《金融期货与期权》（英文版） Chapter 15 Volatility smiles

15.1 Volatility smiles Chapter 15 Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 15.1 Volatility Smiles Chapter 15

152 Put-Call Parity arguments Put-call parity p+soe-ql=c+xer/ holds regardless of the assumptions made about the stock price distribution ·| t follows that pmkt-pbs -Cmkt-Cbs Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 15.2 Put-Call Parity Arguments • Put-call parity p +S0 e -qT = c +X e–r T holds regardless of the assumptions made about the stock price distribution • It follows that pmkt-pbs=cmkt-cbs

153 Implied volatilities 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 The same is approximately true of American options Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 15.3 Implied Volatilities • 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 • The same is approximately true of American options

15.4 Volatility smile A volatility smile shows the variation of the implied volatility with the strike price The volatility smile should be the same Whether calculated from call options or put options Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 15.4 Volatility Smile • A volatility smile shows the variation of the implied volatility with the strike price • The volatility smile should be the same whether calculated from call options or put options

15.5 The volatility Smile for Foreign Currency Options (Figure 15.1, page 332) Implied Volatility Strike Price Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 15.5 The Volatility Smile for Foreign Currency Options (Figure 15.1, page 332) Implied Volatility Strike Price

156 Implied distribution for Foreign Currency Options The implied distribution is as shown in Figure 15.2, page 332 Both tails are heavier than the lognormal distribution It is also" more peaked than the lognormal distribution Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 15.6 Implied Distribution for Foreign Currency Options • The implied distribution is as shown in Figure 15.2, page 332 • Both tails are heavier than the lognormal distribution • It is also “more peaked than the lognormal distribution

The Volatility Smile for Equity 5.7 Options(Figure 15.3, page 334) melle Volatility Strike Price Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 15.7 The Volatility Smile for Equity Options (Figure 15.3, page 334) Implied Volatility Strike Price

158 Implied distribution for equity Options The implied distribution is as shown in Figure 15. 4, page 335 The right tail is less heavy and the left tail is heavier than the lognormal distribution Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 15.8 Implied Distribution for Equity Options • The implied distribution is as shown in Figure 15.4, page 335 • The right tail is less heavy and the left tail is heavier than the lognormal distribution

15.9 Other Volatility smiles What 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, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 15.9 Other Volatility Smiles? What 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

15.10 Possible causes of volatility Smile Asset price exhibiting jumps rather than continuous change Volatility for asset price being stochastic (One reason for a stochastic volatility in the case of equities is the relationship between volatility and leverage) Options, Futures, and other Derivatives, 5th edition 2002 by John C. Hull

Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 15.10 Possible Causes of Volatility Smile • Asset price exhibiting jumps rather than continuous change • Volatility for asset price being stochastic (One reason for a stochastic volatility in the case of equities is the relationship between volatility and leverage)