%

0.0%

Di�

-2.3%

-0.8%

+0.3%

+1.1%

+0.44

+0.16

-0.01

-0.13

+182

.0%

177.1%

162.4%

148.2%

PutS

prea

d95

/80

-2.1%

-1.3%

-0.7%

-0.2%

0.88

0.77

0.78

0.79

162.6

%14

1.2%

114.6

%85

.6%D

ives

ted

Equi

ty-0

.1%0.0

%0.0

%0.0

%0.4

80.5

40.6

60.7

60.0

%0.0

%0.0

%0.0

%Di�

-0.02

-1.2%

-0.6%

-0.1%

+0.39

+0.23

+0.12

+0.03

+162

.6%

+141

.2%

+114

.6%

+85.6%

PutS

prea

d95

/85

-2.1%

-1.1%

-0.5%

-0.2%

0.94

0.82

0.84

0.85

143.9

%11

6.6%

89.8%

63.3%

Div

este

dEq

uity

0.0%

0.0%

0.0%

0.0%

0.55

0.63

0.75

0.82

0.0%

0.0%

0.0%

0.0%

Di�

-0.02

-0.01

-0.4%

-0.2%

+0.39

+0.19

+0.09

+0.03

+143

.9%

+116

.6%

+89.8%

+63.3%

PutS

prea

d10

0/90

-3.5%

-1.9%

-0.8%

0.0%

0.82

0.77

0.81

0.82

+442

.0%22

2.6%

139.2

%83

.8%D

ives

ted

Equi

ty-0

.1%-0

.1%0.0

%0.0

%0.2

40.4

60.6

60.8

30.0

%0.0

%0.0

%0.0

%Di�

-3.4%

-1.8%

-0.8%

+0.1%

+0.58

+0.31

+0.15

-0.01

+442

.0%

+222

.6%

+139

.2%

+83.8%

Colla

r95/

105

0.3%

1.9%

1.3%

-0.7%

0.56

0.36

0.28

0.22

106.8

%70

.9%45

.1%25

.8%D

ives

ted

Equi

ty0.0

%0.0

%-0

.1%-0

.1%0.5

80.6

50.4

50.1

20.0

%0.0

%0.0

%0.0

%Di�

+0.4%

+2.0%

+1.3%

-0.7%

-0.02

-0.29

-0.18

+0.10

+106

.8%

+70.9%

+45.1%

+25.8%

Frac

tiona

lPro

tect

ive

Put

-2.1%

-0.6%

0.5%

1.2%

0.81

0.59

0.52

0.45

181.6

%17

5.8%

162.4

%14

0.1%

Div

este

dEq

uity

-0.1%

-0.1%

0.0%

0.0%

0.42

0.46

0.57

0.62

0.0%

0.0%

0.0%

0.0%

Di�

-2.0%

-0.6%

+0.6%

+1.2%

+0.40

+0.13

-0.06

-0.16

+181

.6%

+175

.8%

+162

.4%

+140

.1%

Frac

tiona

lPut

Spre

ad95

/85

-1.8%

-0.9%

-0.4%

0.0%

0.92

0.82

0.85

0.85

143.6

%11

6.1%

89.3%

60.5%

Div

este

dEq

uity

0.0%

0.0%

0.0%

0.0%

0.57

0.65

0.77

0.85

0.0%

0.0%

0.0%

0.0%

Di�

-1.8%

-0.9%

-0.4%

0.0%

+0.35

+0.17

+0.07

0.00

+143

.6%

+116

.1%

+89.3%

+60.5%

56

0 2 4 6 8 10 12

Maturity (months)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Sh

arp

e r

atio

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Diff.

in

Sh

arp

e r

atio

ve

rsu

s D

ive

ste

d E

qu

ity

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



Figure 4.8: Le�: Sharpe ratio of the option-based protection strategies versus maturity. Right:Di�erence in Sharpe ratio between the option-based protection strategies and the divested equitystrategy versus maturity.

0 50 100 150 200 250

Windows (days)

60%

65%

70%

75%

80%

85%

90%

95%

Pro

babili

ty D

ivesting h

as b

etter

Dra

wdow

n

1m Portfolios

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 50 100 150 200 250

Windows (days)

30%

40%

50%

60%

70%

80%

90%

100%

Pro

babili

ty D

ivesting h

as b

etter

Dra

wdow

n

3m Portfolios

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 50 100 150 200 250

Windows (days)

30%

40%

50%

60%

70%

80%

90%

100%

Pro

babili

ty D

ivesting h

as b

etter

Dra

wdow

n

6m Portfolios

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 50 100 150 200 250

Windows (days)

30%

40%

50%

60%

70%

80%

90%

Pro

babili

ty D

ivesting h

as b

etter

Dra

wdow

n

12m Portfolios

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



Figure 4.9: �e probability that the divested equity strategy has be�er peak-to-trough drawdownsthan the option-based protection strategies.

57

0 2 4 6 8 10 12

Maturity (months)

-5%

-4%

-3%

-2%

-1%

0%

1%

2%

Diff. in 9

9th

perc

entile

dra

wdow

n v

ers

us D

ivesting

5-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-8%

-6%

-4%

-2%

0%

2%

4%

Diff. in 9

9th

perc

entile

dra

wdow

n v

ers

us D

ivesting

10-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-10%

-5%

0%

5%

Diff. in 9

9th

perc

entile

dra

wdow

n v

ers

us D

ivesting

20-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-25%

-20%

-15%

-10%

-5%

0%

5%

10%

15%

Diff. in 9

9th

perc

entile

dra

wdow

n v

ers

us D

ivesting

63-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-40%

-30%

-20%

-10%

0%

10%

20%

Diff. in 9

9th

perc

entile

dra

wdow

n v

ers

us D

ivesting

125-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-50%

-40%

-30%

-20%

-10%

0%

10%

20%

30%

Diff. in 9

9th

perc

entile

dra

wdow

n v

ers

us D

ivesting

250-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



Figure 4.10: Di�erence in the 99th percentile peak-to-trough drawdown between the divestedequity strategy and the option-based protection strategies versus maturity. �e di�erence ismeasured in percentage points.

58

0 2 4 6 8 10 12

Maturity (months)

-0.5%

-0.4%

-0.3%

-0.2%

-0.1%

1.1102e-16%

0.1%

Diff. in 5

0th

perc

entile

dra

wdow

n v

ers

us D

ivesting

5-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-0.8%

-0.7%

-0.6%

-0.5%

-0.4%

-0.3%

-0.2%

-0.1%

0%

0.1%

Diff. in 5

0th

perc

entile

dra

wdow

n v

ers

us D

ivesting

10-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-1.2%

-1%

-0.8%

-0.6%

-0.4%

-0.2%

0%

0.2%

Diff. in 5

0th

perc

entile

dra

wdow

n v

ers

us D

ivesting

20-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-2.5%

-2%

-1.5%

-1%

-0.5%

0%

0.5%

Diff.

in

50

th p

erc

en

tile

dra

wd

ow

n v

ers

us D

ive

stin

g

63-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-3%

-2.5%

-2%

-1.5%

-1%

-0.5%

0%

0.5%

Diff. in 5

0th

perc

entile

dra

wdow

n v

ers

us D

ivesting

125-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-4%

-3.5%

-3%

-2.5%

-2%

-1.5%

-1%

-0.5%

0%

Diff. in 5

0th

perc

entile

dra

wdow

n v

ers

us D

ivesting

250-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



Figure 4.11: Di�erence in the 50th percentile peak-to-trough drawdown between the divestedequity strategy and the option-based protection strategies versus maturity. �e di�erence ismeasured in percentage points.

59

0 2 4 6 8 10 12

Maturity (months)

-8%

-7%

-6%

-5%

-4%

-3%

-2%

99th

perc

entile

dra

wdow

n

5-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-12%

-11%

-10%

-9%

-8%

-7%

-6%

-5%

-4%

-3%

99th

perc

entile

dra

wdow

n

10-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-16%

-14%

-12%

-10%

-8%

-6%

-4%

99th

perc

entile

dra

wdow

n

20-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-35%

-30%

-25%

-20%

-15%

-10%

-5%

99th

perc

entile

dra

wdow

n

63-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-50%

-45%

-40%

-35%

-30%

-25%

-20%

-15%

-10%

99th

perc

entile

dra

wdow

n

125-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-60%

-55%

-50%

-45%

-40%

-35%

-30%

-25%

-20%

-15%

-10%

99th

perc

entile

dra

wdow

n

250-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



Figure 4.12: 99th percentile peak-to-trough drawdown for the option-based protection strategiesversus maturity.

60

0 2 4 6 8 10 12

Maturity (months)

-0.7%

-0.6%

-0.5%

-0.4%

-0.3%

-0.2%

-0.1%

0%

50

th p

erc

en

tile

dra

wd

ow

n

5-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-1.1%

-1%

-0.9%

-0.8%

-0.7%

-0.6%

-0.5%

-0.4%

-0.3%

-0.2%

-0.1%

50

th p

erc

en

tile

dra

wd

ow

n

10-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-1.6%

-1.4%

-1.2%

-1%

-0.8%

-0.6%

-0.4%

-0.2%

50

th p

erc

en

tile

dra

wd

ow

n

20-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-3%

-2.5%

-2%

-1.5%

-1%

-0.5%

0%

50

th p

erc

en

tile

dra

wd

ow

n

63-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-3.5%

-3%

-2.5%

-2%

-1.5%

-1%

-0.5%

50

th p

erc

en

tile

dra

wd

ow

n

125-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 2 4 6 8 10 12

Maturity (months)

-4.5%

-4%

-3.5%

-3%

-2.5%

-2%

-1.5%

-1%

-0.5%

50

th p

erc

en

tile

dra

wd

ow

n

250-Day

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



Figure 4.13: 50th percentile peak-to-trough drawdown for the option-based protection strategiesversus maturity.

61

4.2.2 Financial crisis

In addition to the results presented in this section, Figures 6.9 - 6.12 in Appendix visualize the

value development of the di�erent option-based protection strategies over the whole investment

horizon. Figure 6.15 in Appendix visualizes the change in the total cost as a percentage of the

initial investment with respect to option maturity.

62

Tabl

e4.5

:Su

mm

ary

ofan

nual

ized

retu

rns

and

perfo

rman

cem

easu

res

for

the

di�e

rent

stra

tegi

esac

ross

the

four

portf

olio

s.Ea

chst

rate

gyis

benc

hmar

ked

agai

nstt

hedi

vest

edeq

uity

stra

tegy

.A

rithm

etic

retu

rnGe

omet

ricre

turn

Vola

tility

Adj.

Shar

peRa

tio20

-Day

Dra

wdo

wn

(Med

ian)

1m3m

6m12

m1m

3m6m

12m

1m3m

6m12

m1m

3m6m

12m

1m3m

6m12

mPr

otec

tive

Put

-46.6

%-2

8.6%

-28.7

%-2

2.7%

-38.8

%-2

5.6%

-25.4

%-2

0.9%

21.9%

13.0%

11.4%

12.1%

-0.10

-0.04

-0.03

-0.03

-4.9%

-2.3%

-1.9%

-1.3%

Div

este

dEq

uity

-44.4

%-2

7.8%

-27.7

%-2

2.3%

-38.8

%-2

5.6%

-25.4

%-2

0.9%

30.4%

18.4%

18.3%

14.7%

-0.13

-0.05

-0.05

-0.03

-3.7%

-2.3%

-2.3%

-1.8%

Di�

-2.2%

-0.8%

-1.0%

-0.3%

0.0%

0.0%

0.0%

0.0%

-8.5%

-5.3%

-6.9%

-2.6%

+0.03

+0.01

+0.02

+0.01

-1.2%

-0.1%

+0.3%

+0.5%

PutS

prea

d95

/80

-43.5

%-3

0.1%

-40.9

%-4

2.6%

-37.4

%-2

8.2%

-36.1

%-3

7.5%

25.9%

24.4%

28.2%

29.7%

-0.11

-0.07

-0.12

-0.13

-4.7%

-2.7%

-3.2%

-3.2%

Div

este

dEq

uity

-42.6

%-3

0.9%

-41.0

%-4

2.8%

-37.4

%-2

8.2%

-36.1

%-3

7.5%

29.1%

20.6%

27.8%

29.2%

-0.12

-0.06

-0.11

-0.12

-3.5%

-2.5%

-3.4%

-3.5%

Di�

-0.9%

+0.9%

+0.1%

+0.2%

0.0%

0.0%

0.0%

0.0%

-3.2%

+3.8%

+0.4%

+0.5%

+0.01

-0.01

-0.0

-0.0

-1.1%

-0.2%

+0.2%

+0.3%

PutS

prea

d95

/85

-47.8

%-3

4.6%

-45.9

%-4

7.8%

-40.7

%-3

2.0%

-40.0

%-4

1.4%

29.7%

27.8%

31.9%

33.3%

-0.14

-0.10

-0.15

-0.16

-5.0%

-2.8%

-3.5%

-3.7%

Div

este

dEq

uity

-47.0

%-3

5.7%

-46.0

%-4

7.9%

-40.7

%-3

2.0%

-40.0

%-4

1.4%

32.4%

23.9%

31.6%

33.1%

-0.15

-0.09

-0.15

-0.16

-3.9%

-2.9%

-3.8%

-4.0%

Di�

-0.8%

+1.0%

+0.1%

+0.1%

0.0%

0.0%

0.0%

0.0%

-2.7%

+3.9%

+0.3%

+0.2%

+0.01

-0.01

-0.0

-0.0

-1.1%

+0.1%

+0.3%

+0.3%

PutS

prea

d10

0/90

-26.0

%-3

1.8%

-41.6

%-4

9.0%

-25.0

%-2

9.9%

-37.1

%-4

2.2%

23.3%

27.3%

30.9%

34.0%

-0.06

-0.09

-0.13

-0.17

-2.5%

-2.5%

-3.0%

-3.6%

Div

este

dEq

uity

-27.1

%-3

3.1%

-42.2

%-4

9.0%

-25.0

%-2

9.9%

-37.1

%-4

2.2%

17.9%

22.1%

28.7%

34.0%

-0.05

-0.07

-0.12

-0.17

-2.2%

-2.7%

-3.5%

-4.1%

Di�

+1.1%

+1.3%

+0.6%

0.0%

0.0%

0.0%

0.0%

0.0%

+5.4%

+5.2%

+2.1%

0.0%

-0.01

-0.01

-0.01

-0.0

-0.2%

+0.2%

+0.5%

+0.5%

Colla

r95/

105

-34.2

%-1

3.3%

-10.8

%-3

.3%-2

9.7%

-12.6

%-1

0.4%

-3.3%

13.9%

6.5%

5.3%

3.8%

-0.05

-0.01

-0.01

-0.0

-3.9%

-1.3%

-1.0%

-0.6%

Div

este

dEq

uity

-32.8

%-1

3.1%

-10.7

%-3

.4%-2

9.7%

-12.6

%-1

0.4%

-3.3%

21.9%

8.7%

7.1%

2.6%

-0.07

-0.01

-0.01

-0.0

-2.7%

-1.1%

-0.9%

-0.3%

Di�

-1.4%

-0.2%

-0.1%

0.0%

0.0%

0.0%

0.0%

0.0%

-8.0%

-2.2%

-1.9%

+1.3%

+0.2

0.0

0.0

0.0

-1.2%

-0.2%

-0.2%

-0.3%

Frac

tiona

lPro

tect

ive

Put

-45.7

%-3

1.1%

-24.5

%-2

4.6%

-38.0

%-2

7.5%

-22.5

%-2

3.0%

20.2%

13.9%

13.8%

17.6%

-0.09

-0.04

-0.03

-0.04

-4.8%

-2.6%

-2.2%

-2.0%

Div

este

dEq

uity

-43.3

%-3

0.1%

-24.2

%-2

4.8%

-38.0

%-2

7.5%

-22.5

%-2

3.0%

29.6%

20.0%

15.9%

16.3%

-0.13

-0.06

-0.04

-0.04

-3.6%

-2.5%

-2.0%

-2.0%

Di�

-2.4%

-1.0%

-0.3%

+0.2%

0.0%

0.0%

0.0%

0.0%

-9.4%

-6.1%

-2.1%

+1.3%

+0.04

+0.02

0.0

0.0

-1.2%

-0.1%

-0.2%

+0.0%

Frac

tiona

lPut

Spre

ad95

/85

-43.2

%-3

9.9%

-46.0

%-5

0.2%

-37.6

%-3

5.8%

-40.1

%-4

3.1%

27.9%

29.5%

32.3%

35.2%

-0.12

-0.12

-0.15

-0.18

-4.7%

-3.4%

-3.8%

-4.2%

Div

este

dEq

uity

-42.8

%-4

0.5%

-46.1

%-5

0.3%

-37.6

%-3

5.8%

-40.1

%-4

3.1%

29.2%

27.4%

31.7%

35.0%

-0.13

-0.11

-0.15

-0.18

-3.6%

-3.4%

-3.8%

-4.2%

Di�

-0.4%

+0.6%

+0.2%

+0.1%

0.0%

0.0%

0.0%

0.0%

-1.3%

+2.0%

+0.6%

+0.3%

0.0

-0.01

-0.0

-0.0

-1.1%

-0.0%

+0.1%

-0.0%

63

Tabl

e4.6

:Sum

mar

yof

annu

aliz

edal

pha,

beta

,tot

alco

st(a

s%of

initi

alin

vest

men

t)an

dm

axim

umdr

awdo

wn

fort

hepr

otec

tion

stra

tegi

esac

ross

the

four

portf

olio

s.�

eto

talc

osti

smea

sure

das

the

accu

mul

ated

cost

over

the

who

lein

vest

men

thor

izon

divi

ded

byth

ein

itial

inve

stm

ent.

�e

size

ofth

ein

itial

inve

stm

enti

sequ

alfo

rall

stra

tegi

es.

Ann

ualiz

edal

pha

Beta

Tota

lcos

tas%

ofin

itial

inve

stm

ent

Max

imum

draw

dow

n1m

3m6m

12m

1m3m

6m12

m1m

3m6m

12m

1m3m

6m12

mPr

otec

tive

Put

-21.5

%-1

7.0%

-20.2

%-1

5.7%

0.43

0.20

0.15

0.12

22.2%

15.2%

14.1%

14.6%

-55.6

%-3

3.3%

-33.3

%-2

6.5%

Div

este

dEq

uity

-1.8%

-2.0%

-2.0%

-1.8%

0.74

0.44

0.44

0.36

0.0%

0.0%

0.0%

0.0%

-55.8

%-3

3.6%

-33.4

%-2

6.6%

Di�

-19.7%

-14.9%

-18.1%

-13.9%

-0.30

-0.24

-0.30

-0.23

+22.2%

+15.2%

+14.1%

+14.6%

+0.2%

+0.2%

+0.1%

+0.1%

PutS

prea

d95

/80

-9.0%

2.8%

-2.2%

-1.8%

0.60

0.57

0.67

0.70

17.9%

10.7%

8.4%

6.8%

-53.2

%-3

8.55%

-51.0

%-5

3.5%

Div

este

dEq

uity

-1.9%

-2.1%

-2.0%

-1.9%

0.70

0.50

0.67

0.71

0.0%

0.0%

0.0%

0.0%

-53.3

%-3

7.6%

-51.0

%-5

3.5%

Di�

-7.1%

+4.9%

-0.3%

+0.1%

-0.11

+0.07

-0.01

0.00

+17.9%

+10.7%

+8.4%

+6.8%

+0.2%

-1.0%

0.0%

0.0%

PutS

prea

d95

/85

-7.3%

3.4%

-1.7%

-1.5%

0.70

0.66

0.76

0.80

14.9%

8.4%

6.3%

4.9%

-59.4

%-4

5.6%

-58.1

%-6

0.8%

Div

este

dEq

uity

-1.6%

-2.1%

-1.6%

-1.5%

0.78

0.58

0.77

0.80

0.0%

0.0%

0.0%

0.0%

-59.5

%-4

3.8%

-58.1

%-6

0.8%

Di�

-5.8%

+5.6%

0.0%

0.0%

-0.09

+0.08

0.00

0.00

+14.9%

+8.4%

+6.3%

+4.9%

+0.1%

-1.8%

0.0%

0.0%

PutS

prea

d10

0/90

4.9%

5.8%

1.1%

-1.7%

0.53

0.65

0.74

0.82

28.4%

12.4%

8.2%

5.8%

-32.5

%-4

3.01%

-52.8

%-6

2.4%

Div

este

dEq

uity

-2.0%

-2.1%

-1.9%

-1.4%

0.43

0.53

0.70.8

20.0

%0.0

%0.0

%0.0

%-3

2.7%

-40.4

3%-5

2.8%

-62.3

%Di�

+6.9%

+8.0%

+3.0%

-0.4%

+0.10

+0.12

+0.04

-0.01

+28.4%

+12.4%

+8.2%

+5.8%

+0.1%

-2.6%

0.0%

-0.1%

Colla

r95/

105

-19.4

%-6

.9%-6

.2%-0

.6%0.2

60.1

10.0

80.0

56.1

%1.9

%0.3

%-0

.8%-3

9.9%

-15.2

%-1

2.4%

-4.7%

Div

este

dEq

uity

-2.1%

-1%

-0.7%

0.2%

0.53

0.21

0.17

0.06

0.0%

0.0%

0.0%

0.0%

-40.0

%-1

5.3%

-12.5

%-4

.1%Di�

-17.2%

-5.9%

-5.4%

-0.8%

-0.27

-0.10

-0.09

-0.01

+6.1%

+1.9%

+0.3%

-0.8%

+0.1%

+0.1%

+0.1%

-0.6%

Frac

tiona

lPro

tect

ive

Put

-21.7

%-1

7.5%

-12.4

%-6

.7%0.4

10.2

40.2

10.3

121

.2%15

.2%11

.4%8.3

%-5

4.1%

-36.2

%-2

8.4%

-29.2

%D

ives

ted

Equi

ty-1

.8%-2

.1%-1

.9%-1

.9%0.7

20.4

80.3

80.3

90.0

%0.0

%0.0

%0.0

%-5

4.2%

-36.5

%-2

8.9%

-29.7

%Di�

-19.9%

-15.4%

-10.6%

-4.8%

-0.30

-0.25

-0.18

-0.09

+21.2%

+15.2%

+11.4%

+8.3%

+0.2%

+0.4%

+0.5%

+0.5%

Frac

tiona

lPut

Spre

ad95

/85

-5.0%

0.9%

-1.1%

-1.1%

0.66

0.70

0.77

0.85

14.2%

+8.0%

5.1%

3.1%

-53.4

%-5

0.1%

-58.1

%-6

4.1%

Div

este

dEq

uity

-1.9%

-2.0%

-1.6%

-1.2%

0.71

0.66

0.77

0.85

0.0%

0.0%

0.0%

0.0%

-53.6

%-5

0.3%

-58.2

%-6

4.2%

Di�

-3.1%

+2.9%

+0.6%

+0.1%

-0.05

+0.04

+0.01

0.00

+14.2%

+8.0%

+5.1%

+3.1%

+0.2%

+0.2%

+0.2%

+0.1%

64

0 50 100 150 200 250

Windows (days)

0%

10%

20%

30%

40%

50%

60%

70%

80%

Pro

babili

ty D

ive

sting h

as b

etter

Dra

wdow

n

1m Portfolios

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 50 100 150 200 250

Windows (days)

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

Pro

ba

bili

ty D

ive

stin

g h

as b

ett

er

Dra

wd

ow

n

3m Portfolios

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 50 100 150 200 250

Windows (days)

0%

10%

20%

30%

40%

50%

60%

Pro

babili

ty D

ive

sting

has b

etter

Dra

wdow

n

6m Portfolios

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



0 50 100 150 200 250

Windows (days)

0%

10%

20%

30%

40%

50%

60%

70%

80%

Pro

babili

ty D

ivesting h

as b

etter

Dra

wdow

n

12m Portfolios

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



Figure 4.14: �e probability that the divested equity strategy has be�er peak-to-trough draw-downs than the option-based protection strategies.

65

Discussion


5.1.1 Risk-adjusted performance

�e mean of the annualized arithmetic and geometric returns, in Table 4.1, are all negative except

for the collar 95/105 strategy. �is is because the chosen jump process makes the distribution of

stock paths more weighted towards negative paths. In total, 58.7% of the simulated stock paths

yielded a negative annualized geometric return. �us, the strategies were run more o�en on bear

markets. We begin by analyzing the overall performance of the protection strategies.

As presented in Table 4.2, all protection strategies have beta values lower than 1, indicating that

they collect less equity risk premium. Beta values below 1 mean that the strategies earn less

during rising markets and lose less during falling markets. Buying put options and selling call

options yield beta values lower than 1 as they reduce the portfolio’s equity exposure. Table 4.2

also presents negative alphas. When portfolios are net long volatility, i.e. net long positions in

options, the additional cost of volatility risk premium adds negative alpha to the portfolio return,

meaning that the return is reduced by the same amount. �e put spread and collar strategies are

constructed to reduce the negative alpha by adding positive alpha through a short position in a

put and a call, respectively. Obviously, the divested equity strategy has zero alpha, since it only

has equity exposure.

Table 4.1 displays that, on average, the divested equity strategy outperformed the option-based

protection strategies in terms of risk-adjusted return. �e mean Sharpe ratios for the divested

equity strategy are superior to the other strategies in almost all portfolios, except in the 12m

portfolio where the collar 95/105 is slightly be�er. �is is also seen in Figure 4.1, which shows

that all strategies are more likely to underperform against divesting. Although the collar 95/105

has a positive di�erence of 0.02 to divesting, it exhibits a larger variation of Sharpe ratios and

66

is very likely to underperform. O�se�ing the negative alphas in the put spreads and the collar

95/105 by selling puts and calls was, on average, not su�cient to outperform a zero-alpha strategy

on a risk-adjusted basis.

Table 4.1 further shows that, as maturity was increased, the Sharpe ratio of the collar 95/105 and

put spread 100/90 improved. �is is a result of a large decline in volatility for the collar 95/105,

due to a sharp decrease in its beta values. and a slight improvement in the return for the put

spread 100/90, despite an increase in volatility. As evidenced in Table 4.2, the put spread 100/90

is a very expensive strategy for short-dated options. Its cost in the 1m portfolio is almost 2.5 times

more higher than the second most expensive strategy. As such, it also realized the worst return

on the 1m portfolio. Yet, its risk-adjusted return is comparable to the protective put and the

fractional protective put. �e payo�s obtained through protecting from the ATM level during

bear markets, evidently, neutralized the impact from the expensive premiums. �e total cost

drops signi�cantly for longer maturities and thus has a less drag on returns, which explains its

stronger risk-adjusted performance in the portfolios with longer-dated options. In fact, the total

cost of all strategies decreased with increasing maturity, as visualized in Figure 6.13 in Appendix.

Longer maturities led to a lower equity exposure in the protection strategies that provide full

downside protection, namely the collar 95/105, the protective put and the fractional protective

put. �is behaviour was less evident in the put spreads given their partial protection, which

is sensible as longer maturities mean that there is more time for the market to fall beyond the

spreads. However, despite a lower equity exposure, the mean risk-adjusted return on the protec-

tive put worsened as maturity was increased. Table 4.2 reports that its alpha values became more

negative, e�ectively implying that there is no escape from the expensive volatility risk premium.

Table 4.1, or for be�er visualization Figure 6.13 in Appendix, shows that it is the second most

expensive strategy. �e protective put underperformed against all strategies, in accordance with

the criticism earned in previous studies.

Interestingly, the fractional protective put saw an improvement in absolute returns and alpha

values over the regular protective put. Its beta values did, however, not reduce as much with

increasing maturity, because of its partial protection in the �rst months. For example, it takes

twelve months for the fractional protective put to be fully protected in the 12m portfolio. Un-

til then it is vulnerable to potential declines in the market. Consequently, it realized a higher

67

volatility for the longer maturities and therefore saw only a slight improvement in risk-adjusted

return. �e same behaviour was documented for the fractional put spread 95/85. �ese �ndings

suggest that buying overlapping fractional amounts of put options, on average, has a marginal

impact on the risk-adjusted return when implied volatility is assumed to be constant.

In order to understand the di�erence between full downside protection and partial protection

be�er, we split the risk-adjusted performance into two cases. Figures 4.2 and 4.3 depict the

distribution of the Sharpe ratio when the long-term excess return is positive and negative, re-

spectively.

Positive long-term excess return

Figure 4.2 shows that when the long-term excess return is positive, the protective put was the

poorest performer among the di�erent protection strategies. �e high cost of put options have

a noticeable impact on the returns. �e mean Sharpe ratio, however, improved modestly from

0.31 in the 1m portfolio to 0.33 in the 12m portfolio. Yet, not enough to make the protective

put a more a�ractive protection strategy than just simply divesting equity or pursuing another

option-based strategy.

�e fractional protective put shows a marginal improvement by 0.01 in mean Sharpe ratio over

the regular protective put for the longer-dated options (3m, 6m and 12m portfolios). �e put

spread strategies, on average, exhibit be�er risk-adjusted return than both the protective put and

the fractional protective put when excess return is positive. �e put spread 95/85 is the cheapest

alternative and the slightly be�er performer in the 1m and 3m portfolios. Its comparably tighter

spread is apparently be�er suited for short-dated options, as longer maturities would give the

market more time to fall below its protection level. On the other hand, it is the strategy with

the highest equity exposure with a beta value ranging between 0.76 - 0.82, only second to the

fractional put spread 95/85, and therefore earns relatively more equity risk premium when the

stock price rises. Nonetheless, it is the strategy with the highest return volatility on average. In

the 6m and 12m portfolios, the put spread 100/90 demonstrated comparable or marginally be�er

mean Sharpe ratio, because of its comparable beta values and relatively lower return volatility.

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�e put spread 100/90 is the most expensive strategy, but generated alphas that are more or less

on par with the put spread 95/85. Buying ATM puts reduced equity exposure more than buying

5% OTM puts in the 1m and 3m portfolios. �is is sensible because in the shorter term, i.e. one to

three months, the market has less time to fall more than 10%, which is fully within the protection

capacity of the put spread 100/90, while the other put spreads would only cover 5% because of

the 5% OTM puts. In the longer term, from six to twelve months, the market is more likely to

dip beyond the 10% spread, which is also why we see an increase in the beta value for the put

spread 100/90 from 0.60 in the 1m portfolio to 0.79 in the 12m portfolio. �is increase allows the

put spread 100/90 to collect more equity risk premium during periods of positive excess return.

Consequently, its mean Sharpe ratio in Figure 4.2 marginally improved with increasing maturity,

despite an increase in mean volatility.

�e put spread 95/80, on the other hand, sells a 20% OTM put, which adds less positive alpha and

makes it more burdened by the volatility risk premium. �erefore, the alphas for the put spread

95/80 is, on average, slightly worse. Its mean beta values, however, are between the other two put

spreads and its mean total cost is comparable to the put spread 95/85. As such its performance

was not very di�erent from the others, perhaps slightly less a�ractive. Based on the aggregated

results, the investor would be indi�erent between the three put spreads on a risk-adjusted basis

in this idealized environment. However, if the investor is bullish on the underlying stock and if

achieving high absolute returns is the objective, then she should refrain from employing a put

spread 100/90 strategy based on 1m puts.

Similar to the fractional protective put, the risk-adjusted return on the fractional put spread

95/85 displayed a modest improvement of 0.01 over the regular put spread 95/85 as maturity was

increased. As seen in Table 4.1, it yielded slightly be�er returns, but higher return volatilites on

average.

It is not odd that the collar 95/105 strategy generated relatively be�er alpha and lower beta

values compared to the other strategies. �e strategy is designed to cost li�le to nothing and

to limit its returns on both the downside and the upside. As seen in Table 4.2 and Figure 6.13,

both its mean beta values and mean total cost are signi�cantly lower than those for the other

strategies. Its outperformance on a risk-adjusted basis is visualized in Figure 4.2. �e collar 95/105

displays superior mean Sharpe ratio across all four portfolios, with signi�cant improvement as

69

the maturity increases, despite its comparably lower beta. It is not because the collar yield higher

returns, but because its return volatility, on average, is much lower than the other strategies. Put

di�erently, the collar earns its returns with lower risk compared to the other strategies.

Overall, the di�erence in risk-adjusted performance between the di�erent protection strategies

is not huge.

Negative long-term excess return

When the long-term excess return is negative, Figure 4.3, the collar 95/105 is by far the best per-

forming strategy due to its more conservative pro�le and signi�cantly lower beta. �e protective

put does a relatively good job and improves signi�cantly as maturity increases. Evidently, the

protective put demonstrates a more a�ractive risk-adjusted performance when the long-term

excess return is negative. Likewise, the fractional protective put exhibits a strong risk-adjusted

pro�le when the excess return is negative - not very di�erent from the regular protective put.

All three bene�t from their full downside protection.

As would be expected, the put spread strategies fare worse due to their limited protection capacity

and higher equity exposure. �e put spread 95/80 stands out with its be�er distribution, as

displayed in Figure 4.3, given its wider spread and thus more extensive protection. �e put

spread 100/90 exhibits a be�er mean adjusted Sharpe ratio than the put spread 95/85 in the 1m

portfolio, despite its signi�cantly higher total cost. Hence, we see a marginal gain in protecting

from the ATM level for short-dated options. �e risk-adjusted performance of the put spread

95/85, put spread 100/90 and fractional put spread 95/85 converges with increasing maturity and

is more or less the same in the 12m portfolio. �e small di�erences in the size of their spreads

do not ma�er over longer horizons, as the market is more likely to fall far beyond them.

Not surprisingly, when the long-term excess return is negative, the most important factor appears

to be the extent of the protection and not the cost of the options. �is is, in particular, con�rmed

by the superior risk-adjusted performance of the protective put and the fractional protective put.

However, combining full downside protection with lower cost, as in a collar, is even be�er.

Based on the aggregated results in Table 4.1 and Figures 4.2 - 4.3, the collar 95/105 strategy

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displayed the most a�ractive risk-reward pro�le. As it caps the upside performance, it is mainly

suitable when a rally is not expected or when a downturn is more likely. �e overweight on

negative returns in the simulated stocks paths therefore favored the collar strategy.

5.1.2 Peak-to-trough drawdown characteristics

Tables 6.1 - 6.4 in Appendix show that the mean peak-to-trough drawdowns for the di�erent

protection strategies at the 99th, 95th and 50th percentiles are all lower than those for the sim-

ulated stock paths. However, the prevailing question is how e�ective the provided protection is

and how it compares to the divested equity strategy.

Tail protection

A rolling protection strategy that buys 1m options is expected to protect the portfolio, at least,

against peak-to-trough drawdowns that persist over a 20-day window. Similarly, buying 3m, 6m

and 12m options are expected to provide e�ective protection, at least, over the 63-day, 125-day

and 250-day windows, respectively. Figures 4.4 and 4.6 show that this, indeed, is the case for the

protective put, the fractional protective put and the collar 95/105 strategy. Both the protective

put and the collar 95/105 outperforms the divested equity strategy at the mean 99th percentile

over windows that align with the maturity of the purchased options. For example, over the 20-

day window, the protective put’s and collar’s mean 99th percentile peak-to-trough drawdowns in

the 1m portfolio are -10.4% and -8.1%, respectively. �ese are 5.0 and 2.0 percentage points be�er

than divesting, respectively. �e fractional protective put exhibits similar strong tail protection

characteristics in the 1m and 3m portfolios, but becomes less e�ective against tail risk in the 6m

and 12m portfolios, both in terms of absolute value and versus divesting. As discussed earlier,

the fractional protective put is more vulnerable to declines in the market for longer maturities.

It must be emphasized, though, that the fractional protective put has slightly be�er tail protection

than the regular protective put in the 1m and 3m portfolios (see Figures 4.4 and 4.6). It gains full

protection faster in these portfolios and therefore sees the bene�t of reduced path dependency.

A fair comparison between them would require that we cut o� the period where it is partially

71

protected. However, the results indicate that there indeed is an improvement over the regular

protective put, given that the fractional protective put has gained full protection shortly a�er the

initial investment.

Figure 4.6 shows that the mean 99th peak-to-trough drawdowns for the collar 95/105 consis-

tently improves with increasing maturity, while the protective put peaks over the windows that

align with the maturity length. �e quality of the protective put’s protection improves when the

option maturity is most closely aligned with the length of the peak-to-trough drawdown cycle.

Overall, as displayed in Figure 4.4, the outperformance in terms of percentage points of the collar

95/105 and the protective put against the divested equity strategy grows with increasing matu-

rity, suggesting that these strategies are more suitable for tail protection than divesting when

long-dated options are used.

�e put spreads, overall, provide very poor tail protection. As seen in Figure 4.6, their mean

99th peak-to-trough drawdowns are generally the worst across all windows. Due to their par-

tial protection the drawdowns become worse with increasing maturity. Figure 4.4 further shows

that they consistently underperformed against the divested equity strategy. According to these

results, the put spreads are unarguably not suitable as tail protection strategies. It should, how-

ever, be noted that the put spread 100/90 had be�er mean 99th percentile drawdowns in the 1m

and 3m portfolios compared to the other put spreads. �e put spread 95/80 displays a superior

performance for longer maturities. Evidently, protecting from the ATM level with a 10% spread

appeared to be be�er for short-dated options, while wider spreads are more suitable for longer

maturities. �e fractional put spread 95/85 did not see an improvement over the other put spread

strategies. In fact, it performed worse than those on average.

An observation that is true for all strategies is that divesting provides a be�er protection against

5- to 10-day mean 99th percentile drawdowns.

�e discussed �ndings are also visualized in Figures 6.1 - 6.4 in Appendix. As evidenced by the

�gures, the collar 95/105, the protective put and the fractional protective put are signi�cantly

more likely to yield be�er tail protection than the put spread strategies. �e put spreads consis-

tently show a large variation in its 99th percentile drawdown, fortifying the great uncertainty in

their protection capacity.

72

Modest market declines

Figure 4.5 reports that the protective put, overall, falls short at the mean 50th percentile versus

the divested equity strategy, indicating that its protection is most o�en e�ective against tail risk

drawdowns and not during modest market dips, in agreement with the conclusions drawn by

Israelov (2017). As evidenced in Figure 4.7, the collar’s mean 50th percentile is the lowest across

all windows and portfolios. In particular, in the 12m portfolio, it consistently outperformed the

divested equity strategy across all three percentiles. �is is explained by its far lower mean beta

value of 0.14 (compare to the 0.25 for the divested equity strategy). Likewise, it exhibits a lower

mean beta value in the 3m and 6m portfolios, which makes it very conservative and less exposed

to market downturns.

�e put spread strategies’ mean 50th percentile drawdowns fared be�er against the divested

equity strategy, compared to the protective put and the fractional protective put. �e partial

protection is a good idea when the market does not fall beyond the extent of the spread, which

would correspond to a modest volatility environment in the real-world equity markets. Still, the

mean 50th peak-to-trough drawdowns for the put spreads, as displayed in Figure 4.7, are fairly

una�ractive.

Unlike the other strategies, the put spreads show no dependency on the choice of maturity. �e

mean 50th percentile for the protective put, the fractional protective put and the collar improves

with increasing maturity, both in terms of absolute value and versus divesting, i.e. their protec-

tion against modest dips in the market improves.

5.2 Backtesting

5.2.1 Risk-adjusted performance

Table 4.3 and Figure 4.8 display that all protection strategies underperformed against the divested

equity strategy in the 1m portfolio in terms of mean Sharpe ratio. All but the collar 95/105

consistently improved for longer maturities, but not su�ciently to outperform divesting. �e

73

poor performance across all strategies can be mainly a�ributed to their much higher volatility -

because of their signi�cantly higher beta values - compared to the divested equity strategy. �us,

an investor that seeks monthly protection would have achieved a be�er risk-adjusted return by

statically reducing the portfolio’s exposure to the S&P 500 index.

Among the 1m portfolios, the collar 95/105 had the best Sharpe ratio at 0.35 and just 0.03 below

the divested equity strategy. Its superior risk-adjusted return to the other protection strategies

can also be seen through its less volatile value development over time in Figure 6.5 in Appendix.

As evidenced by Table 4.4, its beta value is lower than the divested equity strategy and all the

other protection strategies. Moreover, it has the lowest total cost, resulting from the short 5%

OTM call which by theory should be closely priced with the long 5% OTM put due to the parabola-

like volatility skew near the ATM level. We see, though, that, in practice, the puts are more

expensively priced, leading to a total net cost of 106.8% of the initial investment. �e collar

95/105 was 41% cheaper than the protective put, but as much as 95% cheaper, on average, in

the Monte Carlo simulation in which both the call and the put were priced based on the same

volatility. �is underscores the signi�cance of variations in volatility for option-based protection

strategies. Furthermore, despite its lower equity exposure, the collar 95/105 realized a higher

volatility than divesting. Consequently, its risk-adjusted return was only inferior to the divested

equity strategy.

�e collar 95/105 was most closely followed by the fractional and the regular put spread 95/85,

which realized a Sharpe ratio of 0.29 and 0.28, respectively. �eir underperformance against the

divested equity strategy was more severe, though, given their negative alphas and signi�cantly

higher equity exposure - their beta values are ca 60-70% higher than those for the divested port-

folio - which e�ectively made them more risky. �e put spread 95/80 was not far behind with a

Sharpe ratio of 0.26, but with comparably lower volatility than both the fractional and regular

put spread 95/85. It was, however, ca 13.0% more expensive than the 95/85 spreads, due to its

wider spread. Evidently, the drag on returns because of higher cost had a bigger impact on the

risk-adjusted return than its comparably lower volatility.

It is striking how much of an impact the total cost of the strategies have on the risk-adjusted

return. It is particularly evident in the 1m portfolios. �e put spread 100/90, the protective put

and the fractional protective put were the most expensive strategies among the 1m portfolios,

74

where, in particular, buying ATM puts led to a total cost of 442.0% over the initial investment. �is

is ca 143.0% more than the cost for the protective put, which is the second most expensive strategy

among the 1m portfolios. Obviously, selling 10% OTM puts as a measure to reduce the cost did

not make much of a di�erence. �e most expensive strategies were also those who yielded the

worst Sharpe ratios and with most severe underperformance against divesting. In Figure 6.5 in

Appendix, the drop in the portfolio value due to higher cost is clearly visible. We found in the

Monte Carlo simulation that the total cost is less important than the extent of protection when

the long-term return is negative. We �nd now that this is not the case when the long-term return

is positive - the S&P 500 have a positive annualized return in the examined period.

Figure 4.8 shows that all strategies, except for the collar 95/105, consistently improved their risk-

adjusted return with increasing maturity. �e highest Sharpe ratios were achieved by the collar

95/105, which outperformed the divested equity strategy in both the 3m and 6m portfolios at

Sharpe ratios of 0.52 and 0.45, respectively. Its risk-adjusted return dramatically worsened in the

12m portfolio, sinking down to only 0.10 and making it the worst performing strategy for 12m

options. �e results are sensible given that the S&P 500 index enjoyed a huge upswing over the

long-term, which meant that the sold long-dated calls in the collar 95/105 were more likely to

expire ITM, e�ectively reducing the return on the portfolio. �is cap on the upside reduced beta

to 0.22 in 12m portfolio from the 0.56 in the 1m portfolio. Conversely, the higher equity exposure

in the 1m, 3m and 6m portfolios allowed the collar 95/105 to participate more in the upturn. See

Figures 6.6 - 6.8 in Appendix.

�e protective put and the fractional protective put were the only ones that outperformed the

divested equity strategy in the 12m portfolio. �eir beta values consistently dropped with in-

creasing maturity, making them less exposed to the volatility in the S&P 500 index, while their

alpha values improved and turned positive. Overall, the fractional protective put showed a slight

improvement in risk-adjusted performance over the regular protective put. Hence, while there

are some merits of diversifying the puts over overlapping cycles, they are not very convincing on

a risk-adjusted basis. Figures 6.6 - 6.8 in Appendix show that the value of the fractional protective

put portfolio is consistently, slightly above the protective put.

�e Sharpe ratios for the put spreads also improved with increasing maturity, but none of the

strategies outperformed the divested equity strategy in any of the portfolios. �e investor would

75

have achieved a be�er risk-adjusted return by divesting rather than sacri�cing some protection

for lower premiums. On the other hand, the risk-adjusted returns for the put spread 95/80 and put

spread 95/85 are higher than those for the protective put, which can be a�ributed to their higher

beta values and the long-lasting upswing in the market a�er the �nancial crisis. Figures 6.5 - 6.8

in Appendix show that the higher equity exposure allows them to grow faster with a rallying

market despite their greater loss during the �nancial crisis. In particular, based on the Sharpe

ratios, the investor would choose the put spread 95/85 over the protective put in the 1m, 3m and

6m portfolios. �e put spread 100/90 only became competitive in the 6m and 12m portfolios, and

was particularly the best alternative among the put spreads in the 12m portfolio. Less frequent

purchases of ATM puts reduced the total cost from 442.0% in the 1m portfolio to 83.8% in the 12m

portfolio. Similarly, the negative alpha improved from -3.5% to 0.0%, implying that the expensive

volatility paid for the shorter-dated options diminished with increasing maturity. �is led to a

Sharpe ratio of 0.40, which is on par with the fractional put spread 95/85 and only exceeded by

the fractional protective put with a Sharpe ratio of 0.41.

�e fractional put spread 95/85, like the fractional protective put, only led to a marginal im-

provement over the regular put spread 95/85. �e Sharpe ratio improved at most by 0.01 across

all portfolios. �e e�ect on risk-adjusted return, as stated earlier, is modest.

By taking a look at the Black-Scholes-Merton formulas in 2.2.3, we conclude that, all else being

equal, the price of an option rises with increasing time to maturity. �e option writer demands a

higher cost for the additional risk that comes with longer time to maturity. Yet Figure 6.14 shows

that the total cost for all strategies decreased with increasing maturity. �is is because rolling

purchases of long-dated options reduce the time value cost, which is higher for shorter-dated

options since the time value decays faster near expiry. �us, we see that the choice of maturity

has a material impact on the returns through the total cost incurred.

5.2.2 Peak-to-trough drawdown characteristics

As seen in Figure 4.9, the probability that divesting has a be�er drawdown than the option-based

protection strategies is generally high across all portfolios, with some exceptions. �e collar

95/105 stands out in the 3m and 6m portfolios, where it is more likely to yield be�er drawdowns

76

than divesting. In the 12m portfolio, the put spread 100/90 is more likely to yield be�er drawdown

than divesting over the shorter windows (5, 10 and 20 days). Indeed, protecting from the ATM

level over shorter windows with reduced total cost should cover more of the losses.

Tail protection

Figure 4.10 shows that the 99th percentile peak-to-trough drawdown provided by the option-

based protection strategies versus divesting was overall poor, except for the collar 95/105, which

outperformed the divested equity strategy in the 1m, 3m and 6m portfolios, bene�ting from the

full downside protection and its comparably lower beta value. Table 6.5 in Appendix, in partic-

ular, reports that the collar 95/105 is be�er than divesting at the 99th percentile peak-to-trough

drawdown over the 20-day, 63-day and 125-day windows in the 1m portfolio and over all win-

dows in the 6m portfolio. Despite its underperformance against divesting in the 12m portfolio,

we see that the peak-to-trough drawdowns in Figure 4.12 look a�ractive in comparison to the

other strategies. Looking at these numbers blindly gives an uncomplete picture of its perfor-

mance. We recall from the risk-adjusted performance analysis that the collar 95/105 strategy’s

Sharpe ratio in the 12m portfolio was 0.10. Figure 6.8 in Appendix, indeed, shows that its draw-

downs are signi�cantly smaller than the other protection strategies, but so is its total return over

the whole investment horizon.

�e two expensive strategies, namely the protective put and the put spread 100/90, performed

signi�cantly worse versus divesting in the 1m portfolio and had more or less the worst peak-

to-trough drawdowns across all windows. For example, Table 6.5 reports that 99th percentile

peak-to-trough drawdown, in the 1m portfolio, for the protective put over the 20-day window

was -15.4% versus -6.4% achieved by divesting. �is result is in line with the conclusions drawn

by Israelov (2017), but di�erent from our Monte Carlo results. In the idealized environment, the

protective put was, on average, be�er than the divested equity strategy at the 99th percentile

peak-to-trough drawdowns over the 20-day and 63-day windows. Also, the fact that the put

spread 100/90 fared the worst against divesting in Figure 4.10 contradicts the Monte Carlo results.

�e backtesting was performed on a period in which the long-term return was positive, while

the Monte Carlo simulation was overweight on periods with negative long-term return. �is is

an important di�erence that has a material impact on the performance of both the protective put

77

and the put spread 100/90.

However, we note, in Figure 4.12, that the size of the 99th percentile drawdowns for the put

spread 100/90 in the 1m portfolio was not the worst. It outperformed the protective put over all

windows and was the second best performer over the shorter windows (5, 10 and 20 days). �is

contradicts the Monte Carlo results, where all put spreads consistently underperformed at the

mean 99th percentile. �is is evidently not the case when the implied volatility is non-constant

and the long-term return is positive.

For the longer maturities, the performance of both the protective put and the put spread 100/90

versus divesting improved signi�cantly. �e protective put, in particular, outperforms the di-

vested equity strategy over all windows in the 12m portfolio and is fairly competitive in the 6m

portfolio. For example, the 99th percentile peak-to-trough drawdown over the 20-Day window

decreased by 42% (or 6.5 percentage points) from -15.4% to -8.9% by switching from 1m to 3m puts.

Conversely, in absolute terms, the peak-to-trough drawdowns for the put spread 100/90 did not

improve much and remained one of the most una�ractive alternatives. �is is in agreement with

our earlier conclusion that the put spreads are not ideal for tail protection when longer-dated

options are used.

�e fractional protective put fared marginally be�er against divesting compared to the regular

protective put. Figures 6.5 - 6.8 shows that the return on the S&P 500 in the �rst two years is

positive. Since the fractional protective put is partially hedged in the �rst months, this would

have a positive e�ect on the performance of the strategy. However, Figure 4.10 shows that the

di�erence between the fractional protective put and the regular protective put in absolute terms

is small and is only noticeable in the 12m portfolio, where the former is be�er over the longer

windows.

�e other put spreads consistently underperformed against the divested equity strategy across

all windows and maturities. �ey did, however, outperform the protective put in absolute terms

over the longer windows in the 1m portfolio. Since protective put is a relatively more expen-

sive strategy for short-dated options, this demonstrates the detrimental e�ect of higher cost not

just on risk-adjusted performance, but also on the tail protection. As seen in Figure 4.12, the

improvement of the peak-to-trough drawdowns of the put spreads with increasing maturity in

78

absolute terms is modest. For example, between the 1m and 12m portfolios the put spread 95/85

saw an improvement of 10.2 percentage points over the 250-day window in the 99th percentile

peak-to-trough drawdown, while the protective put improved as much as 33.5 percentage points.

�e fractional put spread 95/85 only showed a marginal improvement in the peak-to-trough

drawdowns over the regular put spread 95/85 for the longer-dated options. Like the put spreads,

its tail protection is uninspiring.

Based on these results, having full downside protection when a tail risk event occurs is generally

be�er than being partially protected. Shorter-dated options, in particular 1m options, should be

avoided.

Modest market declines

Figure 4.11 displays that no option-based strategy, besides from the collar 95/105, outperforms

the divested equity strategy at 50th percentile drawdowns. �e collar 95/105 is a slightly be�er

alternative over the 5, 10, 20 and 63 days in the 3m and 6m portfolios. Overall, the investor should

divest its equity exposure instead of employing an option-based strategy for protection against

modest market declines.

In Figure 4.13, the absolute 50th percentile peak-to-trough drawdowns are universally best in the

collar 95/105 strategy. �e fractional protective put and the regular protective put are the worst,

in agreement with our �ndings in the Monte Carlo simulation. �e high cost of full downside

protection makes itself visible in the peak-to-trough drawdowns during modest declines. �e

put spreads do generally a be�er job in protecting against modest market declines than tail risk

events. �ey bene�t from their reduced cost pro�le, due to the partial protection. We see, in

particular, that, among the put spread strategies, the put spread 100/90 is the best option for

the longer maturities. Evidently, when the cost burden is not too high, it is be�er to protect

against modest declines from the ATM level. We recall that the total cost of the put spread

100/90 decreases signi�cantly with increasing maturity and becomes comparable to the other

strategies (see Figure 6.14).

79

5.2.3 Financial crisis

Figures 6.9 - 6.12 in Appendix shows that, over the examined period of the �nancial crisis, the

protection strategies outperformed the underlying S&P 500 index, but their risk-adjusted perfor-

mance versus the divested equity strategy, as documented in Table 4.5, shows mixed results. �e

protective put, the fractional protective put and the collar 95/105 improved on a risk-adjusted

basis with increasing maturity, where the la�er overall exhibits the most a�ractive Sharpe ra-

tios. �is is expected in a prolonged bear market, because the longer maturities puts the investor

outside the falling market for a longer period of time. �e Sharpe ratios of the put spreads, con-

versely, worsen with increasing maturity. �e market has more time to fall beyond the extent of

their spreads, leading to worse returns and increased volatility.

�e risk-adjusted performance of the protective put was superior to the divested equity strategy

across all four portfolios. �e fractional protective put was, likewise, superior in the 1m and 3m

portfolios, but comparable to divesting in the 6m and 12m portfolios. �e collar 95/105 was only

superior to divesting in the 1m portfolio, but on par with divesting for the longer-dated options.

�e put spreads were only competitive in the 1m portfolio. We see that, in general, option-based

strategies that buy 1m options made a be�er alternative to divesting during the �nancial crisis.

Furthermore, the put spread 100/90, despite being inferior to divesting, yielded a more a�ractive

risk-adjusted return in the 1m portfolio than the protective put and the fractional protective put.

In fact, it would be the best choice for 1m options. We will discuss why.

It should be noted that all but the put spread 100/90 among the 1m portfolios underperformed

against the S&P 500 until October 2008 (see Figure 6.9 in Appendix). Evidently, all protection

strategies paid o� when the bigger crash occurred. Before the big crash, protecting from the

5% OTM level had signi�cantly greater drag on returns than the rolling cost of ATM puts. �e

outperformance of the put spread 100/90 is further validated in Table 4.6 by its annualized alpha

of 4.9%, which is the only positive alpha among all 1m portfolios. �e collar 95/105 did not bene�t

from its comparably lower equity exposure - compare its beta value of 0.26 versus 0.53 for the

put spread 100/90. �e collar 95/105 realized an annualized alpha of -19.4%. Likewise, the alphas

for the protective put and the fractional protective put amounted to -21.5% and -21.7%. Although,

they might have been a be�er choice than divesting, their weak performance in absolute terms

80

in the 1m portfolio is striking.

Table 4.2 reports the maximum drawdowns for the di�erent strategies. Since the divested equity

strategy was sized to achieve the same geometric return as the other protection strategies, the

di�erence between their maximum drawdowns is near zero. �is happens because the largest

drawdown aligns with the whole examined period. �e maximum drawdown, however, sheds

light on the relative performance of the di�erent option-based protection strategies. In agreement

with the previous paragraph, we see that the put spread 100/90 yielded the lowest maximum

drawdown among the 1m portfolios. �e di�erence is striking, compare its maximum drawdown

of -32.5% to -39.9% for the collar 95/105 and -55.6% for the protective put. �e other put spreads

showed maximum drawdowns between -53.0 % and -60.0%. As documented in Table 4.5, the

median 20-day peak-to-trough drawdowns for the put spread 100/90 were also be�er than those

for the other strategies. �ese results underscores that, if the investor considers 1m options

for protection, what is most important during a prolonged bear market is not a lower total net

cost nor a full downside protection, but the chosen strike level. Indeed, the put spread 100/90

only protects up to a 10% dip, but prices seldom fall more than 10% over the short-term. A 10%

protection is therefore su�cient as long as a sudden crash does not occur, in which case a full

protection is certainly the more important factor. We see in Figure 6.9 that the protective put,

the fractional protective put and the collar 95/105 managed the crash in October 2008 with less

volatility, as con�rmed in Table 4.5.

It should be noted that almost all option-based protection strategies in the 1m portfolio, except for

the collar 95/105, exhibit worse median 20-day peak-to-trough drawdown compared to divesting.

Figure 4.14 show that, in the 1m portfolio, divesting is more likely to yield a be�er drawdown

over the shorter windows. However, over longer windows, all strategies but the put spread

100/90 have a signi�cantly higher probability to provide be�er protection. In particular, over the

250-day window, the protective put, the fractional protective put and the other put spreads are

expected to yield a be�er protection 100% of the time.

With increasing maturity, the protective put, the fractional protective put and the collar 95/105

are by far the best performers, see Figures 6.10 - 6.12. �eir equity exposure decreased signi�-

cantly as the maturity was increased and led to improved maximum drawdowns. �e fractional

protective put, however, saw its beta value increase between the 6m and 12m portfolios and pro-

81

vided comparably less e�ective protection in terms of its drawdown characteristics, due to its

partial protection in the �rst months. It should be noted that it slightly outperformed the regular

protective put in the 1m portfolio in terms of both risk-adjusted return and peak-to-trough draw-

downs. Hence, to fully bene�t from pursuing the fractional protective put instead of the regular

protective put, the portfolio must gain full protection before the bear market commences.

Conversely, the maximum and median 20-day peak-to-trough drawdowns for the put spreads

worsened with increasing maturity. As it is more likely that prices will fall more than 10-20%

in a prolonged bear market, as evidenced by the �nancial crisis, the protective armor of the put

spreads was more easily penetrated. �e shortcomings of the put spreads and the fractional

protective put are combined in the fractional put spread 95/85. �e partial protection, and thus

more exposure to the market, makes this the worst performing strategy during the �nancial

crisis.

Figure 4.14, interestingly, shows that all option-based protection strategies exhibit best draw-

down characteristics in the 12m portfolio. �ey are collectively - with no exceptions - more

likely to provide be�er protection than divesting over the longer drawdown windows. �is im-

plies nothing about the tail protection and should not be confused with the results discussed in

the preceding paragraphs.

Overall, for the longer-dated options, a full downside protection is more important than the

strike level. Indeed, the collar 95/105 is perfectly suitable for a prolonged bear market, which, in

addition to its full downside protection, allows it to collect the premiums from the sold calls that

expire worthless.

5.3 Conclusions

�e use of derivatives for portfolio insurance is just one approach of a myriad of protection

strategies. Likewise, option-based protection strategies can be designed in many di�erent ways

to provide the desired volatility exposure. However, almost all of them require some degree

of sacri�ce in terms of returns, either by reduction in expected upside capture or an explicit

cost for protection. What makes a suitable protection strategy, whether it is to statically reduce

82

equity exposure or to use an overlay of options, was shown to depend on the degree of realized

volatility in the underlying asset, the desired time frame for provision of downside protection

and ultimately the type of drawdown that the investor is more concerned with. For example, the

e�ectiveness of the protection strategies were shown to be di�erent between a sudden crash and

modest drawdowns. Moreover, the choice of maturity was shown to have a material impact on

both the risk-adjusted return and the drawdown characteristics.

�e choice of maturity for a rolling protection depends on the time horizon. As the time horizon

increases, the aggregate cost of protection increases and at the same time the need for a protection

decreases, since equity returns are expected to be positive in the long-term. If protection is only

needed for the next month to hedge against an expected drawdown, then buying a 1m put option

would do the job. However, if the investor seeks protection against volatility over the next few

years, then perhaps longer maturities should be preferred. We report that rolling protection

strategies become cheaper for longer maturities. �e investor should, in general, refrain from

buying 1m options across all strategies.

Based on our discussion around the results, an uninformed investor would generally be be�er o�

divesting its equity position than pursuing an option-based protection strategy. However, if the

uninformed investor seeks the comfort in options’ convex payo� pro�le, which automatically

reduces the equity exposure as markets decline, she should employ a collar 95/105 strategy for

options with maturities 3m or 6m. �e longer the maturity, the higher is the potential for bet-

ter risk-adjusted return and drawdown characteristics. A problem with the collar 95/105 arises

when the expected long-term return on the equity market is positive. In such case, for too long

maturities, e.g. 12m, the sold call options are more likely to expire ITM and thus reduce returns.

It should therefore be preferred when a rally in the equity market is unlikely. �e low cost and

low beta of the collar 95/105 makes it a suitable protection strategy for investors who are more

concerned with the downside risk than the upside potential. In contrast to divesting and the

other protection strategies, it displayed, on average, superior protection against both tail risk

events and modest drawdowns.

�e put spread strategies are generally not ideal for portfolio insurance. �eir strongest perfor-

mance was documented when the long-term equity return was positive, both in the backtesting

and the Monte Carlo simulation, in which case the need for protection decreases. It is therefore

83

harder to justify the application of a put spread as a protection strategy. �e strategies were

shown to provide be�er protection against modest drawdowns than the protective put, but ex-

hibited very poor tail protection and were o�en a worse alternative than divesting across all ma-

turities. �e only exception is a put spread 100/90 strategy that buys and sells 1m options, which

showed an e�ective protection against protracted drawdowns and a promising risk-adjusted re-

turn during the �nancial crisis. We �nd, in particular, that during protracted drawdowns, e.g. a

crisis, the importance of higher strike level outweighs the cost of the strategy. Buying 1m ATM

puts is, conversely, far from promising when returns are positive due to their high cost, which

makes the put spread 100/90 not feasible as a long-term rolling protection strategy. We conclude

that there is very li�le to gain from the put spread strategies as far as tail protection is concerned.

Usually, an investor who seeks uncapped upside potential and full downside protection - the ideal

portfolio insurance structure - would refer to the popular protective put strategy. Our �ndings

show that the risk-adjusted performance and drawdown characteristics of the protective put

are only a�ractive during tail risk events. Compared to divesting and the other option-based

protection strategies, the protective put is less a�ractive during modest drawdowns and never

when excess return is positive, due to the high cost burden from expensive premiums. Its risk-

adjusted performance and e�ectiveness against tail protection improves with longer maturities,

as demonstrated in the Monte Carlo simulation. As such, a protective put that buys 12m options

did fairly well during the �nancial crisis and over the whole backtesting period.

�e fractional strategies were introduced to mitigate path dependency. As they are only partially

protected in the �rst months, their performance was, in general, weaker for longer maturities

during periods of negative excess return. �is was demonstrated in both the Monte Carlo sim-

ulation, which was overweight on negative returns, and the backtesting on the �nancial crisis.

It takes more time for the strategies to become fully protected for longer-dated options and are

therefore more exposed to market risk. However, the fractional strategies showed a slight im-

provement over their regular counterparts during periods of positive excess return and, in par-

ticular, when no signi�cant drawdowns had occurred before they were fully protected. Overall,

the results were not too convincing.

Similar to the previous studies, we conclude that no option-based protection strategy can domi-

nate the other in all market situations. It is, though, encouraging to report that the divested equity

84

strategy is not completely unbeatable. Our results show that, although marginally, both be�er

risk-adjusted return and drawdown characteristics can be achieved by employing an option-

based strategy.

Lastly, as a proposal for future research, the fractional strategies introduced in this paper need

further examination. Also, it would be interesting to contrast the protective put to protection

strategies that employ linear derivatives, such as swaps, futures and forwards.

85

Appendix

6.1 Modelling examples

6.1.1 Protective Put

Below is a description of how the protective put portfolio is modelled based on the assumptions

and delimitations in Section 3.2 with some additional simpli�cations. �e risk-free rate and the

expected dividend yield are set to zero for simplicity.

Let Sindex(t) represent the price of the S&P 500 Index (including dividends) and p(t,Tk) the Black-

Scholes-Merton price of a put option at time t with strike price K and time of maturity Tk .

During the investment period, the portfolio will buy n put options at times t = τ0, τ1, ..., τn−1with corresponding maturities T1,T2, ...,Tn, where Tk − τk−1 = T (for k = 1, ...,n) is constant, i.e.

the time to maturity, T , is the same for all puts. Since a new put option is purchased every time

the current one expires, it follows that τ1 = T1, τ2 = T2, ..., τn−1 = Tn−1. �e �rst put option with

maturity T1 is bought at time t = τ0 = 0 and is rolled over into a new one with maturity T2, as

it matures at time t = τ1 = T1. Hence, at each maturity Tk the portfolio rolls over into a new put

option with maturity Tk+1, except at time t = Tn, which is the end of the investment horizon. �e

investment horizon for the portfolio is thus given by T ∗ =∑ni=1(Tk−τk−1) ==

∑ni=1(Tk−Tk−1) =

n · T .

Assume further that the total net asset value (NAV) of the portfolio is invested in the S&P 500

Index. For simplicity, the put options are �nanced with a �nancing rate of 0% and their cost is

incurred at the time of maturity - not at the purchase date. �us, for i = 2, ...,n, the total cost of

86

the put options incurred by time t is

C(t) =

0, 0 ≤ t < T1∑i−1j=1p(Tj−1,Tj ), Ti−1 ≤ t < Ti∑nj=1p(Tj−1,Tj ), t = Tn

and the accrued payo�s from the puts by time t is

Φ(t) =

0, 0 ≤ t < T1∑i−1j=1(K − Sindex(Tj ))+, Ti−1 ≤ t < Ti∑nj=1(K − Sindex(Tj ))+, t = Tn

�en, the value of a portfolio, Π(t), at time t, which implements a protective put strategy, follows

the price process

Π(t) =

Sindex(t) + p(t,T1), 0 ≤ t < T1

Sindex(t) + p(t,Ti) +∑i−1j=1(K − Sindex(Tj ))+ −

∑i−1j=1p(Tj−1,Tj ), Ti−1 ≤ t < Ti

Sindex(Tn) +∑nj=1(K − Sindex(Tj ))+ −

∑nj=1p(Tj−1,Tj ), t = Tn

for i = 2, ...,n.

If, for example, the investment horizon is 5 years, T ∗ = 5, and the time to maturity Ti −Ti−1 is 1

month, then the investor will buy in total 12× 5 = 60 put options during the entire period.

�e modelling of the other option strategies in Table 3.1 is similar conceptually, but some of them

also include a short position in an option, which would result in negative accrued payo�s and

positive costs (income).

6.1.2 Divested Equity

For the modelling of the divested equity portfolio, let wr and wc represent the proportion of the

NAV invested in the risky asset and cash, respectively. Moreover, it holds that wr +wc = 1 and

87

thus 0 ≤ wr ,wc ≤ 1. �e concept of the divested equity portfolio is based on keeping a certain

percentage of the NAV invested in the risky asset, e.g. 75% in the risky asset (wr = 0.75) and

the remaining 25% in cash (wc = 1−0.75 = 0.25). �is e�ectively requires frequent rebalancing

of the portfolio at predetermined intervals to recapture the original exposure to the risky asset.

For simplicity, let the NAV be equally sized to one unit of the risky asset at time t = 0, i.e. S0.

If Vt,r and Vt,c represent the value of the position in the risky asset, St , and the cash position at

time t, respectively, then the initial value of the portfolio is:

V0,r = wrS0

V0,c = S0 −wrS0 = (1−wr )S0 = wcS0

Π0 = V0,r +V0,c

where Π0 is the portfolio value at time t = 0. Between the rebalancing dates {T0 = 0,T1, ...,Tn},

the dynamics of this portfolio follows the process dΠt = dVt,r = wrdSt and dVt,c = 0, where

t ∈ (Ti−1,Ti) for i = 1, ...,n. At every rebalancing date, t = Ti > 0, the portfolio is rebalanced to

recapture its original exposure to the risky asset:

VTi ,r = wrΠTi

VTi ,c =ΠTi −VTi ,r

for i = 1, ...,n.

6.2 Tables

6.2.1 Monte Carlo Simulation

88

Tabl

e6.1:

1mpo

rtfol

ios:

Mea

nva

lueo

fthe

99th

,95t

han

d50t

hpe

rcen

tiles

ofth

epea

k-to

-trou

ghdr

awdo

wns

fort

hesim

ulat

edpr

otec

tion

stra

tegi

es,

excl

udin

gca

sesw

here

the

geom

etric

retu

rnon

the

dive

sted

equi

typo

rtfol

ioco

uld

notb

em

atch

ed.

5-D

ay10

-Day

20-D

ay63

-Day

125-

Day

250-

Day

99th

95th

50th

99th

95th

50th

99th

95th

50th

99th

95th

50th

99th

95th

50th

99th

95th

50th

Sim

ulat

edst

ocks

-7.4%

-4.8%

-1.2%

-12.7

%-7

.0%-2

.0%-2

3.8%

-10.2

%-3

.0%-3

0.5%

-25.1

%-5

.9%-3

4.7%

-30.7

%-8

.8%-3

9.3%

-35.9

%-1

3.8%

Prot

ectiv

ePu

t-5

.4%-3

.9%-1

.0%-7

.6%-5

.6%-1

.7%-1

0.4%

-7.9%

-2.8%

-17.3

%-1

4.3%

-5.6%

-22.8

%-1

9.7%

-8.2%

-28.3

%-2

5.6%

-12.0

%D

ives

ted

Equi

ty-3

.8%-2

.5%-0

.6%-6

.8%-3

.6%-1

.0%-1

5.4%

-5.4%

-1.6%

-18.8

%-1

5.1%

-3.1%

-21.0

%-1

8.4%

-4.7%

-23.8

%-2

1.5%

-7.5%

Di�

-1.7%

-1.4%

-0.4%

-0.8%

-1.9%

-0.7%

+5.0%

-2.5%

-1.3%

+1.5%

+0.7%

-2.5%

-1.8%

-1.3%

-3.5%

-4.6%

-4.1%

-4.5%

PutS

prea

d95

/80

-6.0%

-3.9%

-1.0%

-10.4

%-5

.6%-1

.8%-1

8.5%

-8.1%

-2.9%

-24.5

%-2

0.0%

-5.6%

-29.0

%-2

5.5%

-8.2%

-33.9

%-3

0.8%

-12.8

%D

ives

ted

Equi

ty-5

.2%-3

.2%-0

.8%-1

0.1%

-4.6%

-1.3%

-18.7

%-6

.9%-2

.0%-2

3.1%

-19.0

%-3

.9%-2

6.0%

-23.0

%-5

.9%-2

9.5%

-26.9

%-9

.6%Di�

-0.8%

-0.8%

-0.3%

-0.3%

-1.0%

-0.5%

+0.2%

-1.2%

-0.9%

-1.4%

-1.0%

-1.7%

-3.0%

-2.5%

-2.3%

-4.4%

-3.9%

-3.1%

PutS

prea

d95

/85

-6.1%

-4.0%

-1.0%

-10.7

%-5

.6%-1

.8%-1

9.0%

-8.1%

-2.9%

-24.9

%-2

0.7%

-5.6%

-29.4

%-2

5.9%

-8.2%

-34.2

%-3

1.1%

-12.7

%D

ives

ted

Equi

ty-5

.3%-3

.2%-0

.8%-9

.9%-4

.6%-1

.3%-1

7.4%

-6.9%

-1.9%

-21.8

%-1

8.0%

-3.9%

-24.8

%-2

1.8%

-5.9%

-28.3

%-2

5.7%

-9.6%

Di�

-0.9%

-0.8%

-0.3%

-0.8%

-1.0%

-0.5%

-1.6%

-1.2%

-0.9%

-3.1%

-2.7%

-1.8%

-4.6%

-4.1%

-2.3%

-6.0%

-5.4%

-3.0%

PutS

prea

d10

0/90

-4.8%

-2.8%

-0.7%

-8.3%

-4.0%

-1.3%

-16.2

%-5

.8%-2

.2%-2

0.3%

-17.2

%-4

.5%-2

3.8%

-21.3

%-6

.4%-2

8.0%

-25.7

%-9

.9%D

ives

ted

Equi

ty-4

.5%-2

.8%-0

.7%-8

.2%-4

.0%-1

.1%-1

4.7%

-6.1%

-1.8%

-18.8

%-1

5.5%

-3.6%

-21.7

%-1

9.1%

-5.5%

-25.3

%-2

2.9%

-9.0%

Di�

-0.3%

-0.1%

0.0%

-0.1%

0.0%

-0.2%

-1.5%

+0.3%

-0.5%

-1.5%

-1.7%

-0.9%

-2.1%

-2.2%

-0.9%

-2.8%

-2.8%

-0.9%

Colla

r95/

105

-4.4%

-3.1%

-0.6%

-6.0%

-4.4%

-1.0%

-8.1%

-6.3%

-1.7%

-13.9

%-1

1.2%

-3.7%

-18.0

%-1

5.3%

-5.5%

-21.9

%-1

9.6%

-8.3%

Div

este

dEq

uity

-3.2%

-2.3%

-0.6%

-5.1%

-3.3%

-0.9%

-10.1

%-4

.8%-1

.4%-1

3.8%

-10.9

%-2

.8%-1

6.2%

-13.9

%-4

.2%-1

8.9%

-16.9

%-6

.4%Di�

-1.1%

-0.8%

0.0%

-0.9%

-1.1%

-0.1%

+2.0%

-1.5%

-0.3%

-0.1%

-0.3%

-0.9%

-1.8%

-1.4%

-1.4%

-2.9%

-2.6%

-1.9%

Frac

tiona

lPro

tect

ive

Put

-5.3%

-3.9%

-1.0%

-7.3%

-5.5%

-1.8%

-9.8%

-7.8%

-2.9%

-16.8

%-1

4.0%

-5.7%

-22.5

%-1

9.4%

-8.3%

-28.1

%-2

5.4%

-11.9

%D

ives

ted

Equi

ty-3

.7%-2

.5%-0

.6%-6

.7%-3

.6%-1

.0%-1

5.2%

-5.3%

-1.5%

-18.7

%-1

4.9%

-3.1%

-20.9

%-1

8.3%

-4.7%

-23.6

%-2

1.3%

-7.5%

Di�

-1.6%

-1.4%

-0.4%

-0.7%

-1.9%

-0.8%

+5.4%

-2.5%

-1.3%

+1.8%

+0.9%

-2.6%

-1.6%

-1.1%

-3.6%

-4.5%

-4.1%

-4.5%

Frac

tiona

lPut

Spre

ad95

/85

-0.06

-3.9%

-1.1%

-10.7

%-5

.6%-1

.8%-1

9.1%

-8.0%

-3.0%

-25.1

%-2

0.9%

-5.8%

-29.7

%-2

6.2%

-8.4%

-34.7

%-3

1.6%

-12.9

%D

ives

ted

Equi

ty-5

.5%-3

.3%-0

.8%-1

0.5%

-4.9%

-1.3%

-18.5

%-7

.3%-2

.1%-2

3.2%

-19.0

%-4

.1%-2

6.3%

-23.1

%-6

.3%-2

9.9%

-27.3

%-1

0.3%

Di�

-0.5%

-0.6%

-0.3%

-0.2%

-0.7%

-0.5%

-0.7%

-0.7%

-0.9%

-2.0%

-1.8%

-1.7%

-3.4%

-3.1%

-2.1%

-4.8%

-4.4%

-2.6%

89

Tabl

e6.2:

3mpo

rtfol

ios:

Mea

nva

lueo

fthe

99th

,95t

han

d50t

hpe

rcen

tiles

ofth

epea

k-to

-trou

ghdr

awdo

wns

fort

hesim

ulat

edpr

otec

tion

stra

tegi

es,

excl

udin

gca

sesw

here

the

geom

etric

retu

rnon

the

dive

sted

equi

typo

rtfol

ioco

uld

notb

em

atch

ed.

5-D

ay10

-Day

20-D

ay63

-Day

125-

Day

250-

Day

99th

95th

50th

99th

95th

50th

99th

95th

50th

99th

95th

50th

99th

95th

50th

99th

95th

50th

Sim

ulat

edst

ocks

-7.4%

-4.8%

-1.2%

-12.7

%-7

.0%-2

.0%-2

3.8%

-10.2

%-3

.0%-3

0.5%

-25.1

%-5

.9%-3

4.7%

-30.7

%-8

.8%-3

9.3%

-35.9

%-1

3.8%

Prot

ectiv

ePu

t-5

.1%-3

.5%-0

.8%-7

.4%-5

.0%-1

.4%-1

0.6%

-7.2%

-2.3%

-15.5

%-1

2.7%

-5.1%

-19.9

%-1

7.5%

-7.7%

-25.5

%-2

3.3%

-11.3

%D

ives

ted

Equi

ty-3

.7%-2

.4%-0

.6%-6

.7%-3

.5%-1

.0%-1

4.3%

-5.2%

-1.5%

-17.8

%-1

4.4%

-3.0%

-20.2

%-1

7.7%

-4.6%

-23.0

%-2

0.9%

-7.4%

Di�

-1.5%

-1.1%

-0.2%

-0.8%

-1.5%

-0.4%

+3.7%

-2.0%

-0.8%

+2.3%

+1.7%

-2.1%

+0.3%

+0.2%

-3.1%

-2.5%

-2.4%

-3.9%

PutS

prea

d95

/80

-5.8%

-3.6%

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8.0%

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8.7%

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3.7%

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8.8%

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ives

ted

Equi

ty-5

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7.4%

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7.9%

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2.0%

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6.1%

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Di�

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PutS

prea

d95

/85

-6.0%

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8.9%

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9.8%

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4.7%

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ives

ted

Equi

ty-5

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PutS

prea

d10

0/90

-5.3%

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2.6%

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5.8%

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9.6%

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Div

este

dEq

uity

-4.9%

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0.1%

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3.1%

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6.8%

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Colla

r95/

105

-3.2%

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0.5%

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3.7%

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Div

este

dEq

uity

-2.5%

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2.3%

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4.4%

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0.0%

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Frac

tiona

lPro

tect

ive

Put

-5.2%

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5.2%

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9.7%

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5.5%

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1.7%

Div

este

dEq

uity

-3.9%

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8.2%

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0.7%

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Frac

tiona

lPut

Spre

ad95

/85

-5.9%

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9.4%

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0.1%

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5.0%

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0.1%

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%D

ives

ted

Equi

ty-5

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0.0%

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3.0%

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6.3%

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0.2%

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0.4%

Di�

-0.5%

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-1.5%

-1.3%

-1.3%

-1.9%

-1.8%

-1.8%

-2.7%

-2.7%

-1.9%

90

Tabl

e6.3:

6mpo

rtfol

ios:

Mea

nva

lueo

fthe

99th

,95t

han

d50t

hpe

rcen

tiles

ofth

epea

k-to

-trou

ghdr

awdo

wns

fort

hesim

ulat

edpr

otec

tion

stra

tegi

es,

excl

udin

gca

sesw

here

the

geom

etric

retu

rnon

the

dive

sted

equi

typo

rtfol

ioco

uld

notb

em

atch

ed.

5-D

ay10

-Day

20-D

ay63

-Day

125-

Day

250-

Day

99th

95th

50th

99th

95th

50th

99th

95th

50th

99th

95th

50th

99th

95th

50th

99th

95th

50th

Sim

ulat

edst

ocks

-7.4%

-4.8%

-1.2%

-12.7

%-7

.0%-2

.0%-2

3.8%

-10.2

%-3

.0%-3

0.5%

-25.1

%-5

.9%-3

4.7%

-30.7

%-8

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9.3%

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%-1

3.8%

Prot

ectiv

ePu

t-5

.0%-3

.3%-0

.6%-7

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.8%-1

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1.0%

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-15.6

%-1

2.7%

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-19.0

%-1

6.8%

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-23.9

%-2

2.0%

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%D

ives

ted

Equi

ty-3

.9%-2

.5%-0

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.0%-3

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4.9%

-5.4%

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-18.6

%-1

4.9%

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-21.1

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8.4%

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-24.2

%-2

1.9%

-7.7%

Di�

-1.1%

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0.0%

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+3.0%

+2.2%

-1.3%

+2.1%

+1.6%

-2.3%

+0.3%

-0.1%

-3.1%

PutS

prea

d95

/80

-5.8%

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%-5

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8.1%

-7.7%

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-23.1

%-1

8.7%

-4.7%

-26.6

%-2

3.5%

-7.5%

-30.9

%-2

8.4%

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%D

ives

ted

Equi

ty-5

.2%-3

.2%-0

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7.7%

-6.9%

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-22.3

%-1

8.2%

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%-2

2.4%

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%-2

6.6%

-9.9%

Di�

-0.6%

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0.0%

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-0.8%

-0.5%

-0.8%

-1.3%

-1.2%

-1.5%

-1.8%

-1.9%

-1.8%

PutS

prea

d95

/85

-6.1%

-3.8%

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-10.4

%-5

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9.0%

-8.1%

-2.4%

-24.3

%-1

9.9%

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%-2

4.8%

-7.7%

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9.7%

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%D

ives

ted

Equi

ty-5

.5%-3

.4%-0

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7.7%

-7.2%

-2.1%

-22.6

%-1

8.5%

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%-2

2.8%

-6.4%

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%-2

7.2%

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%Di�

-0.6%

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-1.8%

-1.4%

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-2.5%

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PutS

prea

d10

0/90

-5.8%

-3.6%

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%-5

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8.4%

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%-1

9.2%

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3.8%

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8.3%

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%D

ives

ted

Equi

ty-5

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7.1%

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%-1

7.8%

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%-2

1.9%

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%-2

6.0%

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%Di�

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0.0%

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Colla

r95/

105

-2.4%

-1.4%

-0.2%

-3.6%

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-3.2%

-0.5%

-7.5%

-6.1%

-1.3%

-9.0%

-7.9%

-2.3%

-11.2

%-1

0.1%

-3.8%

Div

este

dEq

uity

-2.0%

-1.4%

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-2.1%

-0.6%

-6.2%

-3.0%

-0.9%

-8.5%

-6.7%

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-10.0

%-8

.5%-2

.6%-1

1.7%

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%-3

.9%Di�

-0.4%

0.0%

+0.2%

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0.0%

+0.3%

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+0.4%

+1.0%

+0.6%

+0.5%

+1.0%

+0.7%

+0.3%

+0.5%

+0.2%

0.1%

Frac

tiona

lPro

tect

ive

Put

-5.1%

-3.5%

-0.7%

-7.5%

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-1.3%

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%-7

.3%-2

.1%-1

6.3%

-13.3

%-4

.7%-1

9.4%

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%-7

.6%-2

4.4%

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%-1

1.6%

Div

este

dEq

uity

-3.9%

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-6.8%

-3.7%

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-14.1

%-5

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.6%-1

7.8%

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%-3

.1%-2

0.4%

-17.7

%-4

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3.5%

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.6%Di�

-1.2%

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+1.6%

+1.0%

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+1.0%

+0.6%

-2.8%

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-1.1%

-4.0%

Frac

tiona

lPut

Spre

ad95

/85

-6.1%

-3.9%

-0.9%

-10.8

%-5

.6%-1

.6%-1

9.9%

-8.3%

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-25.3

%-2

0.7%

-5.1%

-28.9

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5.5%

-8.0%

-33.3

%-3

0.4%

-12.4

%D

ives

ted

Equi

ty-5

.6%-3

.5%-0

.9%-1

0.1%

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-18.4

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.5%-2

.2%-2

3.4%

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6.8%

-23.6

%-6

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0.9%

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%-1

0.7%

Di�

-0.5%

-0.4%

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-0.7%

-0.6%

-0.2%

-1.5%

-0.7%

-0.3%

-1.9%

-1.5%

-0.8%

-2.1%

-1.9%

-1.4%

-2.4%

-2.3%

-1.7%

91

Tabl

e6.4

:12

mpo

rtfol

ios:

Mea

nva

lue

ofth

e99

th,9

5th

and

50th

perc

entil

esof

the

peak

-to-tr

ough

draw

dow

nsfo

rth

esim

ulat

edpr

otec

tion

stra

tegi

es,e

xclu

ding

case

swhe

reth

ege

omet

ricre

turn

onth

edi

vest

edeq

uity

portf

olio

coul

dno

tbe

mat

ched

.5-

Day

10-D

ay20

-Day

63-D

ay12

5-D

ay25

0-D

ay99

th95

th50

th99

th95

th50

th99

th95

th50

th99

th95

th50

th99

th95

th50

th99

th95

th50

thSi

mul

ated

stoc

ks-7

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2.7%

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%-1

0.2%

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-30.5

%-2

5.1%

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%-3

0.7%

-8.8%

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%-3

5.9%

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%Pr

otec

tive

Put

-4.9%

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%-6

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5.8%

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%-3

.7%-1

8.7%

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%-6

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2.8%

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%-1

0.0%

Div

este

dEq

uity

-3.9%

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-7.3%

-3.6%

-1.0%

-15.1

%-5

.4%-1

.6%-1

8.7%

-15.0

%-3

.1%-2

1.2%

-18.5

%-4

.8%-2

4.3%

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-7.8%

Di�

-0.9%

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+0.1%

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0.0%

+2.9%

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+2.5%

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+1.6%

+1.0%

-2.2%

PutS

prea

d95

/80

-6.0%

-3.7%

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%-5

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8.5%

-7.9%

-2.2%

-23.7

%-1

9.1%

-4.6%

-27.2

%-2

3.8%

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-31.3

%-2

8.6%

-11.5

%D

ives

ted

Equi

ty-5

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0.0%

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-18.1

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.1%-2

.1%-2

2.8%

-18.6

%-4

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5.9%

-22.8

%-6

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9.7%

-0.27

-10.3

%Di�

-0.6%

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0.0%

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-0.8%

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-0.5%

-0.5%

-1.2%

-1.0%

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-1.6%

-1.6%

-1.2%

PutS

prea

d95

/85

-6.3%

-3.9%

-0.9%

-11.0

%-5

.7%-1

.5%-1

9.8%

-8.4%

-2.4%

-25.4

%-2

0.7%

-4.9%

-29.1

%-2

5.6%

-7.5%

-33.3

%-3

0.3%

-12.0

%D

ives

ted

Equi

ty-5

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.5%-0

.9%-1

0.4%

-5.1%

-1.4%

-19.0

%-7

.6%-2

.2%-2

4.0%

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%-4

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7.2%

-24.0

%-6

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1.0%

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%-1

0.8%

Di�

-0.6%

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-1.5%

-1.1%

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-2.1%

-1.2%

PutS

prea

d10

0/90

-6.2%

-3.8%

-0.9%

-10.5

%-5

.6%-1

.5%-1

9.0%

-8.2%

-2.3%

-24.6

%-2

0.0%

-4.7%

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%-2

4.7%

-7.2%

-32.3

%-2

9.3%

-11.5

%D

ives

ted

Equi

ty-5

.6%-3

.4%-0

.8%-9

.9%-5

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7.9%

-7.4%

-2.2%

-22.8

%-1

8.7%

-4.3%

-26.0

%-2

2.9%

-6.6%

-29.8

%-2

7.1%

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%Di�

-0.6%

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-0.2%

-1.7%

-1.3%

-0.4%

-2.2%

-1.8%

-0.6%

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Colla

r95/

105

-1.7%

-0.9%

-0.1%

-2.7%

-1.4%

-0.2%

-4.2%

-2.1%

-0.3%

-5.8%

-4.5%

-0.7%

-6.7%

-5.9%

-1.2%

-7.9%

-7.2%

-2.4%

Div

este

dEq

uity

-1.7%

-1.2%

-0.3%

-2.7%

-1.7%

-0.5%

-5.3%

-2.5%

-0.8%

-7.1%

-5.7%

-1.5%

-8.4%

-7.2%

-2.1%

-9.8%

-8.7%

-3.2%

Di�

0.0%

+0.3%

+0.2%

0.0%

+0.4%

+0.3%

+1.0%

+0.4%

+0.5%

+1.3%

+1.1%

+0.8%

+1.7%

+1.3%

+1.0%

+1.9%

+1.5%

+0.8%

Frac

tiona

lPro

tect

ive

Put

-5.2%

-3.4%

-0.7%

-7.8%

-5.0%

-1.2%

-13.3

%-7

.3%-1

.9%-1

8.2%

-14.7

%-4

.2%-2

1.3%

-18.9

%-6

.7%-2

4.9%

-23.0

%-1

0.9%

Div

este

dEq

uity

-4.0%

-2.5%

-0.6%

-7.0%

-3.7%

-1.0%

-14.1

%-5

.5%-1

.6%-1

7.9%

-14.4

%-3

.2%-2

0.4%

-17.8

%-4

.8%-2

3.6%

-21.3

%-7

.6%Di�

-1.2%

-0.9%

-0.1%

-0.8%

-1.3%

-0.1%

+0.9%

-1.9%

-0.3%

-0.3%

-0.4%

-1.0%

-0.9%

-1.1%

-1.9%

-1.4%

-1.7%

-3.3%

Frac

tiona

lPut

Spre

ad95

/85

-6.4%

-4.1%

-1.0%

-11.2

%-6

.0%-1

.7%-2

0.8%

-8.8%

-2.6%

-26.6

%-2

1.7%

-5.2%

-30.4

%-2

6.8%

-7.9%

-34.7

%-3

1.7%

-12.6

%D

ives

ted

Equi

ty-5

.9%-3

.7%-0

.9%-1

0.5%

-5.4%

-1.5%

-19.4

%-8

.0%-2

.3%-2

4.7%

-20.3

%-4

.6%-2

8.3%

-24.9

%-7

.0%-3

2.4%

-29.6

%-1

1.2%

Di�

-0.5%

-0.4%

-0.1%

-0.7%

-0.5%

-0.1%

-1.4%

-0.7%

-0.2%

-1.8%

-1.4%

-0.5%

-2.1%

-1.8%

-0.9%

-2.3%

-2.1%

-1.3%

92

6.2.2 Backtesting

93

Tabl

e6.5

:Sum

mar

yof

peak

-to-tr

ough

draw

dow

nre

turn

sfor

the

1mpo

rtfol

ios.

5-D

ay10

-Day

20-D

ay63

-Day

125-

Day

250-

Day

99th

95th

50th

99th

95th

50th

99th

95th

50th

99th

95th

50th

99th

95th

50th

99th

95th

50th

S&P

500

-7.8%

-4.3%

-0.4%

-11.4

%-5

.7%-0

.7%-1

6.8%

-8.1%

-1.0%

-33.8

%-1

3.9%

-1.5%

-48.6

%-1

6.5%

-1.9%

-58.1

%-3

9.6%

-2.2%

Prot

ectiv

ePu

t-6

.8%-4

.5%-0

.6%-1

0.6%

-6.1%

-1.0%

-15.4

%-9

.1%-1

.5%-3

2.7%

-18.7

%-2

.7%-4

6.1%

-23.6

%-3

.4%-5

7.4%

-43.1

%-4

.4%D

ives

ted

Equi

ty-3

.0%-1

.7%-0

.2%-4

.4%-2

.2%-0

.3%-6

.4%-3

.1%-0

.4%-1

1.9%

-5.2%

-0.6%

-17%

-6.0%

-0.6%

-20.2

%-1

2.7%

-0.8%

Di�

-3.7%

-2.9%

-0.5%

-6.1%

-3.9%

-0.8%

-9.0%

-6.0%

-1.1%

-20.7%

-13.5%

-2.1%

-29.1%

-17.6%

-2.7%

-37.2%

-30.5%

-3.6%

PutS

prea

d95

/80

-6.7%

-4.3%

-0.6%

-10.5

%-5

.8%-1

.0%-1

4.9%

-8.5%

-1.4%

-28.2

%-1

7.6%

-2.5%

-42.9

%-2

2.6%

-3.1%

-54.7

%-3

9.5%

-3.9%

Div

este

dEq

uity

-3.8%

-2.1%

-0.2%

-5.5%

-2.7%

-0.3%

-7.9%

-3.9%

-0.5%

-15.0

%-6

.5%-0

.7%-2

1.5%

-7.6%

-0.8%

-25.7

%-1

6.4%

-1.0%

Di�

-2.9%

-2.2%

-0.4%

-5.0%

-3.1%

-0.6%

-7.0%

-4.7%

-0.9%

-13.3%

-11.1%

-1.8%

-21.5%

-15.0%

-2.3%

-29.0%

-23.1%

-2.9%

PutS

prea

d95

/85

-7.1%

-4.4%

-0.6%

-11.3

%-5

.8%-0

.9%-1

5.6%

-8.5%

-1.3%

-29.7

%-1

7.6%

-2.3%

-47.6

%-2

1.3%

-2.9%

-58.6

%-4

2.3%

-3.5%

Div

este

dEq

uity

-4.3%

-2.3%

-0.2%

-6.2%

-3.1%

-0.4%

-9.0%

-4.4%

-0.5%

-17%

-7.4%

-0.8%

-24.6

%-8

.6%-1

.0%-2

9.5%

-19.0

%-1

.1%Di�

-2.8%

-2.0%

-0.3%

-5.1%

-2.7%

-0.5%

-6.7%

-4.1%

-0.8%

-12.7%

-10.2%

-1.5%

-23.0%

-12.7%

-2.0%

-29.1%

-23.4%

-2.4%

PutS

prea

d10

0/90

-6.7%

-3.5%

-0.5%

-10.0

%-4

.8%-0

.8%-1

3.2%

-6.8%

-1.4%

-31.3

%-1

1.5%

-2.4%

-44.4

%-1

3.9%

-3.1%

-52.6

%-3

4.6%

-3.5%

Div

este

dEq

uity

-1.8%

-1.0%

-0.1%

-2.7%

-1.3%

-0.1%

-3.8%

-1.9%

-0.2%

-6.8%

-3.0%

-0.3%

-9.8%

-3.4%

-0.4%

-11.4

%-7

.0%-0

.4%Di�

-4.9%

-2.5%

-0.4%

-7.3%

-3.5%

-0.7%

-9.4%

-4.9%

-1.2%

-24.5%

-8.5%

-2.1%

-34.6%

-10.5%

-2.7%

-41.2%

-27.7%

-3.1%

Colla

r95/

105

-4.8%

-3.2%

-0.4%

-6.7%

-4.4%

-0.7%

-9.2%

-6.5%

-1.1%

-16.3

%-1

3.0%

-1.9%

-23.1

%-1

7.4%

-2.5%

-32.8

%-2

4.9%

-3.2%

Div

este

dEq

uity

-4.6%

-2.5%

-0.2%

-6.7%

-3.3%

-0.4%

-9.6%

-4.7%

-0.6%

-18.3

%-7

.9%-0

.9%-2

6.5%

-9.3%

-1.0%

-31.7

%-2

0.5%

-1.2%

Di�

-0.2%

-0.7%

-0.2%

-0.1%

-1.1%

-0.3%

+0.4%

-1.8%

-0.5%

+2.0%

-5.1%

-1.0%

+3.4%

-8.1%

-1.5%

-1.1%

-4.4%

-2.0%

Frac

tiona

lPro

tect

ive

Put

-6.6%

-4.4%

-0.6%

-10.4

%-5

.9%-1

.0%-1

5.0%

-8.8%

-1.5%

-31.9

%-1

8.4%

-2.7%

-45.0

%-2

3.2%

-3.3%

-56.1

%-4

2.4%

-4.3%

Div

este

dEq

uity

-3.2%

-1.8%

-0.2%

-4.7%

-2.3%

-0.3%

-6.8%

-3.3%

-0.4%

-12.8

%-5

.6%-0

.6%-1

8.1%

-6.5%

-0.7%

-21.7

%-1

3.7%

-0.8%

Di�

-3.4%

-2.7%

-0.4%

-5.7%

-3.6%

-0.7%

-8.2%

-5.5%

-1.1%

-19.1%

-12.9%

-2.1%

-26.9%

-16.7%

-2.6%

-34.4%

-28.7%

-3.5%

Frac

tiona

lPut

Spre

ad95

/85

-6.9%

-4.3%

-0.5%

-11.0

%-5

.7%-0

.9%-1

5.3%

-8.3%

-1.3%

-29.0

%-1

7.3%

-2.3%

-46.5

%-2

1.0%

-2.9%

-57.4

%-4

1.5%

-3.5%

Div

este

dEq

uity

-4.4%

-2.4%

-0.2%

-6.5%

-3.2%

-0.4%

-9.3%

-4.6%

-0.6%

-17.8

%-7

.7%-0

.8%-2

5.7%

-9.0%

-1.0%

-30.9

%-1

9.9%

-1.2%

Di�

-2.5%

-1.8%

-0.3%

-4.6%

-2.5%

-0.5%

-6.0%

-3.7%

-0.7%

-11.2%

-9.6%

-1.4%

-20.8%

-12.0%

-1.9%

-26.5%

-21.5%

-2.3%

94

Tabl

e6.6

:Sum

mar

yof

peak

-to-tr

ough

draw

dow

nre

turn

sfor

the

3mpo

rtfol

ios.

5-D

ay10

-Day

20-D

ay63

-Day

125-

Day

250-

Day

99th

95th

50th

99th

95th

50th

99th

95th

50th

99th

95th

50th

99th

95th

50th

99th

95th

50th

S&P

500

-7.8%

-4.3%

-0.4%

-11.4

%-5

.7%-0

.7%-1

6.8%

-8.1%

-1.0%

-33.8

%-1

3.9%

-1.5%

-48.6

%-1

6.5%

-1.9%

-58.1

%-3

9.6%

-2.2%

Prot

ectiv

ePu

t-4

.8%-3

.2%-0

.5%-6

.4%-4

.5%-0

.8%-8

.9%-6

.3%-1

.3%-1

7.7%

-10.7

%-2

.4%-2

4.0%

-14.4

%-3

.1%-2

9.7%

-21.2

%-4

.0%D

ives

ted

Equi

ty-3

.4%-1

.9%-0

.2%-4

.9%-2

.4%-0

.3%-7

.0%-3

.5%-0

.4%-1

3.4%

-5.8%

-0.6%

-19.0

%-6

.7%-0

.7%-2

2.7%

-14.3

%-0

.9%Di�

-1.5%

-1.4%

-0.3%

-1.5%

-2.1%

-0.5%

-1.9%

-2.9%

-0.9%

-4.3%

-4.9%

-1.7%

-5.0%

-7.7%

-2.4%

-7.0%

-6.9%

-3.1%

PutS

prea

d95

/80

-5.7%

-3.5%

-0.5%

-8.3%

-4.6%

-0.8%

-12.3

%-6

.7%-1

.2%-2

0.7%

-11.4

%-2

.1%-3

2.7%

-15%

-2.8%

-41.3

%-3

0.3%

-3.5%

Div

este

dEq

uity

-4.2%

-2.3%

-0.2%

-6.1%

-3.0%

-0.4%

-8.8%

-4.3%

-0.5%

-16.7

%-7

.3%-0

.8%-2

4.1%

-8.5%

-0.9%

-29.0

%-1

8.6%

-1.1%

Di�

-1.5%

-1.2%

-0.3%

-2.2%

-1.6%

-0.4%

-3.5%

-2.4%

-0.7%

-3.9%

-4.1%

-1.3%

-8.6%

-6.5%

-1.9%

-12.4%

-11.7%

-2.4%

PutS

prea

d95

/85

-6.2%

-3.6%

-0.5%

-9.1%

-4.8%

-0.8%

-14.0

%-6

.9%-1

.1%-2

5.5%

-11.6

%-2

.0%-3

7.5%

-14.7

%-2

.5%-4

4.9%

-33.2

%-3

.2%D

ives

ted

Equi

ty-4

.9%-2

.7%-0

.3%-7

.2%-3

.6%-0

.4%-1

0.3%

-5.1%

-0.6%

-20.0

%-8

.6%-0

.9%-2

8.8%

-10.0

%-1

.1%-3

4.5%

-22.5

%-1

.3%Di�

-1.3%

-0.9%

-0.2%

-1.9%

-1.2%

-0.3%

-3.7%

-1.8%

-0.5%

-5.5%

-3.0%

-1.0%

-8.7%

-4.7%

-1.4%

-10.4%

-10.7%

-1.8%

PutS

prea

d10

0/90

-6.0%

-3.3%

-0.4%

-8.9%

-4.4%

-0.6%

-12.8

%-6

.2%-1

.0%-2

7.2%

-9.5%

-1.9%

-38.3

%-1

2.0%

-2.4%

-41.5

%-3

1.9%

-3.0%

Div

este

dEq

uity

-3.5%

-2.0%

-0.2%

-5.2%

-2.6%

-0.3%

-7.4%

-3.7%

-0.4%

-14.1

%-6

.1%-0

.7%-2

0.1%

-7.1%

-0.8%

-24.1

%-1

5.3%

-0.9%

Di�

-2.4%

-1.3%

-0.2%

-3.7%

-1.8%

-0.3%

-5.3%

-2.5%

-0.6%

-13.1%

-3.4%

-1.2%

-18.2%

-4.8%

-1.6%

-17.5%

-16.7%

-2.1%

Colla

r95/

105

-3.2%

-2.0%

-0.2%

-4.3%

-2.8%

-0.4%

-5.8%

-3.9%

-0.6%

-8.6%

-6.8%

-1.0%

-12.1

%-9

.1%-1

.4%-1

5.3%

-11.8

%-2

.0%D

ives

ted

Equi

ty-5

.1%-2

.8%-0

.3%-7

.4%-3

.7%-0

.4%-1

0.7%

-5.2%

-0.6%

-20.7

%-8

.8%-1

.0%-2

9.7%

-10.3

%-1

.1%-3

5.7%

-23.3

%-1

.4%Di�

+1.9%

+0.8%

0.0%

+3.1%

+0.9%

0.0%

+4.8%

+1.3%

0.0%

+12.1%

+2.0%

0.0%

+17.7%

+1.2%

-0.3%

+20.4%

+11.5%

-0.6%

Frac

tiona

lPro

tect

ive

Put

-4.8%

-3.2%

-0.5%

-6.6%

-4.4%

-0.8%

-8.9%

-6.3%

-1.3%

-17.5

%-1

0.6%

-2.3%

-23.7

%-1

4.3%

-3.0%

-29.4

%-2

1.0%

-4.0%

Div

este

dEq

uity

-3.6%

-2.0%

-0.2%

-5.3%

-2.6%

-0.3%

-7.6%

-3.7%

-0.4%

-14.3

%-6

.2%-0

.7%-2

0.4%

-7.3%

-0.8%

-24.5

%-1

5.6%

-0.9%

Di�

-1.2%

-1.2%

-0.3%

-1.3%

-1.8%

-0.5%

-1.4%

-2.6%

-0.8%

-3.2%

-4.4%

-1.6%

-3.2%

-7.0%

-2.2%

-4.9%

-5.4%

-3.0%

Frac

tiona

lPut

Spre

ad95

/85

-6.3%

-3.5%

-0.4%

-9.0%

-4.8%

-0.7%

-14.0

%-6

.9%-1

.1%-2

5.4%

-11.5

%-1

.9%-3

7.3%

-14.6

%-2

.5%-4

4.7%

-33.0

%-3

.1%D

ives

ted

Equi

ty-5

.1%-2

.8%-0

.3%-7

.4%-3

.7%-0

.4%-1

0.7%

-5.2%

-0.6%

-20.8

%-8

.8%-1

.0%-2

9.8%

-10.4

%-1

.2%-3

5.8%

-23.4

%-1

.4%D

i�-1.2%

-0.7%

-0.2%

-1.6%

-1.1%

-0.3%

-3.3%

-1.6%

-0.5%

-4.6%

-2.7%

-1.0%

-7.4%

-4.2%

-1.4%

-8.8%

-9.6%

-1.7%

95

Tabl

e6.7

:Sum

mar

yof

peak

-to-tr

ough

draw

dow

nre

turn

sfor

the

6mpo

rtfol

ios.

5-D

ay10

-Day

20-D

ay63

-Day

125-

Day

250-

Day

99th

95th

50th

99th

95th

50th

99th

95th

50th

99th

95th

50th

99th

95th

50th

99th

95th

50th

S&P

500

-7.8%

-4.3%

-0.4%

-11.4

%-5

.7%-0

.7%-1

6.8%

-8.1%

-1.0%

-33.8

%-1

3.9%

-1.5%

-48.6

%-1

6.5%

-1.9%

-58.1

%-3

9.6%

-2.2%

Prot

ectiv

ePu

t-4

.3%-2

.9%-0

.4%-5

.6%-4

.0%-0

.7%-8

.0%-5

.7%-1

.1%-1

3.1%

-9.7%

-1.9%

-23.0

%-1

2.6%

-2.5%

-30.4

%-2

2.1%

-3.0%

Div

este

dEq

uity

-4.0%

-2.2%

-0.2%

-5.9%

-2.9%

-0.3%

-8.5%

-4.2%

-0.5%

-16.1

%-7

.0%-0

.8%-2

3.2%

-8.2%

-0.9%

-27.8

%-1

7.8%

-1.1%

Di�

-0.3%

-0.7%

-0.2%

+0.3%

-1.1%

-0.3%

+0.5%

-1.5%

-0.6%

+3.0%

-2.7%

-1.2%

+0.2%

-4.4%

-1.6%

-2.6%

-4.3%

-1.9%

PutS

prea

d95

/80

-6.0%

-3.4%

-0.4%

-8.7%

-4.6%

-0.7%

-12.4

%-6

.5%-1

.0%-2

3.8%

-11.2

%-1

.7%-3

5.2%

-13.5

%-2

.1%-4

3.5%

-29.1

%-2

.7%D

ives

ted

Equi

ty-5

.2%-2

.8%-0

.3%-7

.6%-3

.8%-0

.4%-1

0.9%

-5.4%

-0.7%

-21.3

%-9

.0%-1

.0%-3

0.6%

-10.6

%-1

.2%-3

6.7%

-24.0

%-1

.4%Di�

-0.8%

-0.5%

-0.1%

-1.1%

-0.8%

-0.2%

-1.5%

-1.1%

-0.4%

-2.6%

-2.1%

-0.7%

-4.6%

-2.9%

-1.0%

-6.8%

-5.1%

-1.2%

PutS

prea

d95

/85

-6.5%

-3.5%

-0.4%

-9.6%

-4.7%

-0.7%

-14.0

%-6

.6%-1

.0%-2

6.6%

-11.4

%-1

.7%-3

9.0%

-13.8

%-2

.0%-4

6.9%

-32.5

%-2

.5%D

ives

ted

Equi

ty-5

.9%-3

.2%-0

.3%-8

.6%-4

.3%-0

.5%-1

2.4%

-6.1%

-0.7%

-24.2

%-1

0.3%

-1.1%

-35.1

%-1

2.1%

-1.3%

-42.1

%-2

7.8%

-1.6%

Di�

-0.7%

-0.3%

-0.1%

-1.1%

-0.5%

-0.1%

-1.6%

-0.6%

-0.3%

-2.4%

-1.1%

-0.5%

-4.0%

-1.7%

-0.7%

-4.8%

-4.7%

-0.9%

PutS

prea

d10

0/90

-6.6%

-3.3%

-0.4%

-9.6%

-4.4%

-0.6%

-14.2

%-6

.1%-0

.9%-2

6.7%

-10.7

%-1

.5%-3

6.5%

-13.2

%-2

.0%-4

1.3%

-29.2

%-2

.5%D

ives

ted

Equi

ty-5

.1%-2

.8%-0

.3%-7

.5%-3

.7%-0

.4%-1

0.8%

-5.3%

-0.6%

-21.0

%-8

.9%-1

.0%-3

0.2%

-10.5

%-1

.2%-3

6.2%

-23.6

%-1

.4%Di�

-1.4%

-0.5%

-0.1%

-2.1%

-0.7%

-0.2%

-3.4%

-0.8%

-0.3%

-5.7%

-1.8%

-0.6%

-6.4%

-2.7%

-0.8%

-5.1%

-5.5%

-1.1%

Colla

r95/

105

-2.7%

-1.6%

-0.1%

-3.6%

-2.3%

-0.3%

-5.4%

-3.1%

-0.4%

-7.9%

-6.0%

-0.6%

-12.8

%-7

.4%-0

.8%-1

6.4%

-12.0

%-1

.1%D

ives

ted

Equi

ty-3

.5%-1

.9%-0

.2%-5

.2%-2

.6%-0

.3%-7

.4%-3

.6%-0

.4%-1

4.0%

-6.1%

-0.7%

-20.0

%-7

.1%-0

.8%-2

4.0%

-15.2

%-0

.9%Di�

+0.8%

+0.3%

0.0%

+1.5%

+0.3%

+0.1%

+2.1%

+0.5%

+0.1%

+6.1%

+0.1%

+0.1%

+7.2%

-0.3%

0.0%

+7.6%

+3.2%

-0.2%

Frac

tiona

lPro

tect

ive

Put

-4.4%

-2.8%

-0.4%

-5.7%

-4.0%

-0.7%

-8.0%

-5.7%

-1.0%

-12.8

%-9

.7%-1

.8%-2

2.7%

-12.5

%-2

.4%-3

0.2%

-21.7

%-3

.0%D

ives

ted

Equi

ty-4

.4%-2

.4%-0

.2%-6

.5%-3

.2%-0

.4%-9

.3%-4

.6%-0

.6%-1

7.8%

-7.7%

-0.8%

-25.7

%-9

.0%-1

.0%-3

0.9%

-19.9

%-1

.2%Di�

+0.1%

-0.4%

-0.2%

+0.8%

-0.8%

-0.3%

+1.3%

-1.1%

-0.5%

+5.1%

-2.0%

-1.0%

+3.1%

-3.5%

-1.4%

+0.6%

-1.7%

-1.8%

Frac

tiona

lPut

Spre

ad95

/85

-6.7%

-3.5%

-0.4%

-9.7%

-4.8%

-0.7%

-14.0

%-6

.8%-1

.0%-2

6.9%

-11.4

%-1

.6%-3

9.0%

-14.0

%-2

.0%-4

7.2%

-32.5

%-2

.5%D

ives

ted

Equi

ty-6

.1%-3

.3%-0

.3%-8

.8%-4

.4%-0

.5%-1

2.8%

-6.3%

-0.8%

-25.1

%-1

0.6%

-1.2%

-36.4

%-1

2.5%

-1.4%

-43.6

%-2

8.9%

-1.7%

Di�

-0.7%

-0.2%

-0.1%

-0.9%

-0.4%

-0.1%

-1.1%

-0.5%

-0.2%

-1.8%

-0.8%

-0.5%

-2.7%

-1.4%

-0.6%

-3.6%

-3.5%

-0.8%

96

Tabl

e6.8

:Sum

mar

yof

peak

-to-tr

ough

draw

dow

nre

turn

sfor

the

12m

portf

olio

s.5-

Day

10-D

ay20

-Day

63-D

ay12

5-D

ay25

0-D

ay99

th95

th50

th99

th95

th50

th99

th95

th50

th99

th95

th50

th99

th95

th50

th99

th95

th50

thS&

P50

0-7

.8%-4

.3%-0

.4%-1

1.4%

-5.7%

-0.7%

-16.8

%-8

.1%-1

.0%-3

3.8%

-13.9

%-1

.5%-4

8.6%

-16.5

%-1

.9%-5

8.1%

-39.6

%-2

.2%Pr

otec

tive

Put

-4.1%

-2.5%

-0.4%

-5.9%

-3.6%

-0.6%

-8.1%

-5.1%

-0.9%

-11.1

%-8

.3%-1

.7%-1

5.4%

-11.3

%-2

.1%-2

3.9%

-17.3

%-2

.5%D

ives

ted

Equi

ty-4

.6%-2

.5%-0

.2%-6

.7%-3

.3%-0

.4%-9

.7%-4

.8%-0

.6%-1

8.6%

-8%

-0.9%

-26.8

%-9

.4%-1

.0%-3

2.2%

-20.8

%-1

.2%Di�

+0.5%

0.0%

-0.1%

+0.9%

-0.3%

-0.2%

+1.6%

-0.3%

-0.3%

+7.5%

-0.2%

-0.8%

+11.4%

-1.9%

-1.1%

+8.3%

+3.5%

-1.3%

PutS

prea

d95

/80

-6.0%

-3.4%

-0.4%

-8.7%

-4.6%

-0.6%

-12.6

%-6

.8%-0

.9%-2

3.3%

-11.2

%-1

.5%-3

4.5%

-13.7

%-1

.9%-4

3.3%

-28.9

%-2

.3%D

ives

ted

Equi

ty-5

.9%-3

.2%-0

.3%-8

.6%-4

.3%-0

.5%-1

2.5%

-6.1%

-0.7%

-24.4

%-1

0.4%

-1.1%

-35.3

%-1

2.2%

-1.4%

-42.4

%-2

8%-1

.6%Di�

0.0%

-0.2%

-0.1%

-0.1%

-0.3%

-0.1%

-0.1%

-0.6%

-0.2%

+1.1%

-0.9%

-0.4%

+0.9%

-1.5%

-0.6%

-0.8%

-0.9%

-0.7%

PutS

prea

d95

/85

-6.7%

-3.6%

-0.4%

-9.5%

-4.9%

-0.6%

-14.3

%-7

.1%-0

.9%-2

7.1%

-12.0

%-1

.5%-3

9.9%

-14.6

%-1

.8%-4

8.4%

-33.1

%-2

.2%D

ives

ted

Equi

ty-6

.4%-3

.5%-0

.3%-9

.4%-4

.7%-0

.6%-1

3.6%

-6.7%

-0.8%

-26.9

%-1

1.3%

-1.2%

-38.8

%-1

3.3%

-1.5%

-46.5

%-3

1.0%

-1.8%

Di�

-0.3%

-0.1%

0.0%

-0.2%

-0.3%

-0.1%

-0.7%

-0.4%

-0.1%

-0.2%

-0.7%

-0.3%

-1.1%

-1.3%

-0.4%

-1.9%

-2.0%

-0.5%

PutS

prea

d10

0/90

-6.7%

-3.4%

-0.4%

-9.5%

-4.5%

-0.6%

-13.9

%-6

.4%-0

.9%-2

7.5%

-10.5

%-1

.4%-3

9.4%

-12.4

%-1

.7%-4

6.3%

-33.1

%-2

.0%D

ives

ted

Equi

ty-6

.5%-3

.5%-0

.3%-9

.5%-4

.7%-0

.6%-1

3.8%

-6.7%

-0.8%

-27.2

%-1

1.4%

-1.3%

-39.2

%-1

3.4%

-1.5%

-46.9

%-3

1.4%

-1.8%

Di�

-0.2%

+0.1%

0.0%

0.0%

+0.2%

0.0%

-0.1%

+0.3%

-0.1%

-0.3%

+0.9%

-0.1%

-0.2%

+1.0%

-0.2%

+0.6%

-1.7%

-0.2%

Colla

r95/

105

-2.5%

-1.3%

-0.1%

-3.8%

-1.7%

-0.2%

-5.5%

-2.6%

-0.2%

-10.0

%-5

.1%-0

.4%-1

0.8%

-7.5%

-0.6%

-13.3

%-9

.9%-0

.9%D

ives

ted

Equi

ty-0

.9%-0

.5%0.0

%-1

.3%-0

.6%-0

.1%-1

.8%-0

.9%-0

.1%-3

.2%-1

.4%-0

.1%-4

.4%-1

.6%-0

.1%-4

.9%-2

.8%-0

.2%Di�

-1.6%

-0.8%

-0.1%

-2.5%

-1.1%

-0.1%

-3.7%

-1.7%

-0.2%

-6.8%

-3.7%

-0.3%

-6.4%

-5.9%

-0.4%

-8.4%

-7.1%

-0.7%

Frac

tiona

lPro

tect

ive

Put

-4.0%

-2.4%

-0.3%

-5.3%

-3.4%

-0.6%

-7.1%

-4.8%

-0.8%

-9.5%

-7.7%

-1.5%

-13.4

%-1

0.3%

-2.0%

-21.8

%-1

5.4%

-2.6%

Div

este

dEq

uity

-4.8%

-2.6%

-0.3%

-7.0%

-3.5%

-0.4%

-10.1

%-5

.0%-0

.6%-1

9.5%

-8.4%

-0.9%

-28.1

%-9

.8%-1

.1%-3

3.7%

-21.9

%-1

.3%Di�

+0.8%

+0.2%

-0.1%

+1.7%

+0.1%

-0.1%

+3.0%

+0.1%

-0.2%

+10.0%

+0.7%

-0.6%

+14.6%

-0.5%

-0.9%

+11.9%

+6.5%

-1.3%

Frac

tiona

lPut

Spre

ad95

/85

-6.8%

-3.5%

-0.4%

-9.4%

-4.8%

-0.6%

-14.1

%-6

.9%-0

.9%-2

7.2%

-11.6

%-1

.5%-3

8.5%

-13.8

%-1

.9%-4

6.7%

-30.4

%-2

.3%D

ives

ted

Equi

ty-6

.6%-3

.6%-0

.4%-9

.7%-4

.8%-0

.6%-1

4.1%

-6.9%

-0.8%

-27.8

%-1

1.7%

-1.3%

-40.2

%-1

3.7%

-1.5%

-48.0

%-3

2.2%

-1.8%

Di�

-0.2%

+0.1%

0.0%

+0.2%

0.0%

-0.1%

-0.1%

0.0%

-0.1%

+0.6%

+0.1%

-0.2%

+1.7%

-0.1%

-0.3%

+1.3%

+1.8%

-0.5%

97

6.3 Figures

6.3.1 Monte Carlo Simulation

-35% -30% -25% -20% -15% -10% -5% 0%

Peak-to-trough drawdown

0

10

20

30

40

50

60

Pro

babili

ty D

ensity F

unction

5-Day

Protective Put, (-5.4%, 0.8%)

Put Spread 95/80, (-6%, 2.4%)

Put Spread 95/85, (-6.1%, 3.3%)

Put Spread 100/90, (-4.8%, 3.2%)

Collar 95/105, (-4.4%, 0.7%)

Frac. Protective Put, (-5.3%, 0.8%)

Frac. Put Spread 95/85, (-6%, 3.4%)

-45% -40% -35% -30% -25% -20% -15% -10% -5% 0%


0

5

10

15

20

25

30

35

40

45

Pro

babili

ty D

ensity F

unction

10-Day


Put Spread 95/80, (-10.4%, 6%)

Put Spread 95/85, (-10.7%, 7.4%)

Put Spread 100/90, (-8.3%, 7%)

Collar 95/105, (-6%, 1%)

Frac. Protective Put, (-7.3%, 1%)

Frac. Put Spread 95/85, (-10.7%, 7.8%)

-60% -50% -40% -30% -20% -10% 0%


0

5

10

15

20

25

30

35

Pro

babili

ty D

ensity F

unction

20-Day


Put Spread 95/80, (-18.5%, 9.3%)

Put Spread 95/85, (-19%, 11.1%)

Put Spread 100/90, (-16.2%, 12%)

Collar 95/105, (-8.1%, 1.4%)


Frac. Put Spread 95/85, (-19.1%, 11.6%)

-70% -60% -50% -40% -30% -20% -10% 0%


0

2

4

6

8

10

12

14

16

18

Pro

babili

ty D

ensity F

unction

63-Day


Put Spread 95/80, (-24.5%, 8.8%)

Put Spread 95/85, (-24.9%, 10.2%)

Put Spread 100/90, (-20.3%, 11.7%)

Collar 95/105, (-13.9%, 2.6%)


Frac. Put Spread 95/85, (-25.1%, 10.6%)

-80% -70% -60% -50% -40% -30% -20% -10% 0%


0

2

4

6

8

10

12

Pro

babili

ty D

ensity F

unction

125-Day


Put Spread 95/80, (-29%, 8.6%)

Put Spread 95/85, (-29.4%, 9.9%)

Put Spread 100/90, (-23.8%, 11.1%)

Collar 95/105, (-18%, 4%)


Frac. Put Spread 95/85, (-29.7%, 10.2%)

-80% -70% -60% -50% -40% -30% -20% -10% 0%


0

1

2

3

4

5

6

7

8

Pro

babili

ty D

ensity F

unction

250-Day


Put Spread 95/80, (-33.9%, 9.9%)

Put Spread 95/85, (-34.2%, 11.1%)

Put Spread 100/90, (-28%, 11.6%)

Collar 95/105, (-21.9%, 5.7%)

Frac. Protective Put, (-28.1%, 6%)

Frac. Put Spread 95/85, (-34.7%, 11.3%)

Figure 6.1: 1m Portfolios: Probability density function of the 99th percentile peak-to-troughdrawdowns for the di�erent strategies.

98

-35% -30% -25% -20% -15% -10% -5% 0%


0

10

20

30

40

50

60

Pro

babili

ty D

ensity F

unction

5-Day


Put Spread 95/80, (-5.8%, 2.5%)

Put Spread 95/85, (-6%, 3.4%)

Put Spread 100/90, (-5.3%, 3.2%)

Collar 95/105, (-3.2%, 0.7%)


Frac. Put Spread 95/85, (-5.9%, 3.3%)

-40% -35% -30% -25% -20% -15% -10% -5% 0%


0

5

10

15

20

25

30

35

40

Pro

babili

ty D

ensity F

unction

10-Day


Put Spread 95/80, (-10%, 6%)

Put Spread 95/85, (-10.3%, 7.1%)

Put Spread 100/90, (-9.3%, 7.2%)

Collar 95/105, (-4.7%, 1.1%)


Frac. Put Spread 95/85, (-10.4%, 7.5%)

-60% -50% -40% -30% -20% -10% 0%


0

5

10

15

20

25

30

Pro

babili

ty D

ensity F

unction

20-Day


Put Spread 95/80, (-18%, 9.4%)

Put Spread 95/85, (-18.9%, 11.1%)

Put Spread 100/90, (-17.6%, 11.5%)

Collar 95/105, (-6.4%, 1.6%)


Frac. Put Spread 95/85, (-19.4%, 11.6%)

-70% -60% -50% -40% -30% -20% -10% 0%


0

2

4

6

8

10

12

14

16

18

20

Pro

babili

ty D

ensity F

unction

63-Day


Put Spread 95/80, (-22.9%, 9.5%)

Put Spread 95/85, (-24.1%, 10.9%)

Put Spread 100/90, (-22.6%, 11.5%)

Collar 95/105, (-9.4%, 2.1%)


Frac. Put Spread 95/85, (-24.4%, 11.4%)

-70% -60% -50% -40% -30% -20% -10% 0%


0

2

4

6

8

10

12

14

16

Pro

babili

ty D

ensity F

unction

125-Day


Put Spread 95/80, (-26.7%, 9.4%)

Put Spread 95/85, (-27.9%, 10.7%)

Put Spread 100/90, (-25.8%, 11.5%)

Collar 95/105, (-12.1%, 2.6%)


Frac. Put Spread 95/85, (-28.2%, 11%)

-80% -70% -60% -50% -40% -30% -20% -10% 0%


0

2

4

6

8

10

12

Pro

babili

ty D

ensity F

unction

250-Day


Put Spread 95/80, (-31.4%, 10.1%)

Put Spread 95/85, (-32.5%, 11.4%)

Put Spread 100/90, (-29.6%, 12.1%)

Collar 95/105, (-15.2%, 3.9%)


Frac. Put Spread 95/85, (-32.9%, 11.5%)

Figure 6.2: 3m Portfolios: Probability density function of the 99th percentile peak-to-troughdrawdowns for the di�erent option-based protection strategies.

99

-40% -35% -30% -25% -20% -15% -10% -5% 0%


0

10

20

30

40

50

60

70

Pro

babili

ty D

ensity F

unction

5-Day

Protective Put, (-5%, 1%)

Put Spread 95/80, (-5.8%, 2.8%)

Put Spread 95/85, (-6.1%, 3.6%)

Put Spread 100/90, (-5.8%, 3.7%)

Collar 95/105, (-2.4%, 0.6%)


Frac. Put Spread 95/85, (-6.1%, 3.6%)

-40% -35% -30% -25% -20% -15% -10% -5% 0%


0

5

10

15

20

25

30

35

40

Pro

babili

ty D

ensity F

unction

10-Day


Put Spread 95/80, (-10%, 6.2%)

Put Spread 95/85, (-10.4%, 7.2%)

Put Spread 100/90, (-10%, 7.4%)

Collar 95/105, (-3.6%, 1%)


Frac. Put Spread 95/85, (-10.8%, 7.7%)

-60% -50% -40% -30% -20% -10% 0%


0

5

10

15

20

25

Pro

babili

ty D

ensity F

unction

20-Day

Protective Put, (-11%, 3.5%)

Put Spread 95/80, (-18.1%, 9.6%)

Put Spread 95/85, (-19%, 11.2%)

Put Spread 100/90, (-18.4%, 11.5%)

Collar 95/105, (-5.5%, 1.8%)


Frac. Put Spread 95/85, (-19.9%, 11.8%)

-70% -60% -50% -40% -30% -20% -10% 0%


0

2

4

6

8

10

12

14

16

18

20

Pro

babili

ty D

ensity F

unction

63-Day


Put Spread 95/80, (-23.1%, 9.6%)

Put Spread 95/85, (-24.3%, 10.9%)

Put Spread 100/90, (-23.6%, 11.3%)

Collar 95/105, (-7.5%, 2%)


Frac. Put Spread 95/85, (-25.3%, 11.4%)

-80% -70% -60% -50% -40% -30% -20% -10% 0%


0

2

4

6

8

10

12

14

16

18

Pro

babili

ty D

ensity F

unction

125-Day

Protective Put, (-19%, 4.1%)

Put Spread 95/80, (-26.6%, 9.9%)

Put Spread 95/85, (-28.1%, 11%)

Put Spread 100/90, (-27.2%, 11.5%)

Collar 95/105, (-9%, 2.3%)


Frac. Put Spread 95/85, (-28.9%, 11.3%)

-80% -70% -60% -50% -40% -30% -20% -10% 0%


0

5

10

15

Pro

babili

ty D

ensity F

unction

250-Day


Put Spread 95/80, (-30.9%, 10.7%)

Put Spread 95/85, (-32.4%, 12%)

Put Spread 100/90, (-31.2%, 12.5%)

Collar 95/105, (-11.2%, 2.8%)


Frac. Put Spread 95/85, (-33.3%, 12%)


100

-35% -30% -25% -20% -15% -10% -5% 0%


0

10

20

30

40

50

60

70

Pro

babili

ty D

ensity F

unction

5-Day

Protective Put, (-4.9%, 1%)

Put Spread 95/80, (-6%, 3.1%)

Put Spread 95/85, (-6.3%, 3.7%)

Put Spread 100/90, (-6.2%, 3.8%)

Collar 95/105, (-1.7%, 0.6%)


Frac. Put Spread 95/85, (-6.4%, 3.8%)

-40% -35% -30% -25% -20% -15% -10% -5% 0%


0

5

10

15

20

25

30

35

40

Pro

babili

ty D

ensity F

unction

10-Day


Put Spread 95/80, (-10.4%, 6.6%)

Put Spread 95/85, (-11%, 7.7%)

Put Spread 100/90, (-10.5%, 7.4%)

Collar 95/105, (-2.7%, 1.1%)


Frac. Put Spread 95/85, (-11.2%, 7.9%)

-60% -50% -40% -30% -20% -10% 0%


0

5

10

15

20

25

Pro

babili

ty D

ensity F

unction

20-Day


Put Spread 95/80, (-18.5%, 10.1%)

Put Spread 95/85, (-19.8%, 11.5%)

Put Spread 100/90, (-19%, 11.3%)

Collar 95/105, (-4.2%, 1.9%)


Frac. Put Spread 95/85, (-20.8%, 12.2%)

-70% -60% -50% -40% -30% -20% -10% 0%


0

2

4

6

8

10

12

14

16

18

20

Pro

babili

ty D

ensity F

unction

63-Day


Put Spread 95/80, (-23.7%, 9.9%)

Put Spread 95/85, (-25.4%, 11.1%)

Put Spread 100/90, (-24.6%, 11%)

Collar 95/105, (-5.8%, 2%)


Frac. Put Spread 95/85, (-26.6%, 11.6%)

-80% -70% -60% -50% -40% -30% -20% -10% 0%


0

2

4

6

8

10

12

14

16

18

20

Pro

babili

ty D

ensity F

unction

125-Day

Protective Put, (-18.7%, 5%)

Put Spread 95/80, (-27.2%, 10.3%)

Put Spread 95/85, (-29.1%, 11.3%)

Put Spread 100/90, (-28.2%, 11.3%)

Collar 95/105, (-6.7%, 2.1%)


Frac. Put Spread 95/85, (-30.4%, 11.5%)

-80% -70% -60% -50% -40% -30% -20% -10% 0%


0

2

4

6

8

10

12

14

16

18

Pro

babili

ty D

ensity F

unction

250-Day


Put Spread 95/80, (-31.3%, 11.5%)

Put Spread 95/85, (-33.3%, 12.5%)

Put Spread 100/90, (-32.3%, 12.6%)

Collar 95/105, (-7.9%, 2.5%)


Frac. Put Spread 95/85, (-34.7%, 12.5%)


6.3.2 Backtesting

101

2002 2005 2007 2010 2012 2015 2017 202050

100

150

200

250

300

350

Port

folio

valu

e

S&P 500

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105

Fractional Protective Put 1/4 1m

Fractional Put Spread 95/85 1/4 1m

Figure 6.5: Performance of the 1m portfolios.

2002 2005 2007 2010 2012 2015 2017 202050

100

150

200

250

300

350

Port

folio

valu

e

S&P 500

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105




2002 2005 2007 2010 2012 2015 2017 202050

100

150

200

250

300

350

Port

folio

valu

e

S&P 500

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105




2002 2005 2007 2010 2012 2015 2017 202050

100

150

200

250

300

350

Port

folio

valu

e

S&P 500

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105




102

10-2007 01-2008 04-2008 07-2008 10-2008 01-2009 04-200940

50

60

70

80

90

100

Port

folio

valu

e

S&P 500

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



Figure 6.9: Financial crisis: Performance of the1m portfolios.

10-2007 01-2008 04-2008 07-2008 10-2008 01-2009 04-200940

50

60

70

80

90

100

110

Port

folio

valu

e

S&P 500

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105




10-2007 01-2008 04-2008 07-2008 10-2008 01-2009 04-200940

50

60

70

80

90

100

Port

folio

valu

e

S&P 500

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105




10-2007 01-2008 04-2008 07-2008 10-2008 01-2009 04-200940

50

60

70

80

90

100

110

Port

folio

valu

e

S&P 500

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105




103

6.3.3 Total cost of strategies

0 2 4 6 8 10 12

Maturity (months)

-20%

0%

20%

40%

60%

80%

100%

120%

140%

Tota

l cost as %

of in

itia

l in

vestm

ent

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



Figure 6.13: Monte Carlo: Total cost for theoption-based protection strategies.

0 2 4 6 8 10 12

Maturity (months)

0%

50%

100%

150%

200%

250%

300%

350%

400%

450%

Tota

l cost as %

of in

itia

l in

vestm

ent

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



Figure 6.14: Backtesting, whole period: Totalcost for the option-based protection strategies.

0 2 4 6 8 10 12

Maturity (months)

-5%

0%

5%

10%

15%

20%

25%

30%

Tota

l cost as %

of in

itia

l in

vestm

ent

Protective Put

Put Spread 95/80

Put Spread 95/85

Put Spread 100/90

Collar 95/105



Figure 6.15: Backtesting, �nancial crisis: Totalcost for the option-based protection strategies.

104

Bibliography

[1] Annaert, J., Osselaer, S.V., Verstraete, B., 2009. Performance evaluation of portfolio insurance

strategies using stochastic dominance criteria, Journal of Banking & Finance 33 (2), 272–280

[2] Bennet C., 2014, Trading Volatility, 1st ed, CreateSpace Independent Publishing Platform

[3] Bertrand, P. , Prigent, J.-L., 2005 Portfolio insurance strategies: OBPI versus CPPI, Finance 26

(1), 5–32

[4] Bertrand, P. , Prigent, J.-L., 2011, Omega performance measure and portfolio insurance, J. Bank

Finance. 35 (7), 1811–1823

[5] Bjork T., 2009, Arbitrage �eory in Continuous Time, 3rd ed, Oxford University Press

[6] Black, F. & Jones, R., 1987, Simplifying portfolio insurance, �e Journal of Portfolio Manage-

ment, 48-51.

[7] Black, F. & Perold, A.R., 1992, �eory of constant proportion portfolio insurance, �e Journal

of Economics, Dynamics and Control, 16, 403-426.

[8] Brooks, C., Davies, R., and Kim, S.S., 2006, Cross Hedging with Single Stock Futures

[9] Cha�erjee R., 2014, Practical Methods of Financial Engineering and Risk Management, New

York: Springer

[10] Chicago Board Options Exchange, 2012a Equity Options Strategies — Equity Collar,

h�ps://www.cboe.com/strategies/equityoptions/equitycollars/partl.aspx. Who Should Use

Equity Collars?, h�ps://www.cboe.com/strategies/pdf/equitycollarstrategy.pdf.

[11] Hull J., 2012, Options, futures, and other derivatives, 8th ed., England: Pearson

[12] El Karoui, N., Jeanblanc, M., Lacoste, V., 2005. Optimal portfolio management with American

capital guarantee, Journal of Economic Dynamics & Control 25 (3), 449–468

[13] Figlewski S, Chidambaran N. K. and Kaplan S., 1993, Evaluating the Performance of the Pro-

tective Put Strategy, Financial Analysts Journal, Vol. 49, No. 4 (Jul. - Aug., 1993), pp. 46-56+69

105

[14] Fodor, A., Doran, J. S., Carson, J. M. and Kirch, D. P., 2010, On the Demand for Portfolio

Insurance,

[15] Israelsen, C, 2005 A re�nement to the Sharpe ratio and information ratio Journal of Asset

Management, April 2005, Volume 5, Issue 6, pp 423–427

[16] Israelov R., 2017, Pathetic Protection: �e Elusive Bene�ts of Protective Puts

[17] Israelov R., Klein M Risk and Return of Equity Index Collar Strategies �e Journal of Alter-

native Investments

[18] Leland, H.E. , Rubinstein, M. , 1976, �e evolution of portfolio insurance, In: Luskin, D.L. (Ed.),

Portfolio Insurance: A Guide to Dynamic Hedging. Wiley, New York

[19] Pezier, J. , Scheller, J. , 2013, Best portfolio insurance for long-term investment strategies in

realistic conditions Insurance 52 (2), 263–274

[20] Scholz, H., 2007, Re�nements to the Sharpe ratio: Comparing alternatives for bear markets

Journal of Asset Management, January 2007, Volume 7, Issue 5, pp 347–357

[21] Zagst, R., Kraus, J., 2011, Stochastic dominance of portfolio insurance strategies, Annals of

Operations Research 185 (1), 75–103.

[22] Zhu, Y., Kavee, R.C., 1988, Performance of portfolio insurance strategies, �e Journal of Port-

folio Management 14, 48–54

106

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