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:
Forward-Euler for time & central difference for space (FTCS) scheme.
Backward-Euler for time & central difference for space (BTCS) scheme.
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:
Forward-Euler for time & central difference for space (FTCS) scheme.
Backward-Euler for time & central difference for space (BTCS) scheme.
Crank-Nicolson finite difference scheme
The values of the parameters are T = 1; K = 10; r = 0:06; = 0:3.
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