Option Pricing Under Markov Regime-Switching Volatility Model Using Singular Perturbation Theory

As one of the important financial derivative securities, option was first applied in the United States in the mid-1970s. For more than 30 years, as a financial innovation tool of the prevention of risks and effective means of speculation, it has gotten rapid developed. However the option price is difficult to obtain from the market directly. So options pricing has always been concerned as an important problem in financial mathematics. Black-Scholes option pricing model is considered as the core and base of option pricing and is regarded as a useful tool for derivative pricing and risk management since it established. But for a long time, both empirical research and theoretical analysis have discovered and proved the constant volatility of Black-Scholes option pricing model is not a very good description of the financial market. This led us to develop and analyze dynamic volatility model.In fact, volatility is random and is the function of stock price and time. This can explain the implied volatility "smile curve" and "deflection effect". Thus we would correct the Black-Scholes model, and propose new option pricing model that is more adaptive to market changes. By analyzing on financial derivatives market, financial derivatives, and the characteristics of stock market running, this paper proves also that the Black-Scholes option pricing model does not reflect the actual price option. Therefore this paper examines with stochastic volatility option pricing models, and propose Markov regime-switching volatility model. In this model, we use singular perturbation theory and idea of two time scale as well as asymptotic expansions for option pricing. Specifically, we mainly research in the following five areas:First of all, in the analysis of the pricing theory of risk neutral valuation theory and no-arbitrage theory underlying Black-Scholes model structure, analyze the results, we think the classic Black-Scholes model cannot accurately reflect the option price. Therefore this paper first discusses several different dynamic volatility models. Based on the analysis of stochastic volatility models, we discuss the characteristics of mean-reverting stochastic volatility models. Because one of the major factors of controlling an individual stock contributes to the overall stock market, therefore it is necessary to make the main parameters of the stock reflect the movement of stock market. Stock market regime can reflect the fundamental economic status, market investors, and other economic factors. Assuming that the market regimes (modes) switching has finite states, putting it in simple terms, the stock parameters depending on the market regimes (modes) is called regime-switching model. We use Markov chain to explain the regimes switching. For example, a two-State Markov chain{0,1}, can be used to describe the market regime (modes) switching in a "bull market" and "bear market". Since Markov chain regime-switching stochastic volatility model can perfectly simulate the changes caused by some events or policy change, therefore, we introduce a finite-state Markov chain to the stochastic volatility model, design and built models, resulting in a Markov regime-switching option pricing model.Secondly, this paper uses the perturbation theory, singular perturbation method and two time scale method to analyze, and applies them into the volatility model to describe the "smiling curve". By dividing the procedure into slow and fast two-part, we can discuss and simplify the complicated system with two subsystems. This provides clear research thoughts and methods of theoretical research and proof.Thirdly, it might seem that option price can be calculated for explicit solution fast and accurately. But in fact, even the simplest of the Black-Scholes formula also contains calculations of the normal distribution probability. In practice, it is difficult to obtain the explicit solutions. To get the option price, the numerical algorithms are needed. Generally it can be estimated according to numerical integration method or the method to getting the approximation. This paper studies singular perturbation theory and two time scale method and introduces them into the regime-switching diffusion model. We prove that the zero-order term of the expansion is the asymptotic value of general Black-Scholes model price, and higher-order terms are its correction. We also show the convergence of the asymptotic approximation.Finally, based on above theory study, numerical experiments also illustrates our model, Markov regime-switching volatility model, can be a good depiction for volatility "smile" effects, and gets good test results. This paper also studies S&P 500 index for empirical research, using the Black-Scholes model, General stochastic volatility model and our regime-switching diffusion model to compare with the empirical option price respectively, which reveals our new model of option pricing is closer to the true value. Furthermore, we get the Markov regime-switching stochastic volatility model fits the movement of financial market.