Application of Levy processes and Esscher transformed martingale measures for option pricing in fuzzy framework

In this paper we consider the European option valuation problem. We assume that the underlying asset follows a geometric Levy process. The log-price is a sum of a Brownian motion with drift and a linear combination of Poisson processes describing jumps in price. In our approach we use martingale method and theory of fuzzy sets. To obtain the European call and put option pricing formulas we use the mean correcting and the Esscher transformed martingale measures. Application of the first mentioned measure required deep analysis of transformation of characteristics of Levy process. We assume that some model parameters cannot be precisely described and therefore we apply fuzzy numbers. Application of fuzzy arithmetic enables us to consider different sources of uncertainty and introduce experts' opinions or imprecise estimates to the model. In contradistinction to our previous papers, where the European call option price at time zero was analysed, we introduce the valuation expressions of the European call and put options for arbitrary time t. Numerical simulations conducted in the paper are used to analyse and illustrate the theoretical results. This numerical approach is based on L-R numbers and the exact shape of the fuzzy numbers which give us the possibility of comparing behaviour of option prices for various values of the parameters of the underlying asset.

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