Empirical Research For European Option Pricing Models On Levy Processes

Black-Scholes (1973) model has founded a brand-new chapter of assets pricing theory. From then on, many scholars studied the insufficiency to the BS model, carried on the experimental expansion exploration research works, which Include the pricing theory's revisions, the model expansion, parameter estimation's improvement and the technical exploration for pricing realized way and so on. Because the supposition condition is too strict, under the requests of perfect market BS model, this article uses several kind of the quite commonly used Levy model, often called imperfect market model-NIG model, VG model and CGMY model, which carried on the empirical study of the model to Hong Kong market HSI's option pricing process. Because the supposition condition is relatively quite loose, the Levy models can simultaneously describe continuously with the jump process, the phenomenon of peak and thick tail. This article through establishes the BS model, NIG model, VG model and CGMY model separately to HSI's evolvement process, studies the different performances of four kinds of model in Hong Kong market for HSI option pricing:First, shows the fitting effect about the HSI returns distribution and carries on the corresponding statistical test under four kinds of random distributions. Second, simulates the HSI price road under several kinds of models using Monte Carlo produces. Through risk neutral way simulation, thus we can obtain the option expectation value, carries on the non-risk discount gain option again the price. Third, through several kinds of model's characteristic functions, carries on the European option price by integral formula transformation, under the utilization new Fourier numerical method for solution option price inferential reasoning risk neutral model theoretical option price. Above all results carry on statistics comparative analysis.This article model's parameter uses the generalized moment estimate method, and carries on slightly approaches the revision; the technologies for option pricing use the simulation and the numerical methods. Four models, two groups of parameters, two kinds of techniques, carry on the research above separately. The empirical study results showed:First, about the fitting comparison for the HSI logarithm returns ratio distribution, the Levy process's distributions compared to the normal distribution displays superiorly, which indicates the HSI market not the perfect condition. The CGMY distribution characteristic is obvious. Second, if only to the foundation property's logarithm returns ratio distribution statistics analysis, estimated parameters through the characteristic function effective moment, regardless of Monte Carlo simulation or the Fourier numerical method, the Levy model in the HSI option pricing is more accurate than the BS model. Third, this article adopt the method of the gradient descent procedures approach the nonlinear least squares under the moment estimated,use the new parameters for Monte Carlo simulation and the Fourier numerical calculus again, finally demonstrated that the new parametric model on the simulation and the numerical calculus precision greatly enhances, the Levy model's simulation was still superior in the BS model simulation; In the numerical calculus method, in the sample space the BS model is best fitting effect with the formula computation, the Levy model is inferior to the BS model, the Levy process simulation and the numerical calculus relies on the random number, but the BS model has the good close solution, the error is the smallest. Forth, outside the sample the predictive ability aspect, the Levy model must be more precise than basically the BS model, but is quite close including the caper Levy model option price. We demonstrate the intrinsic value option (in the money) uses the numerical method (FFT) well; the virtual values option (out of the money) uses Monte Carlo simulation (MCMC) well.Draws the conclusion based on the above result:The Levy process fits the HSI logarithm returns ratio distribution effect better than normal distribution; Hong Kong market regarding perfect market supposition may similarly be inexistence certain; Levy process Monte Carlo simulation option pricing effect better than the BS model; After the model parameter revises, no matter the simulation and the numerical method may enhance the price forecasting greatly the precision; The Levy process union on the estimate parameter have more accurate predictive ability under the gradient descent approach the nonlinear least squares.