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## Description

1. Consider the following American put option problem:

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

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: with suitable initial and boundary and free boundary conditions:

1. Solve the transformed PDE y = yxx of the above IBVP by using the Backward-Time and Central Space (BTCS) Scheme and the Crank-Nicolson nite di erence scheme.

1. Plot V (S; t) for T = 1; K = 10; r = 0:25; = 0:6; = 0:2, and the payo .

1. Solve the problem by using x and , and x=2 and =2 and calculate the error between these two numerical solution. Plot the error.

1. Also calculate the error mentioned above for di erent values of x=2 and t=2 and plot N versus the maximum absolute error.

1. Consider the following American call option problem:

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

<

>

: with suitable initial and boundary and free boundary conditions:

1. Solve the transformed PDE y = yxx of the above IBVP by using the Backward-Time and Central Space (BTCS) Scheme and the Crank-Nicolson nite di erence scheme.

1. Plot V (S; t) for T = 1; K = 10; r = 0:06; = 0:3; = 0:25, and the payo .

1. Solve the problem by using x and , and x=2 and =2 and calculate the error between these two numerical solution. Plot the error.

1. Also calculate the error mentioned above for di erent values of x=2 and t=2 and plot N versus the maximum absolute error.

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