Pricing Financial Derivatives Based On EVG Model

Along with the development of financial theory, the description of dynamics of stock price comes through Markovian processes to independent increment processes, then to geometrical Brownian motion; which supply a basic mathematical tool for the developing of continuous financial theory. However, more and more research indicate that the market is not so perfect, the assets prices process is not always continuous, the distributions of log return is not always normal but with high skewness and fat tail. As a result, the extension of the original theory, especially the improvement of the Black-Scholes model which is so called "the second revolution of Wall Street" becomes one of the focuses during the latest 20 years.Most researches apply Levy process or other process with jump to simulate the movement of financial market in recent years as the developing of discontinuous stochastic analysis theory. These researches improve the new development of mathematical finance. Especially, by introducing more parameters, Variance Gamma process posses good mathematical properties, and has been proven to be able to explain some economical phenomena: mathematically, different with Brown motion, Variance Gamma is a finite variation process, and the distribution of its increment has high skewness and fat tail; economically, the option pricing model based by Exponential Variance Gamma process are able to overcome the famous problem which is so called "volatility smile" in the classical Black-Scholes option pricing model; in the pricing of Credit Default Swap, Variance Gamma model can demonstrate the credit spreads curve in realistic market commendably.For the reasons as above, we focus on the pricing theory of derivatives based on the assumption of the asset log return following Variance Gamma process. Especially, we extend the convertible bond pricing model with Variance Gamma process. For perpetual American convertible bond, we get the optimal convertible strategy of the holder, the optimal callable strategy of the issue and the explicit solution of the convertible bond. For American con- vertible bond with finite maturity, we give out the variational inequality both in call case and in no call case. We use Multi-stage Compound Option (MCO) method and the Explicit-Implicit difference method to find the discrete solution of the convertible bond. Empirical research demonstrates that, although the difference of different model is smaller than the difference of the convertible bond with different clauses, the convertible bond price and the optimal implement boundary in Exponential Variance Gamma model is different with the classical Black-Scholes model obviously.