期权期货及其衍生品第19弹.ppt
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 maturityThe volatility smile for European call options should be exactly the same as that for European put optionsThe 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 holds for market prices(pmkt and cmkt)and for Black-Scholes-Merton prices(pbs and cbs)As a result,pmkt pbs=cmkt cbsWhen pbs=pmkt,it must be true that cbs=cmktIt follows that the implied volatility calculated from a European call option should be the same as that calculated from a European put 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,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 distributionIt is also“more peaked”than the lognormal distribution,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 changesVolatility of exchange rate is stochastic,Options,Futures,and Other Derivatives,8th 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 John 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.Hull 2012,11,The left tail is heavier than the lognormal distributionThe right tail is less heavy than the lognormal distribution,Reasons for Smile in Equity Options,LeverageCrashophobia,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,12,Other Volatility Smiles?,What is the volatility smile ifTrue distribution has a less heavy left tail and heavier right tailTrue 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 Smiles,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,14,Plot implied volatility against K/S0Plot 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 S0Plot 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 volatility term structureThis shows the variation of implied volatility with the time to maturity of the optionThe 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.Hull 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,and 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 isIs the delta higher or lower than for equities?,Options,Futures,and Other Derivatives,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 Derivatives,8th Edition,Copyright John C.Hull 2012,20,A Geometric Interpretation(Figure 19A.1,page 425),Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,21,Assuming that density is g(K)from Kd to K+d,c1+c3 c2=erT d2 g(K),
