Probability of multiple crossings and pricing of double barrier options

Abstract

This paper derives pricing formulas of standard double barrier option, generalized window double barrier option and chained option. Our method is based on probabilitic approach. We derive the probability of multiple crossings of curved barriers for Brownian motion with drift, by repeatedly applying the Girsanov theorem and the reflection principle. The price of a standard double barrier option is presented as an infinite sum that converges very rapidly. Although the price formula of standard double barrier option is the same with Kunitomo and Ikeda (1992), our method gives an intuitive interpretation for each term in the infinite series. From the intuitive interpretation we present the way how to approximate the infinite sum in the pricing formula and an error bound for the given approximation. Guillaume (2003) and Jun and Ku (2013) assumed that barriers are constant to price barrier options. We extend constant barriers of window double barrier option and chained option to curved barriers. By employing multiple crossing probabilities and previous skills we derive closed formula for prices of 16 types of the generalized chained option. Based on our analytic formulas we compute Greeks of chained options directly.