Pricing barrier options under stochastic volatility framework

Option pricing problem plays an extremely important role in quantitative finance. In complete market, Black-Scholes-Merton theory has been central to the development of financial engineering as both discipline and profession. However, in incomplete market, there are not any replicating portfolios for those options, and thus, the market traders cannot apply the law of one price for obtaining a unique solution. Fortunately, the authors can get a fair price via local-equilibrium principle. In this paper, the authors apply the stochastic control theory to price the exotic option-barrier options, and analyze the relationship between the price and the current positions. The authors get the explicit expression for the market price of the risk. The position effect plays a significant role in option pricing, because it can tell the trader how many and which direction to trade with the market in order to reach the local equilibrium with the market.