Wednesday, January 21, 2009

Monte Carlo Simulation of S&P 500 since 1980

I've had a little bit of time on my hands lately so I've been working on option pricing models. As a result of that I have daily returns of the S&P 500 since 1980. Normally, when running a monte carlo simulation (MCS) I use the normal distribution, but I think the last several months have shown us that financial return distributions are non normal, or non-static. To make up for this I used the actual daily returns and randomized the pattern of returns to generate several time series (may or may not be a good method) for the S&P. The frequency distribution (Using 39 trials) looked like this:

This looked like an opportunity to use a log transformation of the data in order to model it. After the transformation the data looked like this:

Based on the simulation, about half the data was above and half was below the current level of the S&P. However, the deviations from the current level were extreme, 132 at the low end and 11522 at the highest end. The compounding component of returns can take returns to an extreme. The logaverage of the data was 6.9 which translated to 991 in real terms. I also ran a 95% confidence interval to determine where the likely mean would be and it was quite large: 117.76 to 8342.47. To me this says that randomness reigns supreme. It is extremely difficult to make an accurate judgment of expected price in the future. The variance is simply too high over long periods of time.

Incidentally, after writing this post, I did some further research. I have been reading a derivatives book that has shown me that the above is theoretically true. I didn't even need the book to figure it out, just some knowledge of monte carlo simulation and distributions.

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