1. Chapter 23 Volatility

2. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-2 Introduction Implied volatility Volatility estimation Volatility and variance swaps Option pricing under stochastic volatility

3. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-3 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

4. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-4 Implied Volatility (cont’d) Volatility skew Volatility smirk Volatility smile Implied volatilities are not constant across strike prices and over time

5. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-5 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

6. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-6 Measurement and Behavior of Volatility Stock price process  α: continuously compounded expected return  δ: continuously compounded divided yield  σ(s t, x t, t): instantaneous volatility If we observe a series of stock price every h periods, we can compute continuously compounded return

7. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-7 Historical Volatility We observe n continuously compounded stock return over a period of length T and h=T/n

8. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-8 Historical Volatility (cont’d)

9. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-9 Time Varying Volatility: ARCH Model The ARCH(m) model Autoregressive conditional heteroskedasticity model

10. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-10 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

11. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-11 The Garch(m,n) Model The GARCH model  Generalized ARCH model

12. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-12 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

13. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-13 Estimation of ARCH, GARCH Model Maximum likelihood estimation Volatility forecasts  Conditional expectation of volatility

14. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-14 Variance Swaps A forward contract that pays the difference between a forward price F 0,τ (v 2 ) and some measure of the realized stock price variance, v 2, over a period of time multiplied by a notional amount  N: the notional amount of the contract

15. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-15 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

16. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-16 Extension of the Black-Scholes Model Three extensions  The Merton jump diffusion model  The constant-elasticity of variance model  The stochastic volatility model

17. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-17 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

18. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-18 Merton’s Jump Diffusion Model (cont’d)

19. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-19 Merton’s Jump Diffusion Model (cont’d) Implied volatility computed using the Black-Scholes model when option prices are computed using the jump model

20. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-20 Merton’s Jump Diffusion Model (cont’d)

21. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-21 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

22. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-22 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

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

24. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-24 The CEV Call Price (cont’d) For β>2 the CEV call price is

25. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-25 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

26. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-26 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

27. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-27 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

28. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-28 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

29. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-29 The Heston Stochastic Volatility Model (cont’d) Implied volatility computed using the Black-Scholes model when option prices are computed using the Heston model

30. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-30 The Heston Stochastic Volatility Model (cont’d)

31. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 23-31 The Heston Stochastic Volatility Model (cont’d)