Ch19 波动率微笑(Volatility Smiles)曲线(Hull)

Options, Futures, and Other Derivatives, 8thlity Smile is the
Same for European Calls and Put
Put-call parity p + S0e−qT = c +K e–rT holds 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 put option when both have the same strike price and maturity
Options, Futures, and Other Derivatives, 8th Edition,
Copyright © John C. Hull 2012
4
Implied Distribution for Foreign Currency Options
Lognormal Implied
Options, Futures, and Other Derivatives, 8th Edition,
Copyright © John C. Hull 2012
7
Historical Analysis of Exchange Rate Changes
>1 SD >2SD >3SD >4SD >5SD >6SD
The left tail is heavier than the lognormal distribution The right tail is less heavy than the lognormal distribution
8
The Volatility Smile for Equity Options (Figure 19.3, page 414)
Implied Volatility
Strike Price
Options, Futures, and Other Derivatives, 8th Edition,
Copyright © John C. Hull 2012
9
Implied Distribution for Equity Options
Lognormal Implied
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
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 distribution
Chapter 19 Volatility Smiles
Options, Futures, and Other Derivatives, 8th Edition,
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 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
6
Possible Causes of Volatility Smile for Foreign Currencies
Exchange rate exhibits jumps rather than continuous changes Volatility of exchange rate is stochastic
Real World (%) Normal Model (%)
25.04
31.73
5.27
4.55
1.34
0.27
0.29
0.01
0.08
0.00
0.03
0.00
Options, Futures, and Other Derivatives, 8th Edition,
Copyright © John C. Hull 2012
Options, Futures, and Other Derivatives, 8th Edition,
3
The Volatility Smile for Foreign Currency Options
(Figure 19.1, page 411)
Implied Volatility
Strike Price