Jump-Di usion Stock Return Models in Finance: Stochastic Process Density with Uniform-Jump Amplitude

The stochastic analysis is presented for the parameter estimation problem for tting a theoretical jump-di usion model to the log-returns from closing data of the Standard and Poor's 500 (S&P500) stock index during the prior decade 1992-2001. The jump-di usion model combines a the usual geometric Brownian motion for the di usion and a space-time Poisson process for the jumps such that the jump amplitudes are uniformly distributed. The uniform jump distribution accounts for the rare large outlying log-returns, both negative and positive in magnitude. The log-normal, log-uniform jump-di usion density is derived, leading to a jump-di usion simulator approximation for the case the the log-return time is a small fraction of a year. There are ve jump-di usion parameters that need to be determined, the means and variances for both di usion and jumps, as well as the jump rate, given the average log-return time. A weighted least squares is used to t the theoretical jump-di usion model to the S&P500 data optimizing with respect to three free parameters, with the two other parameters constrained by the mean and variance of the S&P500 data. The weight distribution derives from stochastic methods. The ideal tted model determines the three free parameters, but the corresponding simulated results resemble the original S&P500 data better. This stochastic analysis paper is a companion to a computational methods and portfolio optimization paper at this conference.