On barrier option pricing in binomial market with transaction costs

Abstract

A barrier option is the most important exotic option for structured products. Barrier options have special characteristics which distinguish them from vanilla options. The most popular standard barrier options are “knock-out” and “knock-in” options. These options are expired or exercisable automatically when the stock price hits the specified barrier level. Both out-and in-options are divided into down and up options by the level of current stock price compared to the barrier level, so there are eight kinds of basic standard barrier options. In this paper, we developed 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 which corresponding the upper boundary and lower boundary of no-transaction region. We compared the barrier option prices to their Black–Scholes counterparts in the complete market as well as in the frictional market respectively.