The Analysis Of Asian Options Based On Multiscale Stochastic Volatility Under The Ornstein-uhlenbeck Process

Options is one of the core tools of the modern financial market risk management,afterF.Black,M.Schole built the famous Black-Scholes opting pricing model,Many scholars underthe additional to discuss the Asian option pricing problem.but m ost of the research is In the caseof volatility is constant.But the underlying asset of many options contracts Will present thevolatility of randomness and Mean reversion characteristic.Volatility plays an Important role in the pricing of financial derivatives,tradingstrategies,and Risk Control.That is to say if there is no volatility, no financial market.But if themarket’s volatility is too large,and lack of risk management tools,the investors will be Worriedabout the risk,and abandon the transaction,it can make the market less attractive to theinvestors.We usually mentioned the Implied volatility reflects the expectation and judgement ofthe market.It contains the information of future market.All of these will be reflected in theoptions pricing process.Because of the mean reversion characteristics of the O-U process,whichcan make it more reasonable and flexible in the study of portray the price fluctuations infinancial assets;Assume that in the options pricing process,the volatility follows theOrnstein-Uhlenbeck process,then the volatility Will return to its long-term level after departurefrom long-term levels. Consider the multi-scale modeling and related resolution approach,Somescholars studied the problem of pricing arithmetic Asian options when the underlying is drivenby stochastic volatility models with two well-separated characteristic time scales. The inherentlypath-dependent feature of Asian options can Help us very good in these aspects of researchwork.In previous studies,the volatility is modeled by a fast mean-reverting process. We propose amathematical method-Perturbation method。It can help us analysis the process of idealizedsystem when the parameters or the structure in the model have any small perturbations.Asingular perturbation expansion is used to derive an approximation for option prices. In thispaper, we consider an additional slowly varying volatility factor so that the pricing par-tial dierential equation becomes four-dimensional. Using the singular-regular perturbation techniqueintroduced by Fouque-Papanicolaou-Sircar-Solna, we show that the four-dimensional pricingpartial dierential equation can be approximated by solving a pair of one-dimensional partialdierential equations,which takes into account the full term structure of implied volatility.Finally,to illustrate with examples,we find that it is necessary to incorporate thecombination of the fast and the slow volatility factors in order to obtained Asian options priceswhich are consistent with the observed term structure of implied volatility.