Asian Option Pricing With Random Interest Rate Under The Condition Of Fractional Brown Motion Environment

In modern financial market,option is the most dynamic and the most important part of it.So,the pricing theory of option is the main content of financial asset pricing,and is one of the most fundamental and substantial areas in mathematical finance.The first full option formula in the world was published by Black-Scholes and Merton in1973.Nevertheless,the Black-Scholes formula has a very important hypothesis that the asset prices are subject to the geometric Brown movement.A large number of empirical studies found that the actual financial market of the stock price process show some fractal characteristics,such as self-similarity,long-term memory,etc.These characteristics don't conform to the classical B-S hypothesis,but conform to fractional Brown motion.Thus,many scholars suggest replace geometric Brown movement by fractional Brown motion.In today's financial derivatives market,Asian option is one of the most active non-standard options in modern derivatives market.It is a strong path-dependent option that its payoff at maturity depends on the average of the underlying asset price in the entire period.In this paper,we study the pricing problem of geometric average Asian option under the condition that the underlying asset follows fractional Brown motion.Firstly,basing on reliability mathematics,measure alteration and No-arbitrage assumptions,we acquired the formula of geometric average Asian option when the rate was random,which extended the interest rate case of constant or function of t.Secondly,we have considered that the emergent events will also have an impact on the underlying asset.Through establishing jump-diffusion model,defining risk neutrality measure,we finally acquired the pricing formula of geometric Asian options when the rate was random.Finally,we simulated the price of Asian option through Monte Carlo method.Given initial value,we respectively simulated the price of constant rate without jump,Vasicek rate with jump and Vasicek rate with jump.After compared the numeric results,we found the option price was undervalued without consideration of random rate and jump.Consequently,considering random rate and jump is logistic in this paper.