Pricing Basket Options In Stochastic Volatility Model

In recent years, along with the rapid growth in global financial markets, there area great number of exotic options ( called path dependent options or no-standard op-tions ) with ?exible transactions and convenient prices, for example, Basket options,Asian options, Lookback options and so on. Furthermore, many new combined fi-nancial derivatives are continuously introduced by the financial institutions. How tovalue these options has become a key topic in modern financial theoretical researchand practical application. It is well known that in the real financial market a suit-able market model plays an important role for investor's decision, risk managementand hedging. Since the classical Black-Scholes model has many deficiencies in de-scribing the systemic variables, many extensions are widely provided, such as theStochastic Volatility model, Levy processes, the fractional Brownian motion model,etc. The Stochastic Volatility model has been proved to be more suitable than theBlack-Scholes models in which the volatility of the underlying asset is constant.The stochastic volatility model has become a very prominent model, however, thereis little work on option pricing in this case.One of the most active trading options on financial derivatives today is the basketoptions whose prices are more cheaper than the plain vanilla one, and are of lowerrisks. Therefore more and more investors are fond of holding it to hedge. Thebasket options are portfolio options which are composed of a variety of underlyingassets. They are usually less expensive than the total cost of all the individual asset.This is because the basket option is more efficient than single option in investment.With growing demands of the investors, who want to diversify their investment,investors'demand for a portfolio of options is increasing day by day. According to the difference of the income functions, basket options can be divided into two forms:the Geometric Basket Options and the Arithmetic Basket Options. Due to the moreunderlying assets, the pricing model become more complex. Therefore, this articlefocuses on the case of this two kinds of asset portfolio. Respectively, with thearithmetic and geometric income function forms of European and American basketoptions pricing.Chapter 1 provides simply an introduction on the academic literature for optionspricing, especially for pricing the basket option, and the background of this article.In Chapter 2, under the stochastic volatility model the pricing of European bas-ket options are considered. By the derivation of the characteristic function underrisk-neutral measure, the corresponding distribution function for the Geometric-style European call basket option by Shephard's theorem is obtained with applyingthe method of partial differential equations. For the Arithmetic-style European callbasket option, it's difficult to derive the accuracy distribution function of the sum oftwo variables which are both log-normal distributed. The corresponding distributionfunction is obtained by approximating the solution of the arithmetic form of Euro-pean basket call options. And the MonteCarlo simulation method has been used incalculation and analysis.In Chapter 3, the pricing of American basket options are considered under thesame assumption in Chapter 2. The value of American options and the reasonable-ness of implementation in advance have been analyzed. Deduced the best imple-mentation of the boundary of the American basket call options. Then using the Kimintegral equation method in the price of European call option, we have obtained theapproximated solution for the American basket call option price. Finally, the ef-fects of the option value and the optimal exercise boundary are analyzed with somenumerical examples in which these results are compared with the finite differencemethod.Our main conclusions and the further research works are summarized in Chapter4.