We have recently discussed the concept of Brownian Bridge in our CFE class and have implemented a full valuation Monte Carlo Brownian Bridge in Excel spreadsheet to value certain financial products.

What is a Brownian Bridge and why do we need a Brownian Bridge? A Brownian bridge (BB) is a 'tied down" Brownian motion. .

In a Brownian motion the state variable, i.e. the stock price, FX rate, interest rate, is stochastic and evolves over a period of time in a random manner. The randomness is tied to the volatility of the asset and the drift is deterministic. In the short run, the volatility dominates the process and the asset price path is truly stochastic, i.e. random. However, over the long period of time the drift will dominate the volatility and therefore, if there are small errors in estimation of the drift it will lead to large fluctuations in the future price distribution. This is one of the drawbacks of a Brownian motion (as applied to the pricing of financial assets). Over longer horizons the drift of the stochastic process (Brownian motion) becomes a complicating factor.

Besides, in a Brownian motion the final state is uncertain. The asset value can be anything above zero. Theoretically, the asset can go from zero to infinity. Of course, the asset returns are normally distributed (Gaussian distribution) from minus infinity to plus infinity.

Therefore, a Brownian motion may not be very suitable for modelling an asset which has a longer maturity period, say, 5 years, 10 years, etc. and where the final state of the asset is known. Like a discount bond (Treasury bonds). A government bond can have maturities of 5 years, 10 years, 30 years and the final value of the bond is known, i.e. the par (face) value. And, a bond will always redeem at par.

In the case of a long dated discount bond we need to simulate values of the asset over a longer period of time such that the stochastic process is conditional on reaching a given final state. For example, take the case of a discount bond such as a Treasury bond. The Treasury bond will always mature at the face (par) value. If the par value of a discount bond is $100 and given a certain yield to maturity, it is currently trading at $95 (market price), then at maturity it has to redeem at par. A bond always gets redeemed at the par value at maturity. Therefore, if we model a discount bond as a stochastic, random process then this process should be tied to the final state of the process. In other words, this stochastic process will evolve conditional on the (given) final state of the process.

A Brownian bridge (BB) is a "tied down" Brownian motion such that even though the stochastic process, the Brownian motion, evolves in a random manner it is conditional on the final state of the process. Simply speaking, in a Brownian Bridge the initial and the final states of the stochastic process are known; however, in between the initial and the final state the process is random. Also, in an integral set up, the drift of the stochastic process is given at the beginning and is simply the return of the process - natural log of the prices - given the initial and the final values of the asset (say, the bond) which is known at

Say, is a standard Weiner process (random walk) used to model a Geometric Brownian motion (GBM) where is on the interval and the initial value of the Weiner process is 0.

Now consider another stochastic process, which is given by:

, where, and

If this is the case then,

Therefore, if then the stochastic process starts at 0 and ends at 0. The stochastic process is known as a Brownian Bridge

In a generalized case, when and the Brownian Bridge is given by:

Another important point is that while modelling a Discount bond we see that in a Brownian bridge the drift drops out of the integral stochastic equation and it is replaced by a constant drift that is conditional only on the initial and final state of the asset (bond).

Besides Treasury and discount bonds, Brownian Bridges are also used to model Barrier options and other exotic options where the terminal value is known in advance.

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