Option Pricing For A Related Option In A Regime-switching Market Using FFT

As an investment tool of derivative securities, the option emerges from theincreasing development of the market economy. In recent years, options become moreand more important in the financial field, many exchange have a lot of futures andoptions trading. Among them, option pricing is one of the core issues in mathematicalfinance.In1973, Black and Scholes proposed BS model, they make a breakthrough ofoption pricing, which has an epoch-making significance. In1997, American economistsRobert C.Merton and Myron S.Scholes were awarded the Nobel prizes in economics, tocommend them the tremendous contribution in option pricing. They established anddeveloped the Black-Scholes model,,under market pricing changes, they laid thetheoretical basis for the various kinds of options pricing in the emerging derivativefinancial market, Which has include stocks, bonds, currencies, commodities. With thecontinuous development of options, despite the European and American options, theinternational financial derivatives markets emerges a lot of new options,which wascombination, derived out from the standard options. the multi underlying assets optionis a kind among them. In this paper, we studied the underlying assets which have acorrelation between the Share price movement, namely the related assets option pricingmodel.The traditional Black-Scholes Option Pricing formula has been widely used forPricing Option and hedging in finance industry, but this model fails to reflect someempirical phenomena, such as the larger random fluctuations, the non-normal featuresand the implied volatility is not a constant as in Black-Scholes model. And there are alarge number of empirical results indicate that the asset price follows the geometricBrownian motion is not realistic. Over the past three decades many different optionvaluation models have been proposed. Some of these include jump-diffusion modelsLévy processes, stochastic volatility models, GARCH model and others. Recently, therehas been considerable interest in applications of a regime switching model which ismodulated by a continuous time Markov chain to option pricing problem. The states ofthe continuous time Markov chain can be interpreted as the states of the economy. Thetransitions of the states of the economy may be attributed to structural changes of the economy and business cycles. Therefore, based on the previous research result thisthesis discuss the Option Pricing under Regime-Switching models and a two-stateRegime-Switching model of the related assets option pricing model is provided.Firstly, we present a model of the related assets option pricing model, andconcluded the European call option price under the risk neutral conditions, thenintroduce the Regime-Switching factors, through the Regime-Switching Esschertransformation, we can get the European call option price under the risk neutralconditions.Furthermore, through Fourier transform, the option price decision problem istransformed into computing random vector markov chain sojourn time problem. Usingthe method proposed by Carr and Madan, taking the European option price formula as aconvolution, then using the properties that the Laplace transform about the convolutionis equal to the product of the two factors Laplace transform, get the Laplace transformabout the convolution, then using an inverse Laplace transform again, we can get thedisplay form of the markov chain sojourn time joint characteristic function under theRegime-Switching market with two state(m=2), and get the value of the option. Thenthrough fast Fourier inverse transform find out the European call option price, after theprocessing about discretization, we can find out the FFT algorithm expression of theEuropean call option.Finally, in the numerical results and analysis, based on fast Fourier transformalgorithm, we using the MATLAB software concluded the European call option pricewhere the markov chain with two state cases, emphatically comparative and analysis therelationship about the European option price with executive price, options maturity date,volatility. And comparative the option price which without Regime-Switching factors.Getting that Regime-Switching related option model has higher option price, and thegap is increasing with the increase of the due date, the reason is that with Regime-Switching factor the pricing getting the risk compensation.