Brownian Bridge

The Brownian bridge algorithm has been implemented for stress testing within the Risk Management framework. It is used for generation of multidimensional random paths whose initial and ending points are predetermined and fixed.

In the context of stress testing this algorithm is used for efficient generation of specific scenarios subject to certain extreme and generally unlikely conditions. If paths were generated by a conventional Monte-Carlo method only a very small portion of all the paths would satisfy such conditions.

The Brownian Bridge algorithm belongs to the family of Monte Carlo or Quasi-Monte Carlo methods with reduced variance. It generates sample paths which all start at the same initial point and end, at the same moment of time, at the same final point.

Eq. (4) is a product of the two factors: the first is the probability of getting to point xf at time tf starting from point xi at time ti, and the second is the probability of passing through point x at time t given those initial and final points.

It is the second factor in formula (4) that gives the distribution of points on the paths connecting fixed initial and final points, i.e. those generated by the Brownian bridge algorithm. In particular, according to eq. (4) for any given time t between tf and ti the distribution of points x should be normal with the mean

The Brownian Bridge algorithm is also very useful for valuing exotic derivatives via Monte Carlo approach. Most exotic products have callable and barrier features. Both callable and barrier characterizations can terminate a trade earlier.

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