The Approximative Calculation Of Barrier Options Under Stochastic Volatility

Options with the feature of barrier are barrier opinions,dependent on whether some barriers are breached within the lives of the options,is referred as a kind of simplest path-dependent options.Barrier options are geared to the needs of sophisticated investors,because these options are created to provide risk managers with cheaper means to hedge their exposures without paying for the price ranges that they believe unlikely to occur.The Black-Scholes option pricing equation for a stock assumes that the stock price follows the Geometric Brownian Motion and the volatility of the stock price remains constant over the life of the option.Although the latter may be a valid simplifying assumption for short maturity options,it becomes less increasingly plausible as the maturity increases in empirical studies.So more and more studies concentrate on how to extend this assumption.Not only do stochastic volatility models explain the basic shapes of smile patterns,but they also allow for more realistic theories of the "term structure" of implied volatility.In order to generate a pricing formula that is appropriate for the case where volatility follows an autoregressive and mean-reverting stochastic process,a large and growing literature suggests that this case is empirically relevant;we use the function of Ornstein-Uhlenbeck process to describe stochastic volatility.At the same time we can make sure that volatility is positive.Now,it will make the partial differential equation of pricing become more complex and hasn't any closed-form solution.So we will derive an approximate solution of this equation as the price formula of barrier options under the incomplete market by using the solution function expansion according toε^1/2=(α^-1)^1/2 and the methods of solving Poisson equations.The study subject of the text is down-out calls whose approximate pricing formula in the incomplete market are worked out in the text.