Brownian Bridge Correction for Continuously Monitored Barrier Pricing
Summary
The document explains how Brownian bridge conditioning improves Monte Carlo pricing of continuously monitored barrier options. A simulated path may cross a barrier between sampled time points, so checking only those points misses knockouts and creates discretization bias. Conditional on the values at two adjacent time points, the bridge method estimates the probability of an intervening crossing and uses it in the pricing calculation.
For Brownian motion and geometric Brownian motion, crossing probabilities are available in closed form, allowing the method to remove this simulation bias in the stated simple setting without extremely fine time steps. The discussion also notes that it can work effectively for more complex processes, such as Heston, and with Euler-discretized paths that are locally Brownian. Its applicability and accuracy depend on the process and modeling assumptions; the document does not provide a full implementation or quantitative comparison beyond citing examples.
Key ideas
- Checking only simulated time points can miss barrier crossings between them and bias continuous-barrier prices.
- A Brownian bridge estimates crossing probability conditional on the path values at neighboring time points.
- Closed-form crossing probabilities for Brownian and geometric Brownian motion can remove this source of bias.
- The technique can also be useful with more complex models, though its performance depends on the process and discretization.
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# How are Brownian Bridges used in derivatives pricing in practice? # 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. ## Answer by Yian Pap (score 9, accepted) https://quant.stackexchange.com/a/36945 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.
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