Volatility Smiles

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‣ The volatility used to price a low-strike-price option (i.e. adeep our-of-the-money put or a deep in-the-money call) issignificantly higher than that used to price a high strike-priceoption.‣ Once again, we turn to a comparison of the probabilitydistribution of the implied volatility and a lognormal onewith the same mean and standard deviation for an analyticalexplanation.6
‣ To the left (low-strike-price) the implied volatility shows ahigher probability than the lognormal one of equalparameters.‣ This means that these options are priced higher than oneswith larger strike-prices. This is not the case to the right,thus the skew. Reasons Behind the Skew‣ Interestingly, there was no marked volatility skew beforeOctober 1987.‣ It has been suggested that the market crash of ’87 instilledfear into traders who therefore trade lower strike-priceoptions more heavily in order to hedge their positions, thusaffecting the price of these options with their increaseddemand.‣ This explanation has some empirical support. A steepeningin the skew is observed when the S&P 500 (the marketportfolio) declines while a flattening accompanies its rise.‣ This shows that the worse the market is doing, the morepeople trade low-strike-price options in apprehension of amarket crash.‣ Another explanation is that as a company’s equity declines, itbecomes riskier and therefore more volatile.‣ When the opposite happens a company is viewed as safer andtherefore has lower volatility.7
Page 2 and 3: ‣ The differences between these t
Page 4 and 5: Empirical Results‣ We have just s
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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