Study On The Pricing Of The Asian Option In The Affine Jump Diffusion Model

There are a variety of exotic options trading in the financial markets, such as asian option, look-back option and barrier option. Compared with the standard option, the ways of these options trading are more flexible and convenient. The investment income of them is more accordant with the investors’intentions and the price is cheaper. Since the structure of the profit function of exotic option, as a kind of product depended on the path, is complicated, how to give this kind of exotic option pricing has become a hot research topic in the field of financial mathematics. The results of exotic options in the classical Block-Scholes model have been very mature, but the Block-Scholes model has many shortcomings:the ideal assumptions, risk free rate of T, stock dividend q and the volatility of stock returns standard deviation a is constant. However, in the actual financial markets, interest rate, dividend and volatility are not constant but random variables which cannot be observed directly in the financial markets. Therefore, extended or modified Block-Scholes model has become the focus in the current financial mathematicians study. One assumes that the volatility, interest rates are subject to random process variables, and then establishes the affine jump diffusion option pricing model. These models give a good interpretation of the volatility, the interest rate and the randomness in the incomplete market. So they are more consistent with the actual financial market in the description of stock price behavior compared with the classical Block-Scholes model. But the investigations by improved exotic options models are not many. Especially in studying the pricing of Asian Options used the affine jump diffusion model at home and abroad, the results are very few.Asian option as an exotic option appeared in the early 90’s, but it is still one of the most active options in the financial derivatives markets. Unlike european options, asian option is based on the average price of the contract period [0, T]of the underlying assets of the size to decide whether to implement the option contract. The asian option pricing has been a lot of extensive research. Most of them are based on the index meeting the geometric Brown motion,and then market interest rate and volatility is constant. Due to the ideal assumption, it is not consistent with the actual data of financial market and cannot explain these phenomena, such as the peak, fat tail and volatility smile in the financial market.In the present paper, we study the asian option pricing, by employing the mixed form of affine jump diffusion model which takes both the class of stochastic volatility and stochastic interest rates into consideration. The mostly previous hard work, studying the asian option pricing, have done by using the measure transformation and martingale method considered the risk free interest rate. In this paper, we use the random analysis method, such as the joint characteristic function of the mul-tidimensional random variables, Girsanov transform and inverse Fourier transform, to give price of the geometric average asian option, and use the expansion of Edgeworth fourth order series and mathematical integral method to approximate the arithmetic average asian option pricing.Resolved by numerical discussion of the second chapte:the jump risk of stock price and volatility and inter-est rates are of very big effect for the geometric average asian option. So It is obviously that the impact of jump risk on asian option pricing cannot be ignored in the market and should be paid more attentions. In addition, the sensitivity analysis on some parameters of the volatility and inter-est rate have been done. Through the numerical analysis of the third chapter, comparison of Monte Carlo simulations and Edgeworth series expansion approximation simulation method,one can use the approximate simulation method of the Edgeworth series expansion to approximate arithmetic average asian option. Some parameters of volatility and interest rates do numerical analysis.Asian option pricing in a class of stochastic volatility and stochastic interest rate under the mixed form of affine jump diffusion model is designed with high complexity and practicability. The actuarial pricing and calculation of the cash value are very complex. Asian options pricing is of great significance to develop other exotic options pricing in financial markets.