Pricing Of Barrier Options In A Stochastic Volatility Model

In 1973, Black and Scholes, as the pioneers, established the celebrated optionpricing formula. The pricing problems for financial derivatives has become one ofthe key research works in Mathematical Finance. Recently, along with the increasingdevelopment in global financial markets, there are a great number of exotic optionswith ?exible transactions and convenient prices in markets, such as Barrier options,Asian options, Lookback options and so on. Exotic options is a kind of much morecomplex and profitable options than the plain vanilla products with European orAmerican style, and is more attractive to investors. There are abundant exotic op-tions that its trading numbers and transaction amounts are very huge, many newexotic options are continuously produced by financial institutions. So, how to eval-uate these options has become a key topic in modern financial theoretical researchand practical application, its academical value and social economic significance arewell-known. However, in the real financial markets, a suitable market model has anin?uence up on investor's decision-making, risk management, hedging and others.The classical Black-Scholes (BS) model expresses the price of derivative securi-ties as the function of both the underlying asset's price and a constant volatility.Since the BS model is simple and easy to use, therefore it is well attracted, but thevolatility of stock's price is assumed to be a constant which the observational dataare inconsistent with that of the actual market. At the same time, a many empiricalstudies have shown that the market price of the option implies the"smile"effectof the volatility . Therefore, many researchers are trying to modify the BS model,which is made to capture the reality. At present, one of significant modified modelsis called the stochastic volatility model (SV) in which the volatility of stock's price is assumed to be another stochastic process correlated with the underlying stock.In order to overcome the"smile"effect of implied volatility for the stock's price.This thesis considers the pricing problem for European and American-style optionswith single-barrier in the Hull-White SV model. Our main contributions are as fol-lows:Chapter 1 provides firstly an introduction to the significance and necessity ofthe pricing option. Second, we introduce the pricing literature of the European andAmerican barrier options, and the causes for choosing this title, the research contentand our study framework.In Chapter 2, European single-barrier options pricing in the Hull-White SVmodel is considered. in the Hull-White SV model, we discuss first the independencebetween the stock's price process and the volatility process with the correlation co-efficientρ= 0. Applying the martingale methods, the conditional distribution prop-erties and the Taylor expansions for the European barrier options formula in the BSmodel, we obtain an approximated closed form solution of the option price. Second,we consider further the case with the correlation coefficientρ= 0, using the anti-thetic Monte Carlo simulation method and CRR (Cox Ross Rubinstein) method, weprovide some numerical results of option prices, and compared these results witheach others, we analyze also risk management.Based on the same model in chapter 2, the American single-barrier options pric-ing is considered In Chapter 3. First, we use transform analysis to convert the (S,Y )space into (Q, F) space, and apply the tree lattice method to solve the value of(Q, F), then get the value of (S,Y ) with the transform. Second, we use the CRRmethod to solve the American option prices in which the method is improved theprice calculation for the American barrier options. The optimal exercise boundarycurve S? of the American barrier options is also obtained. Finally, we analyze theeffects of the stock's prices to hedge parameters.Our main conclusions and the further research works are summarized in Chapter4.