chapter 23 volatility. copyright © 2006 pearson addison-wesley. all rights reserved. 23-2...
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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
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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
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Implied Volatility (cont’d)
• Volatility skew
• Volatility smirk
• Volatility smile
• Implied volatilities are not constant across strike prices and over time
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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
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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 +−=
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⎥⎦
⎤⎢⎣
⎡− ∑=
=
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
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Historical Volatility (cont’d)
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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
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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
Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-11
∑∑=
−−=
++=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
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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
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Estimation of ARCH, GARCH Model
• Maximum likelihood estimation
• Volatility forecasts
Conditional expectation of volatility
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[ ] 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
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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
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Extension of the Black-Scholes Model
• Three extensions
The Merton jump diffusion model The constant-elasticity of variance model The stochastic volatility model
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%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
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Merton’s Jump Diffusion Model (cont’d)
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Merton’s Jump Diffusion Model (cont’d)
• Implied volatility computed using the Black-Scholes model when option prices are computed using the jump model
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Merton’s Jump Diffusion Model (cont’d)
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Merton’s Jump Diffusion Model (cont’d)
• 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
Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-22
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
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)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
Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-24
)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
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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
Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-26
)(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
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{ } 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
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The Heston Stochastic Volatility Model (cont’d)
• 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
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The Heston Stochastic Volatility Model (cont’d)
• Implied volatility computed using the Black-Scholes model when option prices are computed using the Heston model
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The Heston Stochastic Volatility Model (cont’d)
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The Heston Stochastic Volatility Model (cont’d)
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