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

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

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

The Volatility Smile for Foreign Currency Options (Figure 19.1, page 411) Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull Implied Volatility Strike Price

The Volatility Smile for Foreign Currency Options (Figure 19.1, page 411) Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull If you plot the implied volatilities (IV) against the strike prices, you might get the following U-shaped curve resembling a smile. Hence, this particular volatility skew pattern is better known as the volatility smile. The volatility smile skew pattern is commonly seen in near- term equity options and options in the forex market. Volatility smiles tell us that demand is greater for options that are in-the-money or out-of-the-money

Implied Distribution for Foreign Currency Options Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 20126

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 For log normal distribution: The volatility of the asset is constant The price of the asset changes smoothly with no jumps Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull

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

The Volatility Smile for Equity Options (Figure 19.3, page 414) Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull Implied Volatility Strike Price

A more common skew pattern is the reverse skew or volatility smirk. The reverse skew pattern typically appears for longer term equity options and index options. In the reverse skew pattern, the IV for options at the lower strikes are higher than the IV at higher strikes. The reverse skew pattern suggests that in- the-money calls and out-of-the-money puts are more expensive compared to out-of-the-money calls and in-the-money puts. 10 The Volatility Smile for Equity Options (Figure 19.3, page 414)

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

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

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

Ways of Characterizing the Volatility Smiles Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 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”

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

Volatility Surface Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 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

Greek Letters If the Black-Scholes price, c BS is expressed as a function of the stock price, S, and the implied volatility,  imp, the delta of a call is Is the delta higher or lower than for equities? Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull

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

Determining the Implied Distribution (Appendix to Chapter 19) Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull

A Geometric Interpretation (Figure 19A.1, page 425) Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull Assuming that density is g(K) from K−  to K +  c 1 +c 3 −c 2 = e −rT    g(K)