A geometric Levy model for [formula omitted]-fold compound option pricing in a fuzzy framework

Abstract

In this paper, we study the problem of n-fold compound option valuation using the martingale method and the theory of fuzzy sets. We adopt a geometric Levy process for modeling the underlying asset price dynamics. The log-price is the sum of a Brownian motion with drift and a Poisson process describing jumps in the price. To obtain the pricing formula for n-fold compound option, we use the Esscher transformed martingale measure as the equivalent martingale measure. The major challenge in deriving the closed-form analytic expression for n-fold compound options is to calculate complex multivariate normal integrals; the complexity is due to the sophisticated structure of the options. In order to overcome this difficulty, we prove a mathematical expectation related to multivariate normal variables. To the best of our knowledge, this report describes the first time that such an approach has been adopted, and it is highly useful in the pricing of n-fold compound options and many other financial derivatives. Because the uncertainty in financial markets simultaneously involves randomness and fuzziness, we assume that the underlying asset price follows a fuzzy stochastic process. Under this assumption, the α-cut of the fuzzy price of n-fold compound call options is derived. Considering a decision maker’s subjective judgment, the permissible range of the expected prices of n-fold compound call options with uncertainty is given. Finally, numerical examples are presented to illustrate and analyze the theoretical results.