Computational Finance Lab III Solution

$30.00

Description

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

8

>

@V

+

1

2S2

@2V

+ (r )S

@V

rV = 0; (0; 1) (0; T ]; T > 0

@t

2

@S2

@S

>

>

>

>

>

<

for S = 0;

> V (S; t) = 0;

>

>

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

>

>

>

: with suitable initial/terminal condition V (S; 0) or V (S; T ):

Solve the above Black-Scholes PDE by the following schemes:

  1. Forward-Euler for time & central difference for space (FTCS) scheme.

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

  1. Crank-Nicolson finite difference scheme

The values of the parameters are T = 1; K = 10; r = 0:06; = 0:3 and = 0.

2. Consider the following Black-Scholes PDE for European put:

8

@V

1

2S2

@2V

@V

rV = 0; (0; 1) (0; T ]; T > 0

+

+ (r )S

@t

2

@S2

@S

>

V (S; t) = Ke

r(T t)

S;

for S = 0;

>

>

>

>

>

>

< V (S; t) = 0;

for S

! 1

>

with suitable initial/terminal condition V (S; 0) or V (S; T ):

>

>

>

>

>

>

:

Solve the above Black-Scholes PDE by the following schemes:

  1. Forward-Euler for time & central difference for space (FTCS) scheme.

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

  1. Crank-Nicolson finite difference scheme

The values of the parameters are T = 1; K = 10; r = 0:06; = 0:3.

1