Valuation of Barrier Options in a Black–Scholes Setup with Jump Risk

Abstract

This paper discusses the pitfalls in the pricing of barrier options using approximations of the underlying continuous processes via discrete lattice models. To prevent from numerical deficiencies, the space axis is discretized first, and not the time axis. In a Black–Scholes setup, models with improved convergence properties are constructed: a trinomial model and a randomized trinomial model where price changes occur at the jump times of a Poisson process. These lattice models are sufficiently general to handle options with multiple barriers: the numerical difficulties are resolved and extrapolation yields even more accurate results. In a last step, we extend the Black–Scholes setup and incorporate unpredictable discontinuous price movements. The randomized trinomial model can easily be extended to this case, inheriting its superior convergence properties. JEL classification: C63, G12, G13.