Volatility Smiles

‣ 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