Bilinear collocation method for fuzzy Black-Scholes equation

In this paper, the bilinear collocation method is introduced to solve the fuzzy Black-Scholes equation. In this case, the solution is approximated by the functions which are considered on the rectangle. The method is performed on two sample examples to show the efficiency and importance of the proposed scheme.


Introduction
Solving fuzzy differential equations is one of the important topics in fuzzy analysis [8].In recent years, some numerical and analytical methods were proposed in order to solve this equations such as [1,2,4,5,6,7,10].In this work, the collocation method is applied to solve the Black-Scholes equation in fuzzy case and the solution is approximated by a linear combination of four special base functions obtained from four points in four corner of any rectangle whose sides are parallel with the coordinate axes.We call this scheme as bilinear collocation method.Black-Scholes equation has been solved in some works such as [12,17].Let the following fuzzy Black-Scholes equation, with the following initial condition, where u(x,t) is the European call option price at asset price x and at time t, T is the maturity, R(t) ̸ = 0 (∀t ∈ [0, T ]) is the risk free interest rate, σ (x,t) represents the volatility function of underlying asset and g is inhomogeneous fuzzy term.
At first in section 2, we remind some preliminaries of fuzzy concepts.In section 3, the main idea and the concept of bilinear collocation method for fuzzy Black-Scholes equation are offered.In section 4, two sample examples are solved to illustrate the applicability of this method.
Definition 2.1.A fuzzy parametric number u is a pair (u(r), u(r)) , 0 ≤ r ≤ 1, which satisfies the following requirements : The set of all these fuzzy numbers is denoted by and the scaler multiplication of fuzzy numbers are defined by is the Hausdorff distance between u and v.It is shown that E 1 , D is a complete metric space [18]. .We define the n-th order differential of f as follows: We say that f is strongly generalized differentiable of n-th order at t 0 , if there exists an element f (s) (t 0 ) ∈ E 1 ∀s = 1, . . ., n such that (i) for all h > 0 sufficiently close to 0, there exist f

and the limits
) and the limits or (iv) for all h > 0 sufficiently close to 0, there exist f
Remark 2.1.We note that by the above definition a fuzzy function is i-differentiable or ii-differentiable of order n if f (s) for s = 1, . . ., n is i-differentiable or ii-differentiable.It is possible that the different orders have different kind i or ii differentiability.

Main Idea
In order to implement the bilinear collocation method for fuzzy Black-Scholes equation, we approximate the solution of Eq.(1.1) by the following linear fuzzy combination of functions, where, a j are fuzzy coefficients.By choosing four points in four corners of a rectangle such as x 2 and setting (3.3) in (1.1), the linear fuzzy system of equations is obtained: From (3.4), the following 8 × 8 crisp system can be written: where, A = (a 1 , a 2 , a 3 , a 4 ) T , A = (a 1 , a 2 , a 3 , a 4 ) T , G = (g(t 1 , x 1 ), g(t 1 , x 2 ), g(t 2 , x 1 ), g(t 2 , x 2 )) T , G = (g(t 1 , x 1 ), g(t 1 , x 2 ), g(t 2 , x 1 ), g(t 2 , x 2 )) T and s i j are determinate as: then, the matrix S is a non-singular matrix.

International Scientific Publications and Consulting Services
Proof.According to the hypothesis, if Therefore all elements of the matrix M are nonnegative and S 1 = M, S 2 = 0. Hence, according to the theorem 3.1, ) .Therefore the inverse of S is: ) .
Remark 3.2.We note that if the obtained coefficients a i are fuzzy numbers or in other word, the 8 × 8 extended crisp system has a strong solution then the approximate polynomial with fuzzy number coefficients is obtained that, we say strong fuzzy approximate polynomial.Otherwise, the solution is weak fuzzy approximate polynomial [15].

Numerical examples
In this section, two sample examples are solved via the proposed scheme which was described in the previous section.

Definition 2 . 4 .Definition 2 . 5 .
Let u, v ∈ E 1 .If there exists w ∈ E 1 such that u = v + w, then w is called the H-difference of u, v and it is denoted by u ⊖ v. Let a, b ∈ R and f : (a, b) → E 1 and t 0 ∈ (a, b)