Statistical mechanics of financial markets: Exponential modifications to Black-Scholes

The Black-Scholes theory of option pricing has been considered for many years as an important but very approximate zeroth-order description of actual market behavior. We generalize the functional form of the diffusion of these systems and also consider multifactor models including stochastic volatility. We use a previous development of statistical mechanics of financial markets to model these issues. Daily Eurodollar futures prices and implied volatilities are fit to determine exponents of functional behavior of diffusions using methods of global optimization, Adaptive Simulated Annealing (ASA), to generate tight fits across moving time windows of Eurodollar contracts. These short-time fitted distributions are then developed into long-time distributions using a robust non-Monte Carlo path-integral algorithm, PATHINT, to generate prices and derivatives commonly used by option traders. The results of our study show that there is only a very small change in at-the-money option prices for different probability distributions, both for the one-factor and two-factor models. There still are significant differences in risk parameters, partial derivatives, using more sophisticated models, especially for out-of-the-money options.