Sobol numbers in monte Carlo simulation

Question

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.

Thanks

Answer 1

First let me say that in the Black-Scholes model as you have it, there is of course no need for intermediate steps when pricing vanilla calls, since the SDE has the closed-form solution you included. Intermediate steps would be required for complicated payoffs or other SDEs.

To answer your question though, you do need to use additional dimensions. Think of the option pricing algorithm as an integration over the probability space of stock price paths. Each intermediate tenor introduces a new dimension to that probability space.

The point of these quasirandom sequences is that, in multiple dimensions, they provide more evenly-distributed coverage of the probability space than pseudorandom numbers would give us.

Quasirandom

If we don't make multidimensional draws from the Sobol sequence, we would not benefit from that extra regularity.

Pseudorandom

Note how these pseudorandom draws have multiple spots where samples are right next to each other, along with some very big "holes".

Answer 2

You need to use more dimensions. If the number of dimensions (i.e. steps) is large, you may also have to use a Brownian bridge as described in the book by Joshi or Jäckel.