A Regime Switching Jump Diffusion Model For Option Pricing Under Fuzzy Environments

The Black-Scholes model cannot incorporate the two stylized empirical features: the asymmetric leptokurtic features and the volatility smile. Consequently, many alternative approaches have been proposed in order to capture the dynamics of financial returns. In addition to their own factors of risk assets, the market environment is a key factor which can govern the price movements of individual risky assets.The Markov-modulated regime switching model can provide a better way to describe and explain the market environment. The actual data show that regime-switching models can capture stochastic volatility and, hence, fat tails. Due to lack of knowledge of the financial market, this paper presents a fuzzy jump-diffusion model for the European option pricing, which is a reasonable extension of regime switching model. In this paper, we mainly investigate the option pricing when the underlying risky assets are governed by a Markov-modulated jump diffusion model or a fuzzy jump diffusion model. That is to say, assume that in a given market economy state, the underlying assets are characterized by the jump diffusion model or fuzzy jump diffusion model. When the market economy status changes, the process of asset switches among a number of states in the corresponding model. Based on the analysis above, the paper studied option pricing as follows:(1) The pricing of European option under the regime switching jump diffusion model was studied. Firstly, we use It? formula to prove that when the prices of risky assets subject to regime switching jump diffusion process, its logarithm is additive. And then we derived European option pricing formula by Girsanov theorem and risk-neutral pricing principle. Next,we extent our work to more than two state. Finally, BS model, Merton model and regime switching jump diffusion were compared. And we analyzed the influence of the initial state of the economy, Poisson intensity, sequence of jump amplitude, and Markov generation matrix on option value though numerical analysis.(2) The pricing of European option under the regime switching fuzzy jump diffusion model was studied. The pricing formula was obtained through a series of derivation when the sequence of jump amplitude follows fuzzy jump diffusion process. And then the fuzzy number in the pricing formula was studied. For practical purposes, we take the crisp weighted possibilistic mean of the fuzzy number, pricing formula was obtained after the defuzzification. Finally, the conclusions and the third chapter and conclusion obtained under the fuzzy system were compared by numerical experiment. And we analyzed the influence of the initial state of the economy, the core value of jump amplitude fuzzy number and the left and right spreads of fuzzy number on option value though numerical analysis.