chapter 23 volatility. copyright © 2006 pearson addison-wesley. all rights reserved. 23-2...

Chapter 23

Volatility

Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-2

Introduction

• Implied volatility

• Volatility estimation

• Volatility and variance swaps

• Option pricing under stochastic volatility


Implied Volatility

• The volatility is unobservable

• One can use historical stock return data to calculate stock return volatility—sample standard deviation

• One can also use the observed option price and the Black-Scholes model to back out the volatility—implied volatility (IV)

• IV is the volatility implied by the option price observed in the market


Implied Volatility (cont’d)

• Volatility skew

• Volatility smirk

• Volatility smile

• Implied volatilities are not constant across strike prices and over time


Volatility Index –VIX

• Introduced by Professor Bob Whaley at Duke University in 1993

• It provides investors with market estimates of expected volatility

• It is computed by using near-term S&P 100 index options


Measurement and Behavior of Volatility

• Stock price process

α: continuously compounded expected return δ: continuously compounded divided yield σ(st, xt, t): instantaneous volatility

• If we observe a series of stock price every h periods, we can compute continuously compounded return

)/ln( ththt SS ++ =ε

dZtXSdtSdS tttt ),,()(/ σδα +−=


⎥⎦

⎤⎢⎣

⎡− ∑=

=

n

iinH h 1

2

)1(

12 1ˆ εσ

Historical Volatility

• We observe n continuously compounded stock return over a period of length T and h=T/n


Historical Volatility (cont’d)


2

10

12

2

)/(

)5.0()/ln(

it

m

iit

ttt

thtt

aaq

Eq

hSS

−=

−

−

∑+=

=

+−−=

ε

ψε

εσδα

1,0,01

0 <≥> ∑=

m

iii aandaawhere

Time Varying Volatility: ARCH Model

• The ARCH(m) model

• Autoregressive conditional heteroskedasticity model


Time Varying Volatility: ARCH Model (cont’d)

• Intuition ARCH model suggests that the level

of variance depends on recent past level of variance

• Empirical regularity Volatility is highly persistent

•High volatility tends to be followed by high volatility


∑∑=

−−=

++=n

jjtjit

m

iit qbaaq

1

2

10 ε

njbmiaawhere ji ,,1,0,,,1,0,00 LL =≥=≥>

111

<+∑∑==

n

jj

i

m

ii ba

The Garch(m,n) Model

• The GARCH model Generalized ARCH model


The Garch(m,n) Model (cont’d)

• Intuition Volatility at a point in time depends on

recent volatility and recent squared returns

• Special case GARCH (1,1) model


Estimation of ARCH, GARCH Model

• Maximum likelihood estimation

• Volatility forecasts

Conditional expectation of volatility


[ ] NvFpayoff T ×−= )(ˆ 2,0

2ν

Variance Swaps

• A forward contract that pays the difference between a forward price F0,τ(v2) and some measure of the realized stock price variance, v2, over a period of time multiplied by a notional amount

N: the notional amount of the contract


Variance Swaps (cont’d)

• Measurement issues

How frequently the return is measured Whether returns are continuously compounded

or arithmetic Whether the variance is measured by subtracting

the mean or by simply squaring the returns The period of time over which variance is measured How to handle days on which trading does

not occur


Extension of the Black-Scholes Model

• Three extensions

The Merton jump diffusion model The constant-elasticity of variance model The stochastic volatility model


%8%,30,40$,40$ ==== rKS σ

Merton’s Jump Diffusion Model

• The impact of jump on option prices

• Example T-t=0.25 year λ=0.5% probability per year Call price=$2.81, put price=$2.02

• Without jump Call price=$2.78, put price=$1.99


Merton’s Jump Diffusion Model (cont’d)



• Implied volatility computed using the Black-Scholes model when option prices are computed using the jump model



• If prices of options properly account for the jump

• Yet we use the Black-Scholes model to back out option implied volatility

• Then out-of-the money puts have higher implied volatility than at-the-money ones

• In-the-money calls have higher implied volatility than at-the-money ones


dZSSddS t2/)( βσδα +−=

2/)2()( −= βσσ SS

Constant Elasticity of Variance Model

• Cox (1975) proposed the constant elasticity of variance (CEV) model

• Volatility varies with the level of the stock price

• The instantaneous standard deviation of the stock return

If β<2 volatility decreases with the stock price If β>2 volatility increases with the stock price If β=2 the CEV model reduces to the lognormal process


)2,2

2,2()2,

2

22,2(1 yxQKexyQSe rTT

ββδ

−−⎥⎦

⎤⎢⎣

⎡−

+− −−

)1)(2(

)(2)2)((2 −−

−= −− Tre

rkwhere βδβσ

δ

Treksx )2)((2 βδβ −−−=β−= 2kKy

The CEV Call Price

• For the case β<2, the CEV call price is

• Q(a,b,c) denotes the noncentral Chi-squared distribution function will b degrees of freedom and noncentrality parameter c, evaluated at a


)2,2

22,2()2,

2

2,2(1 xyQKeyxQSe rTT

−+−⎥

⎦

⎤⎢⎣

⎡−

− −−

ββδ

The CEV Call Price (cont’d)

• For β>2 the CEV call price is


Implied Volatility in the CEV Model

• When β<2, the CEV model generates a Black-Scholes implied volatility skew

• Implied volatility decreases with the option strike price


)(tv

dtdZdZE ρ=)( 21

2

1

)())(()(

)()(

dZtvdttvvktdv

dZStvSdtdS

vσ

δα

+−=

⋅⋅+−=

The Heston Stochastic Volatility Model

• The model allows volatility to vary stochastically but still to be correlated with the stock price

when is the volatility


{ } rVVVtvtvvSVr

SVtvVtvVStv

tvvS

SvvvvSS

=+−−+−+

++

βκδ

σρσ

)()]([)(

)()(2

1)(

2

1 22

The Heston Stochastic Volatility Model (cont’d)

• Assuming v(t), the instantaneous stock return variance follows an Itô process, the multivariate Black-Scholes partial differential equation is

• This equation has an integral solution that can be solved numerically



• Heston’s stochastic volatility model offers a closed-form solution for option prices

• Empirical test of Heston’s model

• Bakshi, Cao and Chen (1997, Journal of Finance) They find that the feature of stochastic volatility

can lead to 80% reduction in Black-Scholes model pricing error

The stochastic volatility is of first order importance in comparison to the jump feature



• Implied volatility computed using the Black-Scholes model when option prices are computed using the Heston model

chapter 23 volatility. copyright © 2006 pearson addison-wesley. all rights reserved. 23-2...

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