Barrier Options on Underlyings with Time-Dependent Parameters: A Perturbation Expansion Approach

Abstract

In this paper we work out a perturbation expansion for the prices of barrier options on an underlying that follows a Black-Scholes dynamics with time-dependent parameters. It is to be considered as an extension of earlier work that provides, in a sense, a zero order approximation. Through an elegant argument we work out an easy numerical procedure by which a simple yet powerful correction to the average coefficients formulae commonly used yields prices that lie within basis points of the exact price of the option. The very important issue of Greeks bucketing is discussed as well by means of a numerical example, showing that this approach is viable for the effective management of option positions. This has important consequences on options trading particularly in the FX and credit derivatives markets.

Keywords: option pricing, Black--Scholes, barrier options, FX derivatives, credit derivatives, partial differential equations, perturbation theory