The volatility smile

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  • On Wall Street, they were all known as "quants," traders and financial engineers who used brain-twisting math and superpowered computers to pluck billions in fleeting dollars out of the market. Instead of looking at individual companies and their performance, management and competitors, they use math formulas to make bets on which stocks were going up or down.

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  • It is a pleasure to edit the second volume of papers presented at the Mathematical Finance Seminar of New York University. These articles, written by some of the leading experts in financial modeling cover a variety of topics in this field. The volume is divided into three parts: (I) Estimation and Data-Driven Models, (II) Model Calibration and Option Volatility and (III) Pricing and Hedging. The papers in the section on "Estimation and Data-Driven Models" develop new econometric techniques for finance and, in some cases, apply them to derivatives.

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  • To implement this method a problem is that the observed option prices do not provide a continuous range, so that the resulting RND is not a well-behaved function. We overcome this problem by using the smoothed volatility smile. From the observed option prices, the implied volatilities are extracted by means of the Black-Scholes pricing function. To obtain a smoothed volatility smile we then transform our data set of implied volatilities from the volatility/strike space to the volatility/delta space.

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  • This property is required because the empirically observed densities of returns contrast with the Gaussian model [see Pagan 1996]. This rejection results from two stylised facts. First, large price changes appear more frequently than the normal density would lead to expect. Second, there are indications of significant asymmetry in stock returns. In other words, negative and positive price changes do not have the same probability. These two stylised facts are also apparent in implied volatilities.

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