Jump-Diffusion Stock Return Models in Finance: Stochastic Process Density with Uniform-Jump Amplitude
نویسندگان
چکیده
The stochastic analysis is presented for the parameter estimation problem for fitting a theoretical jump-diffusion 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-diffusion model combines a the usual geometric Brownian motion for the diffusion 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-diffusion density is derived, leading to a jump-diffusion simulator approximation for the case the the log-return time is a small fraction of a year. There are five jump-diffusion parameters that need to be determined, the means and variances for both diffusion and jumps, as well as the jump rate, given the average log-return time. A weighted least squares is used to fit the theoretical jump-diffusion 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 fitted 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.
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