Maximum Likelihood Estimation And Empirical Assessment Of The Gamma Jump-diffusion Process

Black-Scholes option pricing model was proposed by Black and Scholcs in1973. Since its inception, it set off a revolution in financial economics and finance industry. As the constant improvement of financial markets, especially when the big news appear in the financial markets, such as emergencies, natural diasters, adjustment of policies and so on, it was found that B-S model was not able to fully adapt to the financial markets. The main reason of a series of hypotheses of B-S model is not realistic. Hence the researchers conducted a further promotion of B-S model. In1976, Merton proposed logarithmic jump diffusion model. After the Merton’s jump diffusion process, basing on different hypotheses, such as the distributional choice for the jump magnitudes, a large number of jump diffusion models are built. The double exponential jump diffusion model which was proposed by Kou and the Pareto and Beta distributions which was proposed by Ramezani and Zeng are the typical models of jump diffusion model. These models can react the leptokurtic characteristics of the distribution of return on assets.Basing on the jump diffusion models, this paper implies the Gamma distribution for jump magnitudes. Then we compare this modcl(GJD model) with log-normally distributed jump diffusion(LJD model) and the geometric brownian motion(GBM model). We apply these models to the Chinese stock market and index market in this paper. In the empirical assessment of these three jump diffusion models, we choose the data from Shenzhen Stock Exchange and Shanghai Stock Exchange which contain the daily returns for8stocks and2indexes. At the same time, we use the Neldcr-Mcad method to obtain the maximum likelihood estimates of unknown parameters of the three models and utilize the BIC criterion to assess the performance of GJD model relative to log-normally distributed jump diffusion and the geometric brownian motion. We find that GJD model and LJD model perform better than GBM model for both indexes and individual stocks. For the individual stocks, GJD model performs better in stocks with high kurtosis than in stocks with low kurtosis. But in general, LJD model provides a better fit than GJD model.