Complex Barrier Options

Abstract

This article develops a new trinomial tree model for barrier options. It is well-known that for barrier options, the positions of nodes in the tree with respect to the barrier value are critical. We use a time-dependent shift to position the tree optimally with respect to the barrier. The model is very flexible and can be used to price options with time-varying barrier structures. It can be used to price knock-in and knock-out options based on either one or two underlying assets, including those with time-varying barriers - single or double. Traditional lattice models all have difficulties when the underlying asset price is very close to the barrier. This model does not suffer from that limitation. Also, in many applications, the barrier condition is based on the daily or weekly fixings. A simple solution for the discrete-time barrier observation is advanced, which enables us to uncover the price differences among barrier options with different observation frequencies.