Barrier option pricing using adjusted transition probabilities

Abstract

In the existing literature on barrier options much effort has been exerted to ensure convergence through placing the barrier in close proximity to, or directly onto, the nodes of the tree lattice. For a variety of barrier option types we show that such a procedure may not be a necessary prerequisite to achieving accurate option price approximations. Using a trinomial tree model we show that with a suitable transition probability adjustment our "probability adjusted" model exhibits convergence to the barrier option price. We study the convergence properties of several option types, including exponential barrier options, single linear time-varying barrier options, double linear time-varying barrier options, and Bermuda options. For options whose strike prices are close to the barrier we are able to obtain numerical results where other models and techniques typically fail. Furthermore, we show that it is possible to calculate accurate option price approximations with minimal effort for options with complicated barriers that defeat standard techniques. In no single case does our method require a repositioning of the pricing lattice nodes.