# Usage of Brownian Bridge?

I was recommended to read something about Brownian Bridge. Could someone familiar with BB give some recommendation?

It was mentioned that BB benefits in 2 places

1. BB could reduce the simulation paths, this reduces computation effort, especially when the underlying factors are a lot (say 20-30). I noticed that Papageorgiou1 has a paper "The Brownian Bridge Does Not Offer a Consistent Advantage in Quasi-Monte Carlo Integration" (2002). So does this point still hold?

2. BB could reduce the computation effort on path-dependent derivatives. For example, during pricing of a barrier option, a path could be simulated with monthly scenarios of the factors; then BB could be used to estimate the probability of the path "knock-out" of the barrier. Which paper/book would you recommend on this topic?

The Papageorgiou paper is presumably referring specifically to quasi-random sequences used in path generation. Researchers had noticed that, in high dimensions, QR sequences tend to have good space coverage for the first couple of dimensions:

but terrible coverage for the latter dimensions:

(Plots here are points 101-200 from a 32-dimensional QR sequence)

One solution proposed was to make sure most variation was from the first few dimensions by taking terminal path values from them, and then "filling in" the path with lower-quality latter dimensions. In practice I believe this has been superseded by scrambling tricks. See this paper for a decent review.

I don't know of a paper covering bridges for "knock-out" probabilities, which is more of a practitioner's trick than an academic's research project, though the computational aspects might well be covered in something like Hull or Shaw. The math is not particularly tricky.

• thanks for the reply. do you happen to have some links to more detailed materials on the "scrambling tricks"? – athos Apr 17 '14 at 2:27
• For scrambling, a good place to start would be Hinckernell's Quasi-Monte Carlo methods and their randomizations. – Brian B Apr 17 '14 at 12:58

1) Brownian Bridge is used in Quasi Monte Carlo pricing of asian options to reexpress paths in a basis where few selected components/subspaces bring the most contribution, so as to align these to the best distributed dimensions/subspaces of a low discrepancy sequence. This allows for better coverage and thus faster convergence and paths amount reduction. However for different kinds of options such important directions will be different so that BB will perform worse, and Papageorgiou showed a pathologic example where the standard incremental path construction performs better (and BB worst), simply by using a payoff most responsive to the individial increments and nearly indifferent to the average. This can be generalized: given any option payoff and a LDS sequence an optimal path construction will be determined. Or given any path construction and LDS sequence one can find both a "friendly" or an indifferent payoff. Check "New brownian bridge" paper for a first variation on the theme. Also Leobacher and Sloan both wrote on this topic, but that might be beyond your needs.

2) This sounds strange to me: the analytical step with BB will not be easier, especially outside of a GBM setting. If you mean to perform all by simulation then considerations in 1) still hold.

You can find a brief but useful explanation of Brownian bridge techniques in Andersen and Piterbarg (page 125), which includes references for further reading. It's probably the best place to start. They discuss valuing barrier options specifically, and discuss the performance issues mentioned here. Later (pg 647), they use Brownian bridges in constructing a Libor market model, which is a useful example.