Solved-Computational Finance Lab IX -Solution

Description

Consider the following Black-Scholes diffusion equation:

{
dX(t) = Xdt + XdW (t)

X(0) = X0:

(a) Obtain the exact solution of the above SDE.

(b) The values of the parameters are = 0:75, = 0:30 and X0 = 307, and t 2 (0; 1).

(c) Solve the above SDE by the following methods:

i. Euler-Maruyama method.

ii. First-order Milstein Scheme.

(d) Plot the order of convergence in a loglog plot (∆ t vs. the mean error).

2. Consider the following Langevin SDE:

{
dX(t) = X(t)dt + dW (t)

X(0) = X0:

(a) The values of the parameters are = 10, = 1 and X0 = 0, and t 2 (0; 4).

(b) Solve the above SDE by the following methods:

i. Euler-Maruyama method.

ii. First-order Milstein Scheme.

(c) Plot the order of convergence in a loglog plot (∆ t vs. the mean error).

1