Volatility Smiles

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Empirical Results‣ We have just shown that the smile used by traders for foreigncurrency options implies that they consider that thelognormal distribution understates the probability of extrememovements in exchange rates, but is this accurate?‣ A study of 12 different exchange rates over a 10-year periodwas conducted.‣ The first step was to calculate the standard deviation of dailypercentage change in each exchange rate. Then to note howoften the actual percentage of change exceeded one standarddeviation, two, three…etc. The final step was to calculatehow often this would have happened if the percentagechanges had been normally distributed. The lognormalmodel implies that percentage changes are almost exactlynormally distributed over a one day time period.‣ The table provides evidence to support the existence of heavytails and the volatility smile used by traders.4
‣ Note: In the mid-80’s a few traders knew about the heavytails of foreign exchange probability distributions. Theybought deep out-of-the-money call and put options on avariety of different currencies and proceeded to make lots ofmoney. This is because the lognormal model that everyoneelse accepted underpriced those options and more of themended up in-the-money than it expected. By the late-80’s,traders realized that foreign currency should be priced with avolatility smile and the trading opportunity disappeared. Reasons for the Smile in Foreign Currency Options‣ For an asset to have a price with a lognormal distribution twoconditions must be met.‣ Its volatility must be constant and the price of the assetshould change smoothly with no jumps. Neither of these issatisfied with exchange rates.‣ The percentage impact of jumps on both prices and volatilitysmiles becomes less pronounced as the maturity of the optionis increased.‣ This is because the jumps tend to get averaged out and so thejumps appear to be smoother. This makes the derivativeconform to the assumed lognormality more readily. Equity Options‣ Equity (stock) options exhibit a volatility skew5
Page 2 and 3: ‣ The differences between these t
Page 6 and 7: ‣ The volatility used to price a
Page 8 and 9: Equity Options when Expecting a Lar
Page 10 and 11: ‣ The implied volatility of a Eur
Page 12 and 13: ‣ If there is a large one-day ret
Page 14: ‣ Here there is a weighting assig

Volatility Smiles

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