Jump–Diffusion Stock-Return Model with Weighted Fitting of Time-Dependent Parameters

This paper treats jump-diffusion processes in continuous time, with emphasis on jump-amplitude distributions, developing more appropriate models using parameter estimation for the market. The proposed method of parameter estimation is weighted least squares of the difference between theoretical and experimental bin frequencies, where the weights or reciprocal variances are chosen by the theory of jump-diffusion simulation applied to bin frequencies. The empirical data is taken from a decade of Standard & Poor 500 index of stock closings and are viewed as one moderately large simulation. The new developments are the combined use of uniform jump-amplitude distributions, least squares weights and time-varying market parameters, introducing more realism into the model, a Log-Normal-Diffusion, Log-Uniform-Jump financial market model. The optimal parameter estimation is highly nonlinear, computationally intensive, and the optimization is with respect to the three parameters of the log-uniform jump distribution, while the diffusion parameters are constrained by the first two moments of the S&P500 data.