# How are Brownian Bridges used in derivatives pricing in practice?

A similar question has already been asked in the past, unfortunately the 2nd question of the OP was never really addressed.

Most references found on internet on Brownian Bridge and Monte-Carlo simulations seem to relate to Quasi-Monte-Carlo methods and look general and academic in nature. However, in the mentioned question it is said that Brownian Bridge "[...] 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 [a Brownian Bridge] could be used to estimate the probability of the path 'knock-out' of the barrier." This seems to be a "practitioner's trick".

Is anybody familiar with this technique/trick and, if so, could she/he explain it briefly?

Alternatively, does anyone has any reference to this topic, if possible openly available on internet (paper, etc.)?

I have an idea on how the Brownian Bridge could be used to speed-up Monte Carlo pricing for some specific path-dependent payoffs (like barriers) but I was wondering if anybody has more knowledge on this, there does not seem to be references in internet.

• I see @noob2, so for example if we have a barrier product with barrier $B$, you select a "confidence level" $\alpha$ and if $|S(t_i)-B|<\alpha$ or $|S(t_{i+1})-B|<\alpha$ then you simulate for example $S(t_i+(t_{i+1}-t_i)/2)$, if not you assume the barrier is not crossed in the interval $[t_i,t_{i+1}]$, right? – Daneel Olivaw Nov 17 '17 at 17:37
• Yes, something like this. – noob2 Nov 17 '17 at 18:20

Yes, the term Brownian Bridge seems to be used loosely. I assume you are talking about continuously monitored barriers by the way, since you mention the probability of the barrier being crossed in between the path time points. If that's the case then "naive" Monte Carlo simulation will have what is called "simulation bias". That's exactly because the simulated process could hit the barrier in between path time points but the simulation will naturally "miss out" on such events. And somewhat counter-intuitively, this bias can be very large: even if one uses say daily time steps for the simulated paths, this naive MC barrier price can still be 30% or 40% off (!) compared to the true continuous barrier price (see second link below for such examples).

So, I'm guessing that when we are talking about BB reducing the computational effort when pricing barrier options, what is meant is that it can do so by enabling a more accurate price for continuous barriers without the need to use a ridiculous amount of time steps/points. Right, now this is called the Brownian bridge technique because it uses the probability of Brownian motion hitting a point conditional on two fixed end points. Such probability formulas are only available for Brownian Motion and Geometric Brownian Motion, so this technique can be used for something simple like Black Scholes and actually remove the simulation bias completely. But it can also be used for other processes with good results. Anytime we use the Euler discretization scheme for example, the simulated process becomes locally BM.

I will not expand here further but here's a reference that describes this technique and more (and has the probability formulas)

Advanced Monte Carlo methods for barrier and related exotic options - Emmanuel Gobet

And in the link below you can see this technique applied to MC pricing of continuously monitored barriers under Black-Scholes (only of academic interest, perfect accuracy) and the Heston model (of more practical interest, technique still seems very effective).

Monte Carlo pricing of continuous barrier options with Heston

Of course there may be other ways a BB-related method can help with MC simulation of path-dependent options that I'm not aware of.

• Please expand regarding the crossing between the simulated time points. – Gordon Nov 17 '17 at 21:21
• I've updated it, let me know what you think. – Yian Pap Nov 17 '17 at 22:50