Solved-Computational Finance Lab VII -Solution

Description

Solve the following two-point Boundary-Value problem by using finite element method:

d ( (x + 1) du ) + (2 + x2)u(x) = x2 4; x 2 (0; 1)

dx dx
u(0) = u(1) = 0;
by using piecewise linear polynomials and using trapezoidal rule and Simpsons rule for the numerical quadra-ture.

b). Solve the following two-point Boundary-Value problem by using finite element method:

d ((x2 2) du ) + (1 + 2x)u(x) = x2 + 4x 5; x 2 (0; 1)

dx dx
u′(1) = 0;
u(0) = 2;
by using piecewise linear polynomials and using trapezoidal rule and Simpsons rule for the numerical quadra-ture.

c). Consider the following Black-Scholes PDE for European call:

@V 1 2S2 @2V @V rV = 0; (0; 1) (0; T ]; T > 0
+ + (r )S
@t 2 @S2 @S
V (S; t) = 0; for S = 0;

V (S; t) = S Ke r(T t) ; for S
− − ! 1

with suitable initial condition V (S; 0):

1. Solve the transformed PDE yt = yxx with suitable initial and boundary conditions by using finite elements mentioned in problem (a)and the Crank-Nicolson scheme.

2. Plot V (S; t) for T = 1; K = 10; r = 0:06; = 0:3, and the payoff.

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