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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.

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