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...for the linear Black-Scholes equation for pricing Asian options. Future work will focus on the construction of NSFD schemes for spatially twodimensional PDEs and for nonlinear Black–Scholes equations =-=[1, 6]-=- especially for equations with a nonlinear convective term, cf. [18]. Also, we will derive a NSFD scheme for the original BlackScholes equation based on a finite volume formulation, since often a stan...

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... 2xx Tw w xτ σ τ = ∈ ∈ s(19)sSubject tos( ) ( ) ( )1 1 2 2,0 max e e ,0 , k k w x x + − = − ∈ s(20a)s( ) 2 , 0 as , 0, 2 Tw x x στ τ → → ±∞ ∈ s(20b)sAccording to =-=[28]-=- European Call option as the solution to the Black Scholes equation on 0 ,0S t T≤ < ∞ ≤ ≤scan be approximated bys( ) ( ), ~ e as ,r T tV S t S K S− −− → ∞s(21)sEquating both hand sides of Equation (21...

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...uch assumption is that there are no transaction costs in hedging a portfolio. Ever since, there have been several and successful remodelling of Black-Scholes that take into account transaction costs (=-=Ankudinova, 2008-=-). Leland (1985) came out with an approximate option replication model featuring transaction costs in the Black-Scholes model. He considered a model that allowed the transactions only at discrete time...

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...ion v(x, t) = ux(x, t) is introduced, hence we consider ut = βuxx + λv − ru (25) vt = βvxx + λuxx − rv. (26) We state for u(x, t) the initial conditions u(x, 0) = u0(x) and boundary conditions u(0, t) = b0(t), u(1, t) = b1(t). For v(x, t) the initial condition is simply the x-derivative of u(x, t): v(x, 0) = u′0(x). In the sequel we will make frequent use of the standard central difference operator ∆0hui := ui+1 − ui−1 = 2hD0hui and the standard second order difference operator ∆2hui := ui+1 − 2ui + ui−1 = h2D2hui Suppose now the spatial grid is uniform, i.e. N sub-intervals form the interval [0, 1] and h = 1/N . The second order approximation v(h, t) = ∂u ∂x (h, t) ≈ D0hu(h, t) = ∆0h 2h u(h, t). November 21, 2010 15:1 International Journal of Computer Mathematics DremkovaEhrhardt˙revised A high-order compact method for nonlinear Black-Scholes equations of American Options 11 can be improved to fourth order if ∆0h is replaced by ∆ 0 h/(1 + 1 6∆ 2 h) v(h, t) = ∆0h 2h(1 + 16∆ 2 h)u(h, t) + O(h4). Doing so, we obtain a fourth order approximation v(0, t) = 3 h (u(2h, t)− u(0, t))− 4v(h, t)− v(2h, t). Hence, we can approximate the right boundary condition of v at x = 1 as v(1, t) = 3 h (u(1− ...