Jump Diffusion Model Of Geometric Average Asian Option, Single State Binary Tree Method Research

This paper studies the binomial tree methods and convergence of Asian option when the underlying asset in a market which follows a jump diffusion model. Different from the traditional diffusion models, the jump diffusion models assume that the price of underlying asset is influenced by the Brownian motion and Poisson process at the same time. This model can give a fine explanation to sudden changes in the market, for example announced the financial figures, important political events, natural disasters etc. so it is more reasonable than the diffusion model.Asian option are path-dependent options whose payoffs depend not only on the underlying asset prices but also on the average of the underlying asset prices. This paper makes a detailed analysis about binomial tree methods for geometric average Asian options in jump diffusion models. The binomial tree method is a discrete model and numerical algorithm for option pricing. Due to its simplicity and flexibility, it has become the most accepted model in financial community. But binomial tree models of the geometric average Asian options involve two state variables because an additional path-dependent variable is introduced. Usually too much calculation is involved for the binomial model and the method is not feasible. Cheuk et al.[9] gave a one-state binomial model for lookback options in diffusion models. Dai[10] used their method to give the one-state binomial model for geometric average Asian options. In this paper, we generalize the results in Dai[10] to the jump diffusion models. We set binomial tree models with continuous sampling, discretely sampling, floating strike and fixed strike for geometric average Asian options in jump diffusion model. Then we construct one-state variable binomial models for geometric average Asian options in jump-diffusion model by some variable transformations. We also discuss the equivalence of the one-state variable binomial tree methods and certain explicit difference schemes. At last we prove the convergence of the binomial tree methods for american-style Asian options in the framework of viscosity solution.