# 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?

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 $S_T$.

• Shouldn't mu be the time weighted average of log(ST+n) and log(S0)? – DennisVDB Sep 13 '16 at 9:41