Geometric Asian Option Pricing Under The L(?)vy Process With Stochastic Interest Rates And Stochastic Volatility

Asian option as one of the strong path-dependent derivatives, different from European or American option, it is to some extent reduce the risk of market manipulation of the underlying asset at maturity, and for its the averaging feature reducing the volatility inherent in it, generally Asian option price with respect to the American or European options are cheaper options, and therefore Asian option is considered to be one of the basic forms of exotic options, since proposed has been widespread concern of many scientists. But in contrast to the general vanilla options, Asian option pricing to be more difficult. With changes in the market, we need do more research to meet the financial market diversification.In this paper, taking into account the market implied volatility smile, fat tail of the underlying asset and the market incompleteness, we establish a L???vy jump diffusion process and let the constant interest rate and volatility as random variables based on the traditional Black-Scholes model to better meet the market and pricing more precisely. In this report, we will depart the pricing process into two parts: the continuous geometric Asian options and the discrete geometric Asian options; Based on the two types of options above, we establish appropriate partial integro-differential equation and solve the characteristic function of the underlying assets and respectively under the new equivalent martingale measure, and obtain the closed-form solutions using fast Fourier transform; Finally, we take the result of Monte Carlo simulation as a reference to verify the efficiency and precision of the fast Fourier transform approach.