The New Option Pricing With Transaction Costs, Problems And Algorithms

Exotic options are the most important for structured products. In this paper, we firstly provided an efficient algorithm to price the barrier options in the presence of proportional transaction costs, using the optimal portfolio framework, the barrier options prices are computed numerically by use of a Markov chain approximation to the continuous time singular stochastic optimal control problem, for the case of exponential utility. The method results in two option prices (Ask/Bid price) which corresponding the upper boundary and lower boundary of no transaction region. Numerical results show that the more the transaction cost, the wider the no-transaction region. And the analytical option prices are always between in the Ask/Bid spread. The effect of the investor's risk aversion on the underlying assets holdings are also be presented. Moreover, In the framework of stochastic impulse control theory, we also developed another efficient algorithm of the utility based reset option pricing and hedging approach in markets with both fixed and proportional transaction costs, this method detected four boundaries option prices which corresponding the upper boundary and lower boundary of no transaction region and the upper boundary and lower boundary of target region respectively. We propose and implement a numerical procedure for computing reset options prices' and corresponding optimal hedging strategies. Our research substantially reduced the computational time consumption as well as improves the computational efficiency and accuracy considerably. Finally, we elaborate on the options pricing differences between in the complete markets and in the markets with transaction costs.