Solved-Computational Finance Lab VI -Solution

Description

1. Consider the following European call option problem:
> @V 1 @2V @V
<
8 @t + 2 2(S)S2 @S2 + (r )S @S rV = 0; 0 S < 1; t T; : > V (S; t) = VT (S); 0 S < 1: With the following transformation > S
>
= S + Pm ;
8
>
< > = T t;
>
>
>
> V (S; T ) = (S + Pm)V ( ; ):
the above European call option problem becomes the following parabolic PDE:
:
@2
> @V1 V @V

8 @ = 2 2()2 (1 2)@ 2 + (r ) (1 ) @ [r(1 ) + ]V ; 01; 0T;
>
>
>
< >
V ( ; 0) = max(2 1; 0); 0 1;
>
>
>
>
> V (0; )= V (0; 0)e r ; V (1; )= V (1; 0)e ; 0 T:
Solve: the transformed PDE by the following schemes:
(i) Forward-Euler for time & central difference for space (FTCS) scheme.

(ii) Backward-Euler for time & central difference for space (BTCS) scheme.

(iii) Crank-Nicolson finite difference scheme

The values of the parameters are T = 1; r = 0:04; = 0:25; = 0:1.

To solve the system of linear algebraic equations arising from the implicit schemes use the iterative methods (Jacobi method, Gauß-Seidel method and SOR method).

Continued on the next page

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2. Consider the following European put option problem:
> @V 1 @2V @V
< rV = 0; 0 S < 1; t T; 8 @t + 2 2(S)S2 @S2 + (r )S @S : > V (S; t) = VT (S); 0 S < 1: With the following transformation > S
>
= S + Pm ;
8
>
< > = T t;
>
>
>
> V (S; T ) = (S + Pm)V ( ; ):
problem becomes the following parabolic PDE:
the above European put option :
@2
> @V1 V @V

8 @ = 2 2( ) 2(1 2)@ 2 + (r ) (1 ) @ [r(1 ) + ]V ; 01; 0T;
>
>
>
< >
V ( ; 0) = max(1 2 ;0); 0 1;
>
>
>
>
> V (0; )= V (0; 0)e r ; V (1; )= V (1; 0)e ; 0 T:
Solve: the transformed PDE by the following schemes:
(i) Forward-Euler for time & central difference for space (FTCS) scheme.

(ii) Backward-Euler for time & central difference for space (BTCS) scheme.

(iii) Crank-Nicolson finite difference scheme

The values of the parameters are T = 1; r = 0:04; = 0:25; = 0:1.

To solve the system of linear algebraic equations arising from the implicit schemes use the iterative methods (Jacobi method, Gauß-Seidel method and SOR method).

2