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Previously, we have shown that the proper method for estimating parameters from discrete, binned stock log returns is the multinomial maximum likelihood estimation, and its performance is superior to the method of least squares. Useful formulas have been derived for the jump-diffusion distributions. Numerically, the parameter estimation can be a large scale nonlinear optimization, but we have used techniques to reduce the computation demands of multi-dimensional direct search. In this paper, three jump-diffusion models using different jump-amplitude distributions are compared. These are the normal, uniform and double-exponential. The parameters of all three models are fit to the Standard and Poor's 500 log-return market data, constrained by the data first and second moments. While the results for the skew and kurtosis moments are mixed, the uniform jump distribution has superior qualitative performance since it produces genuine fat tails that are typical of market data, whereas the others have exponentially thin tails. However, the log-normal model has a big advantage in computational time of parameter estimation compared with the others, while the double-exponential is most costly due to having one more model parameter.