A Application Of Fuzzy Theory To The European Geometirc Asian Options

Asian option is becoming more and more active in financial market, the study of Asian options pricing is also increasingly. This paper studies the pricing model on European geometric average Asian option with fixed strike price and European geometric average Asian option with floating strike price in the fuzzy environment.In this paper, the pricing formulas of European geometric average Asian option with fixed strike price and European geometric average Asian option with Floating strike price are given out by the method of partial differential, there are the two kinds of European geometric average Asian options pricing formula:(1) The pricing formula of European geometric average Asian option with fixed strike price is as follows:(2) The pricing formula of European geometric average Asian options with floating strike price is as follows: In order to establish the fuzzy model of pricing formulas, This article introduces the definitions of fuzzy set, λ-cut set, normal fuzzy set, convex fuzzy set, upper semicontinuous, fuzzy numbers, trapezoidal fuzzy numbers, fuzzy random variable and so on and describes the expansion principle and the related content and theorem in detailTheory (Extension principle) For a given map f:Uâ†'V,A∈F(U) B∈F (V), the map f:F (U)-> F (V), Aâ†'/(A)is called fuzzy set transformation of positive extending from map f:The map f*1:F (V)â†'F (U) Bâ†'-1(B) is called fuzzy set transformation of opposite extending from map f:The membership functions of(A) and f-1(B) are given as follows:Positive transformation Fuzzy set is usually referred as "Fuzzy transformation, opposite transformation Fuzzy set is referred as Fuzzy inversed transform.Theory (Multiple expansion principle) Let F be the set of all fuzzy subsets on R, suppose f(x1,x2,…,xn) is the real function from R" to R and A1, A2,…, An are n the fuzzy subsets of R. We can get a fuzzy value function f:Fnâ†'F from the real function f(x1,x2,-…,xn) in other way f(A1,A2,…,An) is a fuzzy subset on R, its membership function f(A1,A2,…,An) is determined by the type: Based on preparations for the application of principle of expansion, we set up the pricing models of European geometric average Asian options with fixed strike price and European geometric average Asian options with floating strike price:(1) The fuzzy pricing formula of European geometric average Asian options with fixed strike price is as follows: According to the principle of expansion, V1(S,t,K,r,σ) is a fuzzy number on t and the membership function of Vi(S,t,K,r,σ) is:(2)The fuzzy pricing formula of European geometric average Asian options with floating strike price is as follows: Be similar to (1), V2(S,J,t) is a fuzzy number on t.The membership function of V2(S,t,K,r,σ) is: We give the specific methods and steps of calculation.