Constructing a Brownian Bridge Between Unequal GBM Endpoints
Summary
The document asks how to generate a Brownian bridge between two different simulated price points in a geometric Brownian motion path. The setting is an arithmetic Asian option whose averaging window begins before expiration, leaving a short interval between the last averaging date and the option’s maturity. The author notes that common explanations describe bridges with endpoints at the same value, and asks whether a linear interpolation or a drift fitted to the endpoint change is appropriate when the simulated prices differ.
The text provides the modeling problem and candidate intuitions, but no answer, derivation, or numerical comparison. It therefore serves as a prompt to study conditional path generation between unequal endpoints, rather than as a validated simulation recipe. Any implementation would need to specify the process and conditioning variables, including whether the bridge is constructed for prices or log prices, before choosing an interpolation method.
Key ideas
- The question concerns path generation between distinct simulated values at two times under geometric Brownian motion.
- The motivating use case is an arithmetic Asian option with averaging ending before expiration.
- A straight-line interpolation and an endpoint-fitted drift are raised as possible approaches, not established solutions.
- The document provides no derivation or evidence identifying the correct bridge construction.
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# Brownian Bridge from timestep 1 to timestep @ expiration, proper mathematical way to generate # Brownian Bridge from timestep 1 to timestep @ expiration, proper mathematical way to generate When I was learning finance, we didn't cover the subject of Brownian Bridges. So I am trying to learn the proper way of generating paths when you have an arithmetic Asian option which has an averaging period at some point in the future, call it 0.5 years. Then a time of expiry, call it 0.6 years. What my dilemma is with Brownian Bridges, most of the literature assumes that you either generate a starting T_timestep1 OR T_expiry, and create the bridge that starts and ends with the same value (the GBM simulated price). Now less commonly taught, is transitioning between the T_timestep1 price and T_expiry price, assuming GBM, with the same "shock", there is higher volatility as sqrt(T) is higher at T_expiry. So call these 2 points a and b, where abs(a < b), but your Brownian Bridge shocks assume a 0 shock starting price and a 0 shock ending price. Although I've read as much as I can find on the internet, I still don't know the "correct" way of transitioning between these 2 simulated points. Is the proper way a simple slope between a and b to which you apply the Brownian Bridge shocks? Say simulated price T_timestep1 is 20 and simulated price (with the same shock) is 30 at T_expiry, how do you generate the proper Bridge between a and b? If I was to guess an approach that would theoretically work, I would solve for a drift component that yields a change of 10 (30-20) over the time period, for this example 0.1 years (0.6-0.5). But just using a slope is simple, since I'd need to calculate it for every simulation path. And then I would still have the option of using drift (although simple GBM with drift gives crazy results over long time frames without mean reversion). Just looking for a simple explanation of how the path from a->b is constructed with a Brownian Bridge, when a != b. The Brownian Bridge seems to be explained, wherever I've been able to find an explanation, assuming a = b. Documentation on this subject is severely lacking. So any help is appreciated! Obviously, I haven't taken any financial engineering classes in ages, so excuse my lack of knowledge in this area.
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