Pricing European options under uncertainty with application of Levy processes and the minimal Lq equivalent martingale measure

Abstract

The study involves pricing of European options with usage of stochastic and fuzzy methods. We use a geometric Levy process as the model of the underlying asset, assuming that the log-price of a primary financial instrument is a jump–diffusion with jump part described by a linear combination of time-homogeneous Poisson processes. Analytical option pricing formulas in crisp case, using the minimal L q equivalent martingale measure, are derived. The pricing expressions are achieved employing probability and stochastic analysis. The fuzzy counterparts of some model parameters are explored due to the fact that they are imprecisely evaluated. Applying fuzzy arithmetic, we derive the analytical option pricing expressions with fuzzy parameters. Moreover, we conceptualize a method of decision-making, taking into consideration the obtained fuzzy formulas. At last, we go through numerical examples to illustrate our theoretical results. Our main achievement in the paper is overcoming difficulties related to derivation of the option pricing formulas by advanced analysis of Jacod–Grigelionis characteristics of the log-price process used in our approach for the minimal L q equivalent martingale measure as well as the skilled use of fuzzy arithmetic methods.