期權(quán)期貨及其衍生品第19彈.ppt

1、Chapter 19Volatility Smiles,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,1,What is a Volatility Smile?,It is the relationship between implied volatility and strike price for options with a certain maturity The volatility smile for European call options should be

2、exactly the same as that for European put options The same is at least approximately true for American options,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,2,Why the Volatility Smile is the Same for European Calls and Put,Put-call parity p + S0eqT = c +K erT hold

3、s for market prices (pmkt and cmkt) and for Black-Scholes-Merton prices (pbs and cbs) As a result, pmkt pbs=cmkt cbs When pbs = pmkt, it must be true that cbs = cmkt It follows that the implied volatility calculated from a European call option should be the same as that calculated from a European pu

4、t option when both have the same strike price and maturity,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,3,The Volatility Smile for Foreign Currency Options (Figure 19.1, page 411),Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,4

5、,Implied Distribution for Foreign Currency Options,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,5,Properties of Implied Distribution for Foreign Currency Options,Both tails are heavier than the lognormal distribution It is also “more peaked” than the lognormal di

6、stribution,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,6,Possible Causes of Volatility Smile for Foreign Currencies,Exchange rate exhibits jumps rather than continuous changes Volatility of exchange rate is stochastic,Options, Futures, and Other Derivatives, 8th

7、 Edition, Copyright John C. Hull 2012,7,Historical Analysis of Exchange Rate Changes,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,8,The Volatility Smile for Equity Options (Figure 19.3, page 414),Options, Futures, and Other Derivatives, 8th Edition, Copyright Joh

8、n C. Hull 2012,9,Implied,Volatility,Strike,Price,Implied Distribution for Equity Options,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,10,Properties of Implied Distribution for Equity Options,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C.

9、Hull 2012,11,The left tail is heavier than the lognormal distribution The right tail is less heavy than the lognormal distribution,Reasons for Smile in Equity Options,Leverage Crashophobia,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,12,Other Volatility Smiles?,W

10、hat is the volatility smile if True distribution has a less heavy left tail and heavier right tail True distribution has both a less heavy left tail and a less heavy right tail,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,13,Ways of Characterizing the Volatility

11、Smiles,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,14,Plot implied volatility against K/S0 Plot implied volatility against K/F0 Note: traders frequently define an option as at-the-money when K equals the forward price, F0, not when it equals the spot price S0 Pl

12、ot implied volatility against delta of the option Note: traders sometimes define at-the money as a call with a delta of 0.5 or a put with a delta of 0.5. These are referred to as “50-delta options”,Volatility Term Structure,In addition to calculating a volatility smile, traders also calculate a vola

13、tility term structure This shows the variation of implied volatility with the time to maturity of the option The volatility term structure tends to be downward sloping when volatility is high and upward sloping when it is low,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hu

14、ll 2012,15,Volatility Surface,Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,16,The implied volatility as a function of the strike price and time to maturity is known as a volatility surface,Example of a Volatility Surface(Table 19.2, page 417),Options, Futures, an

15、d Other Derivatives, 8th Edition, Copyright John C. Hull 2012,17,Greek Letters,If the Black-Scholes price, cBS is expressed as a function of the stock price, S, and the implied volatility, simp, the delta of a call is Is the delta higher or lower than for equities?,Options, Futures, and Other Deriva

16、tives, 8th Edition, Copyright John C. Hull 2012,18,Volatility Smiles When a Large Jump is Expected (pages 418 to 420),At the money implied volatilities are higher that in-the-money or out-of-the-money options (so that the smile is a frown!),Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012,19,Determining the Implied Distribution (Appendix to Chapter 19),Options, Futures, and Other Deriv