On the relevance of jumps for the pricing of S&P 500 options : with particular emphasis on the adjustment for systematic risk in jump-diffusion models

Abstract

Jump-diffusions are a class of models that is used to model the price dynamics of assets whose value exhibit jumps. The first part of this thesis discusses the implications of such models for the pricing of derivatives. Particular emphasis is put on explaining the adjustment for systematic risk. Efforts are made to link purely mathematical arguments with economic theory and intuitive explanations. In the second part, the theoretical framework for derivatives pricing are applied to answer the question whether jumps are relevant for the pricing of European options with the S&P 500 index as the underlying asset. Analysis of the distributional properties of log-returns leads to the suggestion of a specific jump-diffusion model for the dynamics of this index. The model is calibrated to market data on a daily basis for a period of 80 trading days prior to and 80 trading days after what is considered the outbreak of the financial crisis of 2008. Obtained values of the jump-diffusion parameters implicit in option prices establish that jumps are relevant for their value.