Option pricing under finite moment log stable process in a regulated market: A generalized fractional path integral formulation and Monte Carlo based simulation

Abstract

In contrast to a non-regulated market, a regulated market can be defined as a market affected by external factors, which cause abnormal behaviors in market prices. Nevertheless, these behaviors are not enough to ignore the fundamental principles of finance, while many econophysicists do so. In this paper, it is considered that returns are driven by a finite moment log stable process in the most general form. Then a potential function is used to model the rest of the regulations. Consequently, the pricing problem is formulated as an integral whose kernel can be found solving an inhomogeneous space-fractional diffusion equation. Given the inhomogeneous equation with the Riesz-Feller fractional derivative and the potential function, a new path integral seems to be necessary to formulate the solution of the kernel equation. Thus, a Generalized Fractional Path Integral will be derived, and an Asymmetric Fractional Path Integral Monte Carlo algorithm will be developed to find the results. Finally, a daily European call option is priced in a real market with a daily price limit rule, and the maximum and minimum price for a typical contract is calculated as some example applications of the proposed approach.