Application Of CGMY Levy Process In Option Pricing Based On FFT

Financial engineering is an essential part for Quantitative Finance recently, which dues to its effective implementation in trading and strategy. Widely usage of computational algorithm makes higher profit in hedge funds rather than other fundamental analysis, but lower trading mistakes, thus enjoying its population among financers. But during the last half centuries, global financial crisis, starting from abuse of CDS are also reminding us of the importance to improve pricing, systematically.Option, different from other derivatives, asymmetric obligations between sellers and buyers makes it much more effective in investing, value finding and risk avoiding. That is the very reason that mature markets are extremely active than others. Modern derivative pricing enjoys its significant progress from 1976, when Black, Scholes and Merton (1976) published a new model, named B-S Model. Based on B-S Model, Greek Letter Hedge Strategies are widely used in the market. But more and more empirical evidences show that it could hard to explain following phenomenon:(1) Implied volatility smiling (2) Stochastic Jump in the movement of stock price (3) Leptokurtosis. Exponential Levy Processing, as a classic modified model for B-S Model, including three parts:linear drift, Gaussian part (which contains Brownian motion), Jump part, could effectively fit historical data of stock price.Carr and Madan (2002) published their new model named CGMY, which is the newest fruit in Exponential Levy Processing in Journal of Business. They added a new parameter in VG model, called Y to simulate finite activities and infinite activities in jump processing, through different values.This article analyzes CGMY Model and put it into European Call Option pricing. In the first part, we review several important definitions and properties in probabilities, stochastic processing and exponential Levy Processing in order to make a strong theoretical support for the second part, which is the mathematical proofs for CGMY Model, including its characteristic function, logarithm stock pricing and European Call Option pricing. The third part describes the quantitative method using in the article, named FFT (Fast Fourier Transform), and transformed MLE in parameter estimation. The last part is the empirical part, which could be divided into two parts:(1) parameter estimation (2) calibration of European Call Option. And we also try to give the MATLAB code in this part, as a cherish asset for the following analysts.