Improved Transparent Boundary Conditions for Pricing American Options Ali Foroush Bastani a , Seyyed Mohammad Mahdi Kazemi a a Department of Mathematics, Inistitute for Advanced Studies in Basic Sciences, Zanjan, Iran bastani@iasbs.ac.ir, m kazemi@iasbs.ac.ir Key words: Black-Scholes equation, American option pricing, Free boundary value problem, Transparent boundary conditions, Chebyshev polynomials, Finite difference method. In this paper, we present a new method for the approximation of transparent boundary conditions, when solving the American option pricing problem in financial mathematics. Using the standard change of variables cited in [1], the free boundary value problem for pricing American options is equivalent to the following problem: ∂u ∂τ = ∂ 2 u ∂x 2 , a<x<x f (τ ), 0 <τ ≤ τ * , u (x, τ )= g(x, 0), a<x<x f (0), u (x f (τ ),τ )= g (x f (τ ),τ ) , 0 <τ ≤ τ * , e (α-1)x f (τ )+βτ ∂u(x f (τ ),τ ) ∂x + αu (x f (τ ),τ ) =1, 0 <τ ≤ τ * , u (x, τ ) → 0 as x → -∞. (1) where g(x, τ )= e -αx-βτ max(e x -1, 0) and the free boundary has changed to x f (τ ) = ln(S f (t)/E). Using this transformation, the free boundary value problem for an American call option which pays dividend and has an unbounded domain, is reduced to a problem in a bounded domain (see §3 in [1]). Hence, the last equation of (1) is changed to: ∂u ∂x x = a = 1 √ π τ 0 ∂u (a, λ) ∂λ dλ √ τ - λ . (2) We call this equation as the transparent boundary condition (TBC) [2]. In this paper, we have proposed some new improvements in Han and Wu’s approach [1] via explicit form of transparent boundary condition by expanding it in a Taylor series: φ(λ)= +∞ k=0 (λ - τ ) k φ (k) (τ ) k! . (3) Obviously, φ(τ )= u(a, τ ) is defined and differentiable on [0,τ * ] (for some τ * ) so our Taylor expansion is meaningful. We prove then that the transparent boundary condition (3) is equivalent to the following equation: ∂u ∂x | x=a = 1 √ π +∞ k=0 (-1) k+1 4 k k!(2k - 1) φ (k) (τ ) · τ k- 1 2 (4) 1 Abstract at NumAn2010 numan2010.science.tuc.gr — Conference in Numerical Analysis, Chania, Greece, Sept 15-18, 2010