I wanted to figure how how much faster the Sobol quasi random numbers convergence to the B&S call price compared with pseudo random numbers. To generate the Sobol numbers I used the randtoolbox in R to generate these numbers. When using just one step, so from t=0 to t=T, it’s easy. I used the following formula to go from s(0) to s(T).
S_t= S_0*exp((μ- σ^2/2)*t+ σW_t, where W_t is a Sobol Random Number
I use the Sobol numbers and therefore the convergence is much faster because these numbers are better normal distributed when using pseudo random numbers.
My problem is the following:
How do I need to generate these numbers if I use intermediate steps in my simulation, do I need to use more dimensions or just generate more Sobol numbers from the same numbers. I’m already stuck for a long time on this. Hopefully somebody can help me, especially using the randtoolbox package from R to generate these numbers.