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Option Pricing for a Exponential Lévy Model in a RegimeSwitching Market Using FFT
Author: WangCuiPing
Tutor: WangChunFa
School: Zhejiang University of Finance
Course: Finance
Keywords: Exponential Lévy Model Jump Diffusion Model RegimeSwitching VG Model CGMY Model Fast Fourier Transform Option Pricing
CLC: F830.9
Type: Master's thesis
Year: 2010
Downloads: 73
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Abstract
As an investment tool of derivative securities, the option emerges from the increasing development of the market economy. As the increase of the risk of commodity trading and the market uncertainty, options have been widely applied as instruments for effective hedging and risk management. The research of option pricing problem has significant theoretical and realistic meanings. The price of option means the value judgment made by both of counterparts, but it is rather difficult to watch it. Hereby, the option valuation is always an important subject in financial engineering.Despite the great success of BlackScholes model (1973) in option pricing, this pure lognormal diffusion model fails to reflect the three empirical phenomena: (1) the large random fluctuations such as crashes and rallies; (2) the nonnormal features, that is, negative skewness and leptokurtic (peakedness) behavior in the stock logreturn distribution; (3) the implied volatility smile, that is, the implied volatility is not a constant as in the BlackScholes model.Therefore, many different models are proposed to modify the BlackScholes model so as to represent the above three empirical phenomena, such as stochastic interest models, stochastic volatility models, diffusion jump models, pure jump models and so on. However, most models considered are time homogeneous and as Konikov and Madan (2002) have shown, the theoretical behavior of the term structure of their moments does not match empirical observations. For example, the variance theoretically increases with a factor t (the length of the holding period), skewness decreases with a factor t1/2, and kurtosis decreases with a factor t, while empirically, these moments do not show patterns of growth or decay that are even close to these factors. Given all this, in order to allow for timeinhomogeneity, there has developed an interest in modeling asset returns using switching processes. RegimeSwitching model is mostly used to the traditional stochastic interest models, stochastic volatility models, but rarely applied to the Lévy model with jumps.In this paper, we present a new model of Lévy process in a RegimeSwitching market based on the traditional Lévy models. Three specific RegimeSwitching diffusion models and two pure jump models. By adopted the methodology of Liu, Zhang and Yin (2006), we give a fast Fourier transform approach to option pricing for regime switching models of the underlying asset process. The Fourier transform of the option price is obtained in terms of the joint characteristic function of the sojourn times of the Markov chain. We present the joint characteristic function in explicit form for twostate (m=2) Markov chains, and in terms of solutions of systems of mdimensional differential equations for mstate case.Firstly, we begin with riskneutral valuation for European option, where the asset price follows a general Lévy process in a regimeswitching market. The Fourier transform of the option price is obtained in terms of the joint characteristic function of the sojourn times of the Markov chain. We present the joint characteristic function in general form for twostate Markov chains, and in terms of solutions of systems of mdimensional differential equations for mstate case. However, the explicit form option price depends on the specific Lévy measure of each regimeswitching models.Furthermore, Fast Fourier transform is adopted for calculating option prices, h where the asset price follows four specific regimeswitching Lévy process respectively. The four models are regimeswitching jump diffusion model with logdouble exponential distribution’s jumpamplitudes , Merton jump diffusion model with regimeswitching, RegimeSwitching jump diffusion model with loguniform jumpamplitudes, RegimeSwitching jump diffusion model with loguniform jump amplitudes , RegimeSwitching pure jump model with VG process and RegimeSwitching pure jump model with CGMY process. The diffusion component is given by Markovswitching geometric Brownian motion and the jump component is modeled by a compound Poisson process with Markovswitching Lévy measure. The Fourier transform of the option price is obtained in terms of the joint characteristic function of the sojourn times of the Markov chain. We present the joint characteristic function in explicit form for twostate (m = 2) Markov chains based on the explicit form of the Lévy measure of the five regimeswitching models respectively.At last, the option prices from the regimeswitching jump diffusion models and the regimeswitching pure jump model are compared with those of regime switching BlackScholes (RSBS) model. As expected, call option prices of Lévy model are higher than those of SBS model with respect to the strike price. The reason is the jump and the regime switching factors increase the risk premium.

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