Philosopher, Juris Doctor, surfer, trader, investor, musician, black belt, sailor, semi-centenarian. He lives on the mountain in Kona, Hawaii, with his family.
The Monte Carlo Method, by James Sogi
Curiosity about Chair's and the Experts' use of and reference to Monte Carlo techniques prompted study of Paul Glasserman's, Monte Carlo Methods in Financial Engineering. Monte Carlo is based on the analogy between probability and volume. The mathematics of measure formalizes the intuitive notion of probability associating an event with a set of outcomes and defining the probability of the event to be its volume or measure relative to that of a universe of possible outcomes. Monte Carlo uses this identity in reverse, calculating the volume of a set by interpreting the volume as a probability. The law of large numbers results in the estimates converging to the correct value. Financial assets price are the expected value and computation of the value as an integral often results in infinite or large numbers. Monte Carlo simulates stochastic paths and seeks the fastest convergence in formulating the problems and may present more competitive valuation methods.
The risk inherent in observed fat tails must be balanced against the need to derive precise probabilities using a normal curve but leaves unaccounted for risk inherent in use of normal distributions. The theorists fit the fat tails randomly and retrospectively to alternate distributions, and there does not seem to be a good set of tools to properly measure the relationship of samples to the population and leaves an inability to predict probabilities. Monte Carlo appears to be useful when volatility measures can be factored in accounting for observed risk from the back door so to speak. Monte Carlo is useful for path dependent prices, and might be helpful in the stop/option/risk/size issue.
Glasserman notes that the normal distribution has shortcomings as a model of changes in market prices virtually in all markets, the distribution of observed price changes displays a higher peak and heavier tails than can be captured with a normal distribution. Choosing a description of market data thus entails a compromise between realism and tractability. Imagine my happy surprise to read that the Student t Distribution, my personal favorite, has a distribution with fatter tails, as demonstrated by the graph of its log densities. The student t distribution is a mixture of normals, giving rise to the idea of a mechanism for generating a richer class of distributions by combining two normals with other random variables and allowing for a distribution with fatter tails, but with fixed distribution characteristics to compare with the samples of occurrences being tested, sort of a super custom student t distribution fitted to observations and factoring in occasional big changes.
Another interesting avenue for multivariate analysis factoring in heavy tails is the t-Copula which is also discussed in Carmona, Statistical Analysis of Financial Data in S Plus which has code for some really cool 3D charts. When I figure this out, I will show them to you. This is something that will be useful when looking at bonds and equities relationship.
Jim Sogi, May 2005