Volatility Smiles and Option Pricing Models YONSEI UNIVERSITY YONSEI UNIVERSITY SCHOOL OF BUSINESS In Joon Kim

1 Black and Scholes model (1/2) n Underlying asset price follows geometric Brownian motion process. Price at time T (Ex: In one month) TransitionDensity Present time t Stochastic Process

2 Black and Scholes model (2/2) n Under the constant interest rate of where : Expectations at time t under the risk-neutral probabilities : European call option value : Stock price at time T : Exercise price n Black and Scholes Model where : cumulative normal distribution

3 Historical Volatility (1/2) n Distribution of TransitionDensity Present time t Stochastic Process

4 Historical Volatility (2/2) n For the lognormal distribution, the maximum likelihood estimator for data spaced at regular time intervals n : weekly data (1/52) / daily data (1/365 or 1/250) n How to choose an appropriate value for ? Interval Volatility

5 Implied Volatility n Assuming the model is correct, we ask the model to tell us about the volatility n The volatility that makes the model price to be equal to the market price. n With more than one data points (n expirations and m exercise prices for each expiration)

6 Validity of Black and Scholes Model n Estimate the volatility and compare the model prices with the market prices n Compare the implied volatility with the realized volatility. n Evaluate the hedging performance of the model

7 Systematic Biases in Black and Scholes model (1/3) n Geometric Brownian motion for the asset price process implies l All options on the same asset should provide the same implied volatility. l Implied volatility: The volatility from Black and Scholes Option pricing formula by inputting the quoted price for a single option as the price of option. n Empirical evidence indicates that implied volatilities computed from the Black and Scholes formula tend to differ across exercise prices and times to expiration. l Rubinstein (1994 : JF) : S&P 500 l Taylor and Xu (1993 : RFM) : Philadelphia Exchange foreign currency option market l Duque and Paxson (1993 : RFM) : London International Financial Futures Exchange l Heynen (1993 : RFM) : European Options Exchange

8 Systematic Biases in Black and Scholes model (2/3) n Prior to October 1987, “smile” => After crash, “sneer” (“smirk”) n Smiles for foreign currency options and smirks for equity options IV K/S IV K/S “Smile” “Sneer”

9 n Implied volatilities tend to decrease as time to expiration increases n Volatility surface Systematic Biases in Black and Scholes model (3/3) IV Time to Expiration

10 Implied Distribution for Foreign Currency Options n Both tails are heavier than the lognormal distribution It is also “ more peaked than the lognormal distribution

11 Implied Distribution for Equity Options n The left tail is heavier and the right tail is less heavy than the lognormal distribution

12 KOSPI200 Options ( 3 월 13 일 종가, 4 월물, 잔존일수 27 일 )

13 KOSPI200 Options ( 3 월 13 일 종가, 5 월물, 잔존일수 63 일 )

14 KOSPI200 Options ( 3 월 27 일 종가, 4 월물, 잔존일수 14 일 )

15 KOSPI200 Options ( 3 월 27 일 종가, 5 월물, 잔존일수 49 일 )

16 Arbitrage Bounds for Options Lower Bound for calls: Lower Bound for puts

17 Explosion of Alternative Models (1/5) n Alternative price processes l Jump-diffusion/pure jump u Merton (1976 : JFE) / Jones (1984 : JFE) [Ex] Merton: : Poisson Process with parameter u Naik and Lee (1990 : RFS) / Bates (1991 : JF) l Constant elasticity of variance u Cox and Ross (1976 : JFE) / MacBeth and Merville (1980 : JF) u Emmanuel and MacBeth (1982 : JFQA) [Ex] Emmanuel and MacBeth: l Implied stochastic processes u Rubinstein (1994 : JF) / Dupire (1994 : Risk) u Derman and Kani (1994 : Risk) u Ait-Sahalia and Lo (1998 : JF)

18 Explosion of Alternative Models (2/5) l Stochastic Volatility u Heston (1993 : RFS) / Hull and White (1987 : JF) [Ex] Heston: u Melino and Turnbull (1990 : JE, 1995 : JIMF) u Scott (1987 : JFQA) / Stein and Stein (1991 : RFS) u Wiggins (1987 : JFE) l Stochastic volatility and jump diffusion u Bates (1996 : RFS) / Scott (1997 : MF) n Stochastic interest rates l Merton (1973 : BJE), Ravinovitch (1989 : JFQA) l Amin and Jarrow (1992 : MF) l Kim (1992 : KJFM) : Instantaneous volatility of excess return

19 Explosion of Alternative Models (3/5) n Hybrid Model l Stochastic volatility and stochastic interest rates u Amin and Ng (1993 : JF) u Bailey and Stulz (1989 : JFQA) u Bakshi and Chen (1997a : JFE, b : JF) l Stochastic volatility, jump diffusion and stochastic interest rates u Scott (1997 : MF) [Ex] Scott: : A martingale process containing Poisson jumps u Bakshi, Cao and Chen (1997 : JF)

20 Explosion of Alternative Models (4/5) n Estimation of parameters (implicit estimation) n Relative valuation vs. Absolute valuation n No extended model consistently outperforms the Black-Scholes Model.

21 Explosion of Alternative Models (5/5) n It is difficult to estimate parameters of the extended models from time series price data of the underlying assets. n Modifying the transition density function cannot improve the empirical performance of option pricing models. n We might need to investigate something else to understand the discrepancy between the Black-Scholes model and the observed option prices.

22 Skewed Pricing Errors (1/3) n Kim, Kim and Ziskind (1994 : AIAPM) n Observed price = “true price” + pricing errors n Pricing errors = l : Market friction and other factors of market imperfection l : Pricing convention ( tick ) l : Lower arbitrage bound to preclude arbitrage S K OTM ITM Bound: Max{0, S - K exp(-rT)}

23 Skewed Pricing Errors (2/3) n The third component of pricing errors behaves like the volatility smile. l For deep OTM calls, 3rd component of error is skewed and positive because the true call values are small and the observed call values must be positive. l For deep ITM options, 3rd component of error is skewed and positive because the true option values should be greater than the lower bound. l 1st and 2nd component of error is symmetric.

24 Skewed Pricing Errors (3/3) n The empirical biases could have been maginfied by the differential sensitivity of volatility to the pricing errors. l It is subject to how much pricing errors affect implied volatilities S/K

25 Implications of Pricing Errors n We need to understand the behavior of the skewed pricing errors in detail. n We need to eliminate the skewed pricing errors before considering alternative models to extend and generalize the Black-Scholes model.

Q & A In Joon Kim YONSEI UNIVERSITY SCHOOL OF BUSINESS