This shows simulations of a fictional stock’s time path. The stock starts at $100 and 200 random simulations are cast for 500 periods into the future. This assumes that the market for the stock is efficient. If it were not efficient, we could not model the time path as a random/stochastic process.

This is the distribution of each simulation’s price at the 500th period. The distribution is centered around the starting price of $100. This is consistent with what we would expect to see in an efficient market: the best predictor (expected value) of the stock’s future price is today’s price. The variance of the distribution increases over the forecasting period. This is also intuitive since we expect greater uncertainty about the price 1 year from today than the price 1 week from today, for example.

The following 4 graphs represent a 35-week forecast of the S&P 500 at different periods of time. The forecast is made by projecting the S&P’s volatility over the previous 35 weeks onto the future 35 weeks. There are 100 gray lines representing 100 random simulations. We expect the actual price path to be bounded by the most extreme (lower and upper) simulations.

I good alternative to using historical volatility to forecast 35 weeks ahead may be to use an implied volatility from the options market. The market price of the option contract that expires 35 weeks later can be used to “reverse-engineer” the market’s expected volatility over the forecast period. This may be the subject of my next post!