## Path Construction

Discuss anything related to quantitative finance which is not directly connected to a particular book here.

### Path Construction

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"?
sruggiero

Posts: 13
Joined: Sat Jul 11, 2009 7:36 pm

### Re: Path Construction

always use 1 not 2. They will give the same variance but 2 requires more steps per path for accuracy.

No one uses Halton. Use Sobol.

Brownian bridge almost always increases the rate of convergence of QuasiMonteCarlo. It has no effect on Monte Carlo. Yes you can construct examples where it doesn't but these are always contrived.

The algorithm in 3a and 3b looks wrong. It's easier first to generate W_{j} for j=1,...,n using Brownian bridge and then set

S_{j} = S_{j-1}exp( ( r-\sigma^2) (t_j - t_{j-1}) + \sigma \sqrt{t_j - t_{j-1}) (W_{j} - W_{j-1}).
mj

Posts: 1380
Joined: Fri Jul 27, 2007 7:21 am

### Re: Path Construction

yeh my bad, in 3a and 3b it should be in terms of W(j) and then plugged into the forumla you give.

Imagining 3a and 3b are in terms of W(j) which one is the correct way of constructing the bridge?

Can I just use the formula to fill in the missing values chronologically, or do I need to find W(T/2) using the formula then use this for W(T/4) and W(3T/4), etc?
sruggiero

Posts: 13
Joined: Sat Jul 11, 2009 7:36 pm

### Re: Path Construction

you should always do W(T/2), W(T/4), W(3T/4) and so on...
mj

Posts: 1380
Joined: Fri Jul 27, 2007 7:21 am

### Re: Path Construction

Hi thanks for the reply, can you have a quick look at this thread going on at willmott:

Is Quasi-Monte Carlo good for early exercise options? With Brownian bridge as well?

I see brownian bridge is gonna be in your new book, any idea of a launch date yet? Can i purchase just this chapter, the volatility derivatives and L&S chapters if they are already written? Theyd really help with my dissertation

Cheers
sruggiero

Posts: 13
Joined: Sat Jul 11, 2009 7:36 pm

### Re: Path Construction

drop me an e-mail
mj