Option Pricing With Application of Levy Processes and the Minimal Variance Equivalent Martingale Measure Under Uncertainty

This paper is dedicated to European option pricing under assumption that the underlying asset follows a geometric Levy process. The log-price of a primary financial instrument has the form of a sum of a drift component, a Brownian component, and a linear combination of time-homogeneous Poisson processes, modeling jumps in price. In our approach we apply stochastic analysis, especially the change of probability measure techniques, as well as fuzzy sets theory. To obtain the option valuation formulas we use the minimal variance equivalent martingale measure, which requires an advanced analysis of transformation of Levy characteristic triplets. We obtain analytical option valuation expressions in crisp case. Moreover, we assume that some model parameters are described in an imprecise way and therefore we use their fuzzy counterparts. Applying fuzzy arithmetic, we take into account various types of uncertainty on the market. As a result, we obtain the analytical option pricing formulas with fuzzy parameters. We also propose a method of automatized decision making, which utilizes the fuzzy valuation formulas. Apart from the general pricing expressions, we provide numerical examples to illustrate our theoretical results.