Additional Information in Higher Order Derivatives of the Black-Scholes Formula

University essay from Lunds universitet/Nationalekonomiska institutionen

Abstract: Volatility smile arising from the Black-Scholes model is a long studied subject in option pricing theory. By analysing higher order derivatives of the model, I hope to put another perspective to the problem. Accoridng to a study on the American S&P 500 future options, about 90% of the risks of holding an option in five days can be removed by using higher order derivatives of Black-Scholes. The objective of this thesis is to investigate whether or not the price changes of Swedish OMX-index options follow the Taylor expansion of Black-Scholes. The method used here is to compare the observed option price changes after five holding days with the price changes predicted by first, second and third order Taylor serie. The findings and conclusions of this study can be summarised as follow: 1. The mean of prediction errors increases as higher order derivatives are used to predict the price changes, suggesting that the OMX-index options do not follow Taylor expansion of Black-Scholes. 2. As the standard deviation of the prediction errors is decreasing when orders of derivatives in use increase, I conclude that there is additional information in the second and third order derivatives of Black-Scholes. 3. Increasing maturity leads to better predictions, suggesting that the Black-Scholes differential equation is better at describing the movement of long maturity options.

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