Volatility Smiles

Fundamentals of Futures and Options Markets, 8th Ed, Ch 19, Copyright John C. Hull 2013Volatility SmilesChapter 1911Fundamentals of Futures and Options Markets, 8th Ed, Ch 19, Copyright John C. Hull 2013Volatility SmileA volatility smile shows, for options with a certain maturity, the variation of the implied volatility with the strike price The volatility smile is the same whether calculated from European call options or European put options. (This follows from put-call parity.) It is also approximately the same when calculated from American options

22Fundamentals of Futures and Options Markets, 8th Ed, Ch 19, Copyright John C. Hull 2013The Volatility Smile for Foreign Currency Options (Figure 19.1, page 419)ImpliedVolatilityStrikePrice33Fundamentals of Futures and Options Markets, 8th Ed, Ch 19, Copyright John C. Hull 2013Implied Distribution for Foreign Currency Options

44Fundamentals of Futures and Options Markets, 8th Ed, Ch 19, Copyright John C. Hull 2013Properties of Implied Distribution for Foreign Currency OptionsBoth tails are heavier than the lognormal distributionIt is also more peaked than the normal distribution 55Fundamentals of Futures and Options Markets, 8th Ed, Ch 19, Copyright John C. Hull 2013Possible Causes of Volatility Smile for Foreign CurrenciesExchange rate exhibits jumps rather than continuous changesVolatility of exchange rate is stochastic 66Historical Analysis of Exchange Rate Changes (Table 19.2, page 420)Fundamentals of Futures and Options Markets, 8th Ed, Ch 19, Copyright John C. Hull 2013Real World (%)Normal Model (%)>1 SD25.0431.73>2SD5.274.55>3SD1.340.27>4SD0.290.01>5SD0.080.00>6SD0.030.0077Fundamentals of Futures and Options Markets, 8th Ed, Ch 19, Copyright John C. Hull 2013The Volatility Smile for Equity Options (Figure 19.3, page 422)ImpliedVolatilityStrikePrice88Fundamentals of Futures and Options Markets, 8th Ed, Ch 19, Copyright John C. Hull 2013Implied Distribution for Equity Options

99Properties of Implied Distribution for Equity OptionsFundamentals of Futures and Options Markets, 8th Ed, Ch 19, Copyright John C. Hull 201310The left tail is heavier than the lognormal distributionThe right tail is less heavy than the lognormal distribution

10Reasons for Smile in Equity OptionsLeverageCrashophobiaFundamentals of Futures and Options Markets, 8th Ed, Ch 19, Copyright John C. Hull 20131111Fundamentals of Futures and Options Markets, 8th Ed, Ch 19, Copyright John C. Hull 2013Other 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 tail1212Ways of Characterizing the Volatility SmilesFundamentals of Futures and Options Markets, 8th Ed, Ch 19, Copyright John C. Hull 201313Plot 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

13Volatility Term StructureIn 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 lowFundamentals of Futures and Options Markets, 8th Ed, Ch 19, Copyright John C. Hull 20131414Fundamentals of Futures and Options Markets, 8th Ed, Ch 19, Copyright John C. Hull 2013Example of a Volatility Surface(Table 19.2, page 424)

1515The Impact of a Large Jump (pages 425 to 426)At the money implied volatilities are higher that in-the-money or out-of-the-money options (so that the smile is a frown!)Fundamentals of Futures and Options Markets, 8th Ed, Ch 19, Copyright John C. Hull 20131616Strike Price0.900.951.001.051.101 mnth14.213.012.013.114.53 mnth14.013.012.013.114.26 mnth14.113.312.513.414.31 year14.714.013.514.014.82 year15.014.414.014.515.15 year14.814.614.414.715.0