Brownian Bridges for Options Expiring Within a Monte Carlo VaR Horizon
Summary
The document explains how to value an option that expires before the end of a Monte Carlo Value at Risk horizon. It proposes conditioning on the simulated stock price at the horizon and using a Brownian bridge to draw an intermediate stock price at the option’s expiration. Under a constant-parameter exponential Brownian motion model, the log price at expiration is normally distributed, with its conditional mean interpolated between the starting and horizon log prices and its variance determined by the time intervals and volatility. The sampled expiration price can then be used to calculate the option payoff.
The approach extends to multiple options on one stock by sampling at each relevant expiration. The answer notes that incorporating covariance across stocks makes bridge calculations substantially more complex, so full path simulation may be preferable. For portfolios where options contribute little to risk, it also mentions setting the bridge variance to zero as a simplifying approximation. These methods depend on the assumed price process and trade off computational cost against path detail.
Key ideas
- A Brownian bridge can estimate an underlying price at an option’s expiration between today and the simulated VaR horizon.
- Under constant-parameter exponential Brownian motion, the intermediate log price is conditionally normal.
- The sampled expiration price determines the option’s value at expiry.
- Multiple expirations require intermediate draws at each relevant date.
- Cross-asset covariance complicates bridge simulation, while setting bridge variance to zero is a simplifying approximation.
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Full text
# Handling option expiration during Monte Carlo simulation
# Handling option expiration during Monte Carlo simulation
I have equity options in my portfolio that can expire during a VaR calculation (with Monte Carlo). For example the time to maturity of my option is T days but I simulate for T+n days (n > 0).
What is the appropriate way of handling these kind of situations?
## Answer by Brian B (score 2, accepted)
https://quant.stackexchange.com/a/30108
If you are willing to spend the calculations, you can use a Brownian bridge to work out the probability distribution for the stock price at option expiration time $T$.
Let's say you have an option on $S$ expiring at $T$ and you have simulated the stock price $S_{T+n}$ for your VAR horizon, starting from today's price $S_0$.
Then (if you are willing to model the stock prices as constant-parameter exponential brownian motions) you can assume that $\log(S_T)$ is gaussian distributed,
$$ \log(S_T) \sim N\left( \mu, v \right) $$
with $\mu$ the time weighted average of $\log(S_{T+n})$ and $\log(S_0)$,
and
$$ v = \frac{(T-0)((T+n) - T)}{(T+n)-0} \sigma^2. $$
From here, you simulate a draw from that gaussian distribution, exponentiate to get $S_T$, and work out option expiration value from $S_T$.
Thus, to handle this option in your algorithm, you have a 2-stage simulation where in each iteration you you simulate a terminal stock value, then you simulate an "in-between" stock value to get option expiration prices.
Note that if you have $N$ options on the same stock, you should use an $N$ stage simulation. Also note that it is possible to include covariance of stock $S_1$ and $S_2$ in bridge calculations, but that at that point the bridge complexity is so high that you might as well just simulate full paths instead.
It is common in risk applications where options are only a small part of the portfolio to assume $v=0$ and just accept the bridge mean as $S_T$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.