Trading and managing volatility surface risks

Axpo Risk Management Workshop21.11.2011

Swiss Re – Centre for Global DialogueStian J. Fröiland

The Classical Black-Scholes-Merton options pricing framework

Assumptions:

Efficient markets – the present price of an instrument impounds, at any given time, all the information contained in past prices (no above-average returns from technical analysis)

Short selling of underlying securities with full use of the proceeds is possible

There are no transactions costs or taxes

All securities and prices are perfectly divisible

Securities trading are continuous

There are no riskless arbitrage opportunities

Cash can be lent and borrowed at a constant risk-free interest rate

The Underlying security follows a geometric Brownian motion process with constant drift and volatility

The Classical Black-Scholes-Merton options pricing framework

: the price of the instrument being modelled

: the incremental change in the price level

: the expected (instantaneous) rate of return on the instrument

: the instantaneous (local) volatility of the instrument

: increments of a standard Wiener process

:tcoeffecienDiffusion

:parameterDrift

t

t

t Ft

t,F

rt,Frpriceasset and timeoft independen Parameters

:asseton Return

:priceasset in Change

t

t

t

tttt

σdWrdtF

dF

dWσFdtrFdF

tP

tdP

r

tdW

Geometric Brownian motion price process for the forward prices:

BSM classical assumptions concerning the drift parameter and diffusion coefficient in the BSM-model:

The BSM classical time-homogeneous Geometric Brownian Motion differential equation for the behaviour

of market prices in a risk-neutral world:

The classical BSM-model assumptions has several beneficial properties

tTtTrFF tT ,

2

1~lnln 2

where (m,s) denotes a normal distribution with mean m and standard deviation s.

1. The expected drift – the risk-free interest rate in a risk-neutral world – expressed as a proportion of the market price of the instrument, is constant – constant percentage return

2. The variance of the percentage return is the same regardless of the market price as well as time (stationary)

3. The distributions of the return FT/Ft and the price level FT:

The return of the process in some time interval T-t is normally distributed

4. The Forward price is log-normally distributed5. Asset returns over non-overlapping time intervals are independent (no

autocorrelation)6. The distribution of future asset prices is only dependent on the present

asset price and not the past history of asset prices (Markovian feature)

Monte Carlo Simulation of a Geometric Brownian Motion –Simulation Paths and Terminal Log-normal Price Distribution

Terminal Price Distribution

TimeTt

Mar

ket

Val

ue

Probability

4045

5055

6065

7075

80

01-Jan

01-Feb

01-Mar

01-Apr

01-May

0

0.1

0.2

0.3

0.4

0.5

Vola

tilit

y

StrikeTime

Kt,Ft ,

Constant Volatility with respect to time, Forward Price and Strike Price

Market reality: non-stationary stochastic volatility surface evolution

0.70.8

0.91

1.11.2

1.3

04/01/0207/01/02

10/01/0201/01/03

04/01/0307/01/03

10/01/0301/01/04

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Relative StrikeTime

Vola

tilit

y

𝜎(𝑡, 𝐹, 𝐾)

So what does the input from the market imply?

Despite these observations of inconsistencies between the BSM-Model and the market, the BSM model is still extensively used. As an imperfect «workaround» option traders as well as risk managers use BSM with different implied volatilities for each strike and expiry; hence multiple local BSM-Models each with a constant volatility are being used to compensate inconsistencies between model and market reality. Hence it has been described as using

«the wrong number in the wrong formula to get the right price.»

Message from the market:

The distribution of the underlying security (German Power Forwards) is NOT lognormally distributed, rather in expectation leptokurtic (fat tale(s) with a predominantly positive skew)! Furthermore it exhibits (strong) mean-reversion features.

The volatility is neither stationary nor deterministic, rather a stochastic variable with a stochactic correlational relationship with the underlying security, time as well as the strike of the option; in fact the volatility surface is non-stationary and stochastic!

Alternative modelling approaches to better incorporate market reality

Some alternative modelling approaches:

Generalized BSM-model with deterministic time-dependent deterministic drift in volatility (Schwartz, 1997)

Generalized BSM-model with volatility smile (Wilmott, 1998)

Local volatility models (Dupiré, 1994)

Higher distributional moments and stochastic volatility models

SABR model(«Managing Smile Risk», P.S. Hagan, D. Kumar, A.S. Lesniewski and D.E. Woodward, 2002) [1]

Important Benchmarks Concerning the Quality of Options Pricing Models

Requirements to an suitable option pricing model:

An options pricing model must be able to predict the probability distribution of the price level of the instrument on which the contingent claim is based

An option pricing model must correctly replicate the market dynamics of the volatility surface (“smile risk”)

Single and self-consistent model-framework upon which exotic options may be priced

For effective book management an option model must provide consistent and stable aggregation and consolidation of the greeks to enable viable hedging of the net exposure of the book

It must be fast and flexible enough to be of practical use to the trader in a high-paced trading situation!

Model consistency from the trading floor into the core of risk management is not a definite requirement, but a great advantage!

The SABR-Model provides feasible solutions to most of these of these requirements!

The SABR Model

The SABR Model is a two-factor Markovian governed by two coupled Brownian motion processes:

𝑑 𝐹 = 𝛼 𝐹𝛽𝑑𝑊1, 𝐹 0 = 𝑓𝑑 𝛼 = 𝜈 𝛼 𝑑𝑊2, 𝛼 0 = 𝛼

where the two processes are correlated by

𝑑𝑊1𝑑𝑊2 = 𝜌𝑑𝑡

𝐹 : Stochastic forward price

𝛼 : Stochastic volatility

𝛽 : Parameter that determines the inclination of the “backbone” along which the implied ATM-Volatility traverses

𝜈 : Volatility of the stochastic volatility - “volvol” (impacts the kurtosis of the probability distribution of the forward price; and the “smile” effect on the volatility curve

𝜌 : correlation between the forward price and the volatility of the forwardprice (impacts the skew of the probabililty distribution of the forward price as well as of the volatility curve )

The SABR Model

By using singular pertubation techniques based on a small volatility expansion to price European options under the SABR model in [1], P.S. Hagan et al. showed that the price of the European options are given by Black’s formula where the implied volatility 𝜎𝐵(𝑓, 𝐾) is given by

𝜎𝐵 𝑓, 𝐾 =𝛼

𝑓𝐾1−𝛽2 1 +

1 − 𝛽 2

24𝑙𝑜𝑔2 𝑓

𝐾+

1 − 𝛽 4

1920𝑙𝑜𝑔4 𝑓

𝐾+ ⋯

∙𝑧

𝑥 𝑧∙

1 +(1 − 𝛽)2

24∙

𝛼2

𝑓𝐾 1−𝛽+

1

4

𝜌𝛽𝜈𝛼

𝑓𝐾1−𝛽2

+2 − 3𝜌2

24𝜈2 ∙ 𝑡𝑒𝑥 + ⋯

where

𝑧 =𝜈

𝛼(𝑓𝐾)(1−𝛽)/2log

𝑓

𝐾

𝑥(𝑧) = 𝑙𝑜𝑔1−2𝜌𝑧+𝑧2+𝑧−𝜌

1−𝜌

𝑓 : current forward price

𝐾 : strike price of the European option

Building and using the SABR Model

Typically the 𝜎𝐴𝑇𝑀 and 𝑓 are updated on a several times per day, depending on the liquidity and volatility of the market, while 𝜌 and 𝜈 on the other hand are updated less frequently. 𝛽 stays constant over several months, however as expiry comes closer it needs to be updated.

The SABR model is being built by the following procedure

1. Upfront fitting the parameter 𝛽

2. With 𝜎𝐴𝑇𝑀 (and hence also 𝛼) and 𝛽 given, the parameters 𝜌 and 𝜈 are fitted using nonlinear regression

3. Hence the volatility curve 𝜎𝐵 𝑓, 𝐾; 𝛼, 𝛽, 𝜌, 𝜈 is defined

4. By changing forward prices 𝑓 and ATM-volatilities 𝜎𝐴𝑇𝑀, the SABR model defines the new volatility curve instantly through 𝜎𝐵 𝑓, 𝐾; 𝛼, 𝛽, 𝜌, 𝜈

Building the SABR model for German power forwards – fitting 𝛽

3.2 3.25 3.3 3.35 3.4-2.8

-2.7

-2.6

-2.5

-2.4

-2.3

-2.2

-2.1

-2

-1.9Fitting beta = 1.7793.

Log(forward)

Log(S

igm

aA

TM

)

Building the SABR model for German power Forwards –

fitting the remaining parameters 𝛼, 𝜌, 𝜈

20 21 22 23 24 25 26 27 28 29 306

7

8

9

10

11

12

13

14SABR fit on the 11-Sep-2002 with forward price 23.95 and ATM Volatility 6.66%

Strike

Vola

tilit

y

Applying the SABR model to German power Forwards –

Evolved SABR-Smile caused by «perturbations» in forward price and

ATM Volatility

20 21 22 23 24 25 26 27 28 29 306

7

8

9

10

11

12

13

14Evolved SABR-Smile on the 22-Sep-2002 with "perturbed" forward price 24.60 and "perturbed" ATM Volatility 7.70%

Strike

Vola

tility

Applying the SABR model to German power Forwards –

Smile evolution with multiple SABR-fits (constant 𝛽; refitting 𝛼, 𝜌, 𝜈)

20 22 24 26 28 30 32 34 36 385

10

15

20

25

30

35

40

SABR fit on the 11-Sep-2002 with forward price 23.95 and ATM Volatility 6.66%

SABR fit on the 20-Jun-2003 with forward price 27.65 and ATM Volatility 10.03%

SABR fit on the 24-Nov-2003 with forward price 33.40 and ATM Volatility 32.39%

Strike

Vola

tilit

y

SABR Sensitivities

The functional parametric form of the the SABR model under Black’s valuation formula implies different greeks compared to the standard greeks:

𝑉𝑜𝑝𝑡𝑖𝑜𝑛 = 𝐵𝑆(𝑓, 𝐾, 𝜎𝐵 𝐾, 𝑓 , 𝑡𝑒𝑥)

with 𝜎𝐵 𝐾, 𝑓 = 𝜎𝐵 𝐾, 𝑓; 𝛼, 𝛽, 𝜌, 𝜈 .

The Vega risk [1]:𝜕𝑉𝑜𝑝𝑡𝑖𝑜𝑛

𝜕𝛼=

𝜕𝐵𝑆

𝜕𝜎𝐵∙𝜕𝜎𝐵(𝐾, 𝑓; 𝛼, 𝛽, 𝜌, 𝜈)

𝜕𝛼

The delta risk [1]:𝜕𝑉𝑜𝑝𝑡𝑖𝑜𝑛

𝜕𝑓=

𝜕𝐵𝑆

𝜕𝑓+

𝜕𝐵𝑆

𝜕𝑓∙𝜕𝜎𝐵(𝐾, 𝑓; 𝛼, 𝛽, 𝜌, 𝜈)

𝜕𝑓

Under a normal risk management (VaR) time horizon 𝛽 will be treated as a given constant, and normally this is also a valid assumption for 𝜌 and 𝜈. However for simulations with longer time horizons changes in 𝜌 and 𝜈 have to be incorporated in the risk model and simulations as vanna (skew risk) and volga (vol-gamma) risks/sensitivities [1]:

𝑣𝑎𝑛𝑛𝑎 =𝜕𝑉𝑜𝑝𝑡𝑖𝑜𝑛

𝜕𝜌=

𝜕𝐵𝑆

𝜕𝜎𝐵∙𝜕𝜎𝐵(𝐾, 𝑓; 𝛼, 𝛽, 𝜌, 𝜈)

𝜕𝜌

𝑣𝑜𝑙𝑔𝑎 =𝜕𝑉𝑜𝑝𝑡𝑖𝑜𝑛

𝜕𝜈=

𝜕𝐵𝑆

𝜕𝜎𝐵∙𝜕𝜎𝐵 𝐾, 𝑓; 𝛼, 𝛽, 𝜌, 𝜈

𝜕𝜈

Using SABR in VaR simulations and calculations

For normal VaR time horizons keep 𝛽, 𝜌 and 𝜈 constant.

1. Simulate the forward price according to the SABR stochastic forward price process

2. Simulate the ATM Volatility according to the correlated SABR stochastic volatility process, hence 𝛼 is given

3. Use the closed, parametric-form formula to calculate volatilities along the volatility curve

4. Use Black’s model with the simulated volatilites as input to generate premium valuations

Hence on this basis consistent and realistic VaR simulations incorporating both forward and volatility surface risks can be generated.

Due to the closed-form of the SABR model the VaR can also be calculated analytically to the same accuracy order as the model specification, 𝒪(𝜀2).

Summary

The SABR model provides a high degree of probabilistic consistency with the real distribution of the underlying forward contract as the SABR implied volatility surface fits the market’s implied volatility surface relatively well

The SABR model correctly replicates the market dynamics of the volatility curve (“smile risk”)

The SABR model provides a consistent framework upon which some exotic options may be priced

The SABR model enables traders and risk managers to consistently evaluate and simulate all sensitivities of a standard (European) option book on any aggregation level

The SABR model is fast (closed-form), flexible and adaptive to a fast-paced market situation, yet at the same time perfectly suitable for risk management purposes; hence it promotes model continuity from the trading floor into the core of risk management!

However, the SABR model has an important limitation:

The SABR model does not guarantee the global absence of arbitrage in it’s local volatility surface solutions; hence it does not provide a global solution for the global implied volatility surface!