| As the financial derivatives get more and more popular in the global financial world,option pricing has become a hot spot. The theory of option pricing originates from thefamous BS formula in1973, however, it has been doubted by lots of phenomenon in themarket, such as the volatility smile, mean reversion, high kurtosis and fat tail, leverageeffect and so on. How to improve the BS model to incorporate more features shown by themarket data, and how to obtain an accurate and efficient numerical algorithm, are what thepaper is trying to answer.Based on the BS model, a European option pricing model including pure jump Levyprocess and mean reversion as well as CEV volatility was established, and it was proofedto be both rational and superior from the mathematical and economical viewpoints. As themarket described by the Levy process is incomplete, the paper is trying to obtain theanalytical solution through the combination of martingale and characteristic functionmethods. The Ito formula under Levy process was obtained, and some substitution wasmade based on it. For the complexity of the model, a degenerate model was dealt with.First a general proposition of the option pricing formula based on the characteristicfunction of the log stock price was proposed, and then the characteristic function of thelog stock price was obtained with the help of stochastic analysis theory as well asprobability theory, so finally we got the European option pricing formula. In the end,assumed the pure jump Levy process to be a specific VG process, some numericalsimulation was executed. First, we gave the detailed implementation steps of the MonteCarlo simulation and the FFT algorithm based on the model, and then the numericalresults were presented. The results show that, compared with the Monte Carlo numericalmethod, FFT algorithm based on the analytical solution was both accurate and efficient.Though the results turned out not as satisfying as expected, it laid a foundation for future’sfurther research. |
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