Hiya, wondered if anyone could kindly clarify a few things for me:

Which of the following ways is best to construct a stock path following the usual dS/S=r*dt+vol*dW(t) in the risk neutral world, for example to price a discretely sampled Asian Option. When I say best I mean which will converge the fastest using monte carlo.

I am also using the Halton algorithm for gerenating uniform "random" numbers before plugging them into the Box-Muller algorithm.

1) S(t_j+1)=S(t_j)*exp((r-(vol^2)/2)*((t_j+1)-(t_j))+vol*sqrt((t_j+1)-(t_j))*N(0,1))

2)Euler discritization: S(t_j+1)-(S(t_j)=r*((t_j+1)-(t_j))+vol*sqrt((t_j+1)-(t_j))*N(0,1)

3) Now for the third way I'm not sure which way of using the Brownian Bridge, is there a big difference between these 2 methods, one is from lecture notes and the other is from Glasserman's book :

3a) Given S0 say at time T0, generate S(T) using S(T)=S0*exp((r-(vol^2)/2)*(T-T0)+vol*sqrt(T-T0)*N(0,1)) then generate the intermediate values, say t_j using S(t_j)=(S0*(T-t_j)+ST*(t_j-T0))/(T-T0)+N(0,1)*sqrt((t_j-T0)*(T-t_j)/(T-T0))

3b) Given S0 at time T0, generate S(T) using the same formula as 3a), then using the same Brownian Bridge formula gerenate

S(T/2) then using this value gerenate S(T/4) and S(3T/4) , etc..assuming that the number of points I need is 2^n.

Is there a big difference between 3a) and 3b)?

Does the Brownian Bridge work well even though I'm using low discrepancy numbers, as whenever I google Brownian Bridge there are quite a few links "The Brownian bridge does not offer a consistent advantage in Quasi-Monte Carlo integration"?