Recursive Formula For Pricing Square Arithmetic Asian Option

With the development of financial market and the increase of investors’divers demand, financial experts show more and more enthusiasm fordeveloping and researching the financial derivative. The pricing problem of therelated products’ price is deeply concerned. Especially, as one of the mainderivatives, options play a significant role in the field of modern finance, whichhas been favored by investors and financial experts.Options are divided into standard and non-standard options. Americanoption and European option belong to standard options, which have thestandard definition and clear meanings. Derived by the standard options, weusually refer to non-standard options as exotic options. Asian option, as a kindof exotic options, is a path dependent option. Its price is associated with thechange of the entire path of the asset price. Compared with the standardoptions, it is quite different. Asian option has the following features:(1) It can hedge risk effectively;(2) It can maintain financial market stability, preventing the manipulationof the underlying asset price;(3) It has cheaper price, and meets the needs of some investors.Due to the above characteristics of Asian option, the share of the Asianoption has been gradually increasing in the trading market. It is particularlyimportant to research the pricing subject. Geometric Asian option’s prices haveanalytical solutions, but there is still no analytical solution for arithmetic Asianoption.We in this paper define a new option-square arithmetic Asian option, andstudy its pricing problems. This option has inherited the advantages of Asianoptions. But in terms of pricing, it is no longer based on arithmetic mean of the underlying asset price, but on the square arithmetic mean of the underlying asset price over a period of time before the expiration. It is also a path dependent option, having no closed form solution.In this article, we give the recursive formula. We have a finite number of observation points, denoted as N. The option payoff under the risk neutral measures is given by Here St(0≤t≤T) is an underlying asset price process,0=T0≤T1···≤TN=T, R is fixed price of square arithmetic Asian option.We in this paper mainly investigate a square arithmetic Asian option price with a continuous underlying asset price process. First we set Yt=S2t, apply the recursive way of arithmetic Asian option (see [14]). Then under the risk neutral measures, we build the relationship between the risk-neutral expectation of the quadratic variation of the return process and European option prices. And the following equation is given: Where g is continuously twice differentiable function, Let c(n)t(y,K) and pt(n)(y,K) be at time t the prices of European call and put option respectively. Subscript t denotes the current time.Taking advantage of Ito formula, we can prove that under the same measure, Yt=St2is still a geometric Brownian motion. Hence the solution can be given. Applying the solution of the stochastic differential equation and independence lemma, we give the specific forms of ct (n)(y,K) and pt (n)(y,K), and finally we obtain the recursively formula of a square arithmetic Asian option price with a continuous underlying asset price process.This paper mainly introduces the definition of square arithmetic Asian option, and gives the recursive formula of pricing. The followings are some of the main conclusions: Proposition1:Let St(O≤t≤T) be the solution of the following stochastic differential equation of the geometric Brownian motion: Under the risk neutral measure, then Yt=St2is still a geometric Brownian motion, and YT=YtEp(2r-σ2)(T-t)+2σ(WT-wT) Theorem1:Under the risk neutral measure Q, we have price formulas of European call option and put option of the underlying asset price Y, as following where (p(x) is the cumulative density function of WTn-Wt-N(0,Tn-t); namely Theorem2:For1≤n≤N-2, define and where andAssume that g(n) is continuously twice differentiable with respect to Xn, and for1≤n≤N-1,Then square arithmetic Asian option price at time0≤t≤T1is given by...