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Trading with a Smile

The Black Scholes Option pricing methodology though a classic in modern finance, hinges on several key assumptions that have been shown to be very frequently violated. The specific parameter of the model that is being considered for the present paper is volatility more specifically at the phenomenon commonly known to as the ‘volatility smile’. This refers to the shape of the curve of implied volatilities of traded options plotted against the option strike prices. Building upon the ‘smile’ we argue that the substantial difference in the implied volatilities of In the Money call options and At the Money call options cannot be justified. The aim is to show that the implied volatilities of In the Money options and At the Money options follow certain distributions, which are not only substantially different from each other, but that only one of them may even seek to approximate the distribution followed by the underlying asset volatility. We hypothesize that the implied volatility distribution of In the Money options would lie substantially to the right of the underlying asset volatility distribution, thus represent a systemic mispricing and existence of a profitable trading opportunity. To do this, we take a set of option prices for In the money call options over a three month period and calculate their implied volatility. Similarly we calculate the implied volatility for a set of at-the-money call options over the same period of time. Using the software, Bestfit, we then extrapolate a volatility-strike price curve. Then we perform the chi-square test of significance for the statistical difference in the two curves to verify the above hypothesis.