1. Consider the following Black-Scholes diffusion equation:
{
dX(t) = Xdt + XdW (t)
X(0) = X0:
Obtain the exact solution of the above SDE.
The values of the parameters are = 0:75, = 0:30 and X0 = 307, and t 2 (0; 1).
Solve the above SDE by the following methods:
Euler-Maruyama method.
First-order Milstein Scheme.
Plot the order of convergence in a loglog plot (∆ t vs. the mean error).
Consider the following Langevin SDE:
{
dX(t) = X(t)dt + dW (t)
X(0) = X0:
The values of the parameters are = 10, = 1 and X0 = 0, and t 2 (0; 4).
Solve the above SDE by the following methods:
Euler-Maruyama method.
First-order Milstein Scheme.
Plot the order of convergence in a loglog plot (∆ t vs. the mean error).
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