A Discrete Time Approach for European and American Barrier Options

The extension of the Black{Scholes option pricing theory to the valuation of barrier options is reconsidered. Working in the binomial framework of CRR we show how various types of barrier options can be priced either by backward induction or by closed binomial formulas. We also consider analytically and numerically the convergence of the prices in discrete time to their continuous{time limits. The arising numerical problems are solved by quadratic interpolation. Furthermore, the case of American barrier options is analyzed in detail. For American barrier call options, binomial formulae and their limit results are given. Finally, the binomial approach is applied to contracts with local and partial barrier checks. 1. Introduction Barrier options are very similar to standard call and put options. However, a nal payoo can only occur if during a monitoring period the price of the underlying asset has { depending on the speciic contract under consideration { either attained or failed to attain a prespeciied upper or lower level, called the "barrier". Such contracts have indeed become the most popular types of exotic options. Merton 1973] and in particular Conze, Viswanathan 1991] have extended the Black{Scholes model to obtain closed formulas for the valuation of several types of barrier options in continuous time. In general, approximate prices for options can be obtained with binomial models even in cases where it is not possible to derive closed formulas. Here we show that prices for the whole class of barrier options can be obtained within the binomial model, if the backward induction algorithm is suitably adjusted. Fortunately, in many cases the application of the reeection principle allows us to obtain binomial formulas and hence to avoid backward induction. Similar to the limit result by Cox, Ross, and Rubinstein 1979] (CRR hereafter) for standard options we recover the well{known continuous time formulas for the price of some barrier options as limits of binomial formulas. The results can be seen as a justiication for using a binomial model as a discrete approximation of the continuous{time setting. However, unfortunately simulations reveal that with an increasing reenement of the binomial lattice option prices converge in a very irregular manner. We explain and solve this problem using quadratic interpolation. The pricing of American options continues to be of great interest to researchers. In the case of barrier options early exercise can be optimal even for call options because losses from the underlying hitting a knock{out …