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Chapter 19 Volatility Smiles Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 20121

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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 2

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Why the Volatility Smile is the Same for European Calls and Put Put-call parity p + S 0 e −qT = c +K e –rT holds for market prices ( p mkt and c mkt ) and for Black-Scholes-Merton prices ( p bs and c bs ) As a result, p mkt − p bs =c mkt − c bs When p bs = p mkt, it must be true that c bs = c mkt 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 3

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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 Volatility Strike Price

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Implied Distribution for Foreign Currency Options Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 20125

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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 Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 6

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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 Edition, Copyright © John C. Hull 2012 7

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Historical Analysis of Exchange Rate Changes Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 20128 Real World (%)Normal Model (%) >1 SD25.0431.73 >2SD5.274.55 >3SD1.340.27 >4SD0.290.01 >5SD0.080.00 >6SD0.030.00

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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

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Implied Distribution for Equity Options Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 201210

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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 distribution The right tail is less heavy than the lognormal distribution

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Reasons for Smile in Equity Options Leverage Crashophobia Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 12

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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, 8th Edition, Copyright © John C. Hull 2012 13

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Ways of Characterizing the Volatility Smiles Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012 14 Plot implied volatility against K / S 0 Plot implied volatility against K / F 0 Note: traders frequently define an option as at-the-money when K equals the forward price, F 0, not when it equals the spot price S 0 Plot 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”

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Volatility Term Structure In addition to calculating a volatility smile, traders also calculate a volatility 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. Hull 2012 15

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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

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Example of a Volatility Surface (Table 19.2, page 417) Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 201217

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