[Lehman Brothers, Harm Stone] Investing in Implied Volatility

8/7/2019 [Lehman Brothers, Harm Stone] Investing in Implied Volatility

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PLEASE SEE ANALYST CERTIFICATION AND IMPORTANT DISCLOSURES BEGINNING ON PAGE 9.INITIAL VERSION OF THIS REPORT HAS BEEN CIRCULATED PRIOR TO 19th FEBRUARY 2004

Equity Derivatives & Quantitative Research

Investing in Implied Volatility

IntroductionVolatility tends to be negatively correlated with equity returns especially at theindex level. Moreover volatility appears to display mean reversion and other

forecastable properties.1

In view of this, investors might wish to invest in

volatility to hedge their equity investments or to take directional positions.

The classic investment in volatility has been a delta hedged straddle or more

recently a volatility/variance swap. Although useful, the payoff of both of theseinstruments has the disadvantage of being tied to the difference between realized

volatility and implied volatility not the level of implied volatility itself. Morerecently volatility indices have become popular. These have been successful in

identifying trends in volatility, but it has not been easy to design products that

represent direct investments in the indices.

This report begins with a short review of the popular approaches to estimating

implied volatility. It reviews the theory behind recent implied volatility indices,

then it shows why investing directly in these indices has proven a challenge.Finally, it ends with possible investible products.

Popular Approaches Estimating Implied Volatility

1 Volatility as an Asset Class, Lehman Brothers 2002

Figure 1: Implied Volatility Report on European Indices for 3rd

February 2004

IndexLatest

Price1 Day Return

3-Month Implied

Volatility

Change in 3-

Month ATM

Implied Volatility

Fixed Strike: Change in

3-Month Implied

Volatility

FTSE 4390.6 0.2% 14.5% -0.28 -0.16ESTOXX 2841.26 -0.4% 18.7% 0.32 0.08

DAX 4057.51 -0.3% 21.5% 0.03 0.02CAC 3638.21 -0.7% 18.4% 0.32 0.00

MIB30 27615 -0.6% 15.1% 0.63 0.42SMI 5735 -0.9% 15.6% 0.01 0.00AEX 351.74 -1.0% 19.9% 0.89 0.34

S&P 500 1136.03 0.1% 14.6% -0.03 0.00Nikkei 10641.92 -1.3% 22.4% -0.18 -0.33

Source: Lehman Brothers, Bloomberg

19th February 2004

Andrew Harmstone

+44 207 [email protected]


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2

Ever since the Black Scholes options pricing model was developed it has been

possible to estimate the level of volatility a stock must have in order to justify its

current price. This estimated volatility is generally called implied volatility. This

is still the most widely used way of measuring volatility. Figure 1 shows avolatility report on the European indices.

Figure 1 shows what is called the At-The-Money (ATM) implied volatility for

each index for a term of three months ahead. This is calculated by choosing theoptions for each index which have strike prices close to the spot price of the

options underlying index and then interpolating. Unfortunately, this estimate isnot stable in the sense that the implied volatility calculated from an option with a

different strike price could be meaningfully different from that calculated from

the ATM option.

Figure 1 shows that the estimated change in volatility can be significantly

influenced by which strike price is chosen. For example, the ATM implied

volatility for the CAC 40 index over a three-month horizon is 18.2%. It rose by0.32% from 2

ndFebruary 2004 to the 3

rdFebruary 2004. But this result holds only

if the strike is used for both days is ATM.

2

Alternatively, it is equally reasonableto use the same strike option on both days. If this is done then Figure 1 shows thatthe implied volatility for the CAC 40 over a three month horizon was unchanged

from 2nd

February 2004 to the 3rd

February 2004. This is shown in the column

labelled Fixed Strike: Change in 3-Month Implied Volatility. As it turns out the

CAC 40 fell by 0.7% on the 2nd

February 2004 and because of this the ATMstrike on the 3

rdFebruary was below that on the 2

nd. Arguably, the rise in the

ATM volatility between the 2nd

and 3rd

of February was an illusion caused by the

market decline making the ATM strike fall. This is because lower strike options

generally have higher implied volatility.

2 The volatilities in Figure 1 are determined from an estimated implied volatility surface that is

determined from actual option prices. This makes it possible to estimate the ATM volatility even

when the index level is not at a particular listed options strike price.


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igure 2: 3M Skew for the EuroSTOXX 50. 100% strike is ATM

10%

15%

20%

25%

30%

35%

80% 90% 100% 105% 110% 120%

Strike

Implieds

Source: Lehman Brothers, Bloomberg

Figure 2, using the EuroSTOXX 50 index as an example, shows the difference in

volatility for different strike prices can be substantial this is the skew in theimplied volatility surface. The skew makes the implied volatility calculation from

an option pricing model like the Black Scholes model unsatisfactory. First of all,

there is the obvious contradiction of estimating volatility that changes overdifferent strike prices with a model that explicitly assumes that volatility remains

constant. Second, direct investment in implied volatility calculated by the modelis difficult. This is because the implied volatility is based on the ATM volatility,

but the ATM option changes every day as markets move. This means that an

investment product tied to the implied volatility defined in this way mustcontinuously rebalance to the current ATM option.

Volatility Indices

Pioneering volatility indices were created by the CBOE and by the DeutscheBrse, among others. These include the old VIX

and the VDAX

. Both of

these indices are based on average implied volatility levels of options that are

close to being at-the-money. These indices were successful in capturing, at leastin a broad sense, the trend in implied volatility. Direct investment in these indices

was hampered, however, by the fact that no one option or portfolio of optionswould track the changes in implied volatility that occurred as the level of themarket changed leading to changes in the ATM strike. Thus even though futures

existed on both indices, market makers found it hard to hedge their positions in

the futures and liquidity stayed low.

Simultaneous with the creation of volatility indices, there developed an Over-the-

Counter (OTC) market in volatility and variance swaps. These were


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generalizations of delta hedged straddles. A variance swap, for example, can be

constructed from a portfolio of options that behaves like a generalized straddle.

Like straddles these options portfolios have positive gamma so that delta

hedging is generally profitable. That is, if markets rise, the delta of the optionstends to rise, leading to sales of futures to reduce the delta down to zero. Similarly

if markets fall, then a positive gamma causes the options delta to drop, leading to

purchases of futures to bring the delta back to its target of zero exposure. Note

this strategy is inherently profitable because it means the delta hedger is sellinghigh and buying low. Moreover, the more volatile the markets are, the more

delta hedging occurs, and the more profit there is from this activity.

The implication is that the higher realized volatility is, the more profitable the

variance swap will be. But, of course, there is no free lunch because any option

position with a positive gamma will also have a negative theta. That is, it willdecline in value over time. It turns out that the rate of decline over time is a

function of the option portfolios implied volatility. Therefore, the net profit

from a variance swap depends on two offsetting factors. The first is the level ofrealized volatility. The second factor is the option portfolios implied volatility

was when it was initiated. Clearly the variance swap payoff is tied to thedifference between implied volatility and realized volatility. Conceptually, itconverts the implied volatility embedded in the options portfolio into realized

volatility generated from delta hedging as the swap goes from inception to

expiration.

Finally, the main advantage of using the portfolio of options in a generalizedstraddle over using a traditional straddle consisting of a long call and a long put

is that the gamma of the generalized straddle is relatively stable when market

prices change. This is clearly not the case with a simple straddle because, if, for

example, prices rise then the call side of the straddle becomes in the money

rapidly and its delta rises up to a maximum of 1 and its gamma falls. But with alower gamma there is less profitable delta hedging. So a straddle rapidly loses its

payoff from high realized volatility in either an upward trending or a downwardtrending market. The variance swap options portfolio is substantially insulated

from this effect because it is explicitly constructed to keep the gamma stable for a

wide range of price levels. So even in up or down markets it retains the ability toconvert realized volatility into profit. Pari passu keeping gamma constant means

its rate of decline over time, or negative theta, stays stable as well. This means

that, unlike a straddle, a variance swap payoff, tied to the delta-hedged

generalized straddle, maintains its desired trade-off between realized volatilityand implied volatility even if market prices move substantially.

Clearly the value of the options portfolio used in a variance swap is a function ofimplied volatility (technically implied variance). Therefore, it seems natural to use

its value as a proxy for the level of implied variance. One of the most remarkable

features of the variance swap option portfolio is that its value at inception (with aminor adjustment) is in fact exactly equal to a very reasonable estimate of implied

volatility. This mathematical fact forms the basis of the most recent generation of

implied volatility indices.


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Latest Generation of Volatility Indices

The new VIX

announced by the CBOE September 2003 is an example of the

latest generation of implied volatility indices. It is an important advance over the

prior generation because it does not use any option pricing model to determine thelevel of implied volatility.

3It provides a single summary estimate of volatility

across the whole volatility surface in a natural way. Finally, it summarizes the

characteristics of the whole volatility surface instead of being tied to ATM or nearATM strike options. It tends to be more correlated with the performance ofvolatility sensitive investment strategies. It accomplishes this because it is based,

in essence, on the value of the generalized straddle used typically in constructing

a variance swap, as discussed above. Note, however, that by nature this index andany other index based on the same methodology is fundamentally tied to implied

variance, not implied volatility. The volatility can always be calculated from the

variance, but the movement in volatility will never be linear with respect to themovement of the underlying portfolio of options that drive the index value.

The underlying principle of the new indices is discussed in the next section; a

more detailed derivation is available on request.

The Key is Reversing the Order of Summation (Integration)!

The generalized straddle is essentially a portfolio of out-of-the money calloptions and out-of the-money put options. The call component, with strikes, Kj,

CallPortfolio, is

CallPortfolio = For all Kj>= Km (K / Kj2) OTMCj.

Km is the lowest strike price that is still higher than the forward price F. K is thedifference between each strike price and for simplicity we assume that it is

constant.

The current value of a call can be calculated as the probability weighted average

of the calls end of period value. Figure 3 illustrates the probability distribution

and the corresponding end of period values.

Suppose that out of a discrete set of end of period stock prices from 0 to some

large number, Sn, the highest discrete stock price that is less than F is Sg. Next,

suppose that the probability that the end of period stock price is Sg is pg. Then the

value of the jth OTM call with strike price Kj, OTMCj, can be written as

(1) OTMCj = i=g+1 to npi * Max(0, Si Kj) where Kj > F, the futures price,

3 A few assumptions about the stock return distribution cannot, however, be avoided. The most

important may be the assumption that the distribution has a finite variance.


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where is the present value discount factor. This means that the call side of thegeneralized straddle is

Figure 3: Estimating the Probability Weighted Value of the OTM 110 Call

Cross-Section: Holding Strike ConstantProb Wgted Ending Option Value is Current Option Price!

0.0

10.0

20.0

30.0

40.0

50.0

50 60 70 80 90 100

110

120

130

140

150

Ending Stock Price

CallValueat

Expiration

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

ProbabilityStock Prob

OTM Call

110

Source: Lehman Brothers

(2A) CallPortfolio = For all Kj>= Km (K / Kj2) {i=g+1 to n pi *Max(0, Si Kj)}.

Similarly, the put component of the generalized straddle is,

(2B) PutPortfolio = For all Kj < Km (K / Kj2) {i=1 to gpi *Max(0, Kj Si)}.

The key step is reversing the order of summation in equations (2A) and (2B). This

changes the right hand side of each equation from a portfolio of options into an

expression that includes the expected log of the normalized end of period stock

price. The reason this is critical is that this expected log is linearly related to thevariance of the stock price.

To see that this is the case, first examine the equations after the order ofsummation has been reversed:

(3A) CallPortfolio = i=g+1 to n pi *{ For all Kj>= Km (K / Kj2) *Max(0, Si Kj)}.

(3B) PutPortfolio = i=1 to gpi * {For all Kj < Km (K / Kj2) *Max(0, Kj Si)}.

The generalized straddle, GenSD, then is

GenSD = i=1 to npi * {For all Kj>= Km (K / Kj2) *Max(0, Si Kj)

+ For all Kj < Km (K / Kj2) *Max(0, Kj Si) }.

But it can be shown that the term in brackets approximates


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S i /F 1 - Log (S i /F).4

This means that the generalized straddle approximates,

i=1 to npi * { S i /F 1 Log (S i /F)}, which simplifies to5,

i=1 to npi * { Log (S i /F)} or, approximately, (-) * Expected Log(S i /F).

That is the generalized straddle equals the negative of the discounted expected log

of the ending stock price normalized by the current futures price.

It is well-known that the expected log itself equals,

Expected Log (S i /F) = Tr Log(F/ S0) - T/2 Variance.

Here, Variance is the annualized variance of the stock price, T is the time to

expiration of the options (as a % of the year) and r is the annualized risk free rate.

Combining these results shows that6

GenSD = CallPortfolio + PutPortfolio

~ T/2 Variance.

In the case where the strike price, Km, does not equal the current futures price an

additional term is required and the relationship becomes,

GenSD = CallPortfolio + PutPortfolio ~ T/2 Variance + (F / Km- 1)2,

where ~ means approximates. Solving for the Variance,

Variance ~ (1/) (2/T) (CallPortfolio + PutPortfolio) - 1/T (F / Km- 1)2.

To get a constant maturity variance it is possible to interpolate the value of theGenSD for two options series chosen such that the weighted average time to

expiration matches the desired constant maturity. Next, a current generation

volatility index annualizes the variance and takes its square root to calculate thedesired annualized volatility.

4 There is another term that reflects the fact that Km does not exactly equal the futures price, F. The

value of this term can be approximated by (F / Km- 1)2

5 The expected value of the stock price, (approximately equal to i=1 to n pi * { S i }), equals thefutures price because by construction the probability distribution is risk-neutral. This causes the

first two terms in the summation to cancel out.6

Tr - Log(F/S0) is close to zero. For example, if there are no dividends and there is a flat risk freerate then, F = (1+r)T S0, making Log(F/S0) = Tlog(1+r). Since Tr is the first term of a Taylors

expansion for Log(1+r), Tlog(1+r) is close to Tr. The difference is of the order of the square of the

risk free rate.


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Conclusion

The latest generation of volatility indices is based on the value of an options

portfolio that behaves like a generalized straddle. The significant innovation isthat the generalized straddle is constructed so that for a wide range of prices for

the underlying, the gamma of the options portfolio is stable. The remarkable

mathematical result is that the value of this generalized straddle, which can be

calculated solely from the market prices of the options in the portfolio,approximates the implied volatility of the underlying.

Nonetheless, the implied variance estimated by the volatility index is not directlyinvestable. For example, it may seem natural simply to buy the options portfolio

that underlies the index. But one day after this portfolio is purchased, the options

experience time decay even if the implied volatility does not drop. Moreover,the options also have a delta and so the value of the options portfolio can rise and

fall with the market even if volatility levels do not change. If delta hedging is

introduced then realized volatility impacts the payoff. So the payoff to a product

that attempts to tie itself directly to an implied volatility index has a payoff that isa hybrid of the changes in implied volatility and other factors such as market

movements or realized volatility combined with decay due to the passage of time.

The reason for this apparent paradox is that implied volatility is strictly a forward

looking concept. One day after it is estimated, implied volatility is corrected by

the actual volatility that occurs in the market place. Implied volatility can becompared to the energy stored in a boulder at the top of a hill. Initially, all the

energy is potential (corresponding to implied volatility). Once the boulder startsdown the hill its energy changes from potential energy to kinetic energy (like

realized volatility). If there is no friction and no force is applied to the boulder

then the total energy level remains constant so that at the bottom of the hill thetotal kinetic energy will be exactly equal to the initial potential energy. This

would correspond to the case where realized volatility was exactly what waspredicted by the implied volatility. But, in reality, a rolling boulder experiences

friction and other forces that affect its energy level so the translation frompotential energy to kinetic is not fully predictable and the total energy in the end

may differ substantially from the potential energy at the top of the hill. Similarly,

the realized volatility will differ from the implied.

Continuing with the analogy, it is not appropriate to estimate the total energy level

of a moving boulder by just calculating its potential energy. Similarly, it is notappropriate to estimate the volatility of a stock index by looking only at implied

volatility once the index starts moving over time. The only time implied volatility

truly corresponds to the total volatility of the stock or stock index is before itstarts moving. That is to say, a product on implied volatility should be to be

forward looking, delivering the implied volatility as of some future date.


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[Lehman Brothers, Harm Stone] Investing in Implied Volatility

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