"Active" Monte Carlo Simulation
In Monte Carlo simulation, stock price paths are randomly generated (in this case, lognormal Brownian motion) according to the assumed volatility of the stock. Each path determines a "mini-history" of the options portfolio and the final result corresponds to a particular outcome. A distinguishing feature of the Monte Carlo simulations in this study is that a hedging trade is executed at each point, depending upon the delta of the portfolio at that point. Thus the simulations are "active", in that they actually mimic the delta-hedging procedure outlined previously. Without this rehedging feature this simulation tool would have little application to option portfolios.
We performed a simulation of 1,000 stock price paths for a very simple portfolio consisting of
a short at-the-money options position on 1,000,000 shares of stock, with 10 days to go to expiry, along with a position in the underlying stock. We simulated daily stock prices and rehedged daily as well. The distribution of outcomes is displayed in figure 1.
The options were sold at an implied volatility equal to that generated during the simulation, namely 40%. Thus the mean should be about zero. In fact the distribution of outcomes has a mean of ($11,552) and a standard deviation of $582,845. The distribution appears to be essentially normal. According to the previous discussion, were we to simulate more frequently (hourly, for example), the distribution would tighten up. In the limit of continuous rehedging it becomes a single spike.