The Pricing of Double Barrier Options and Their Variations

Abstract

This paper derives closed-form solutions for double barrier options under the usual assumptions of the Black-Scholes (1973) model using reflection principle in Brownian motions. While we recover the well-known valuation formulae for standard double barrier options, we show how to apply the same technique to variations of double barrier options. The option payoff function we use is the simplest one that includes call, put, and digital options as special cases. The solution is expressed in general as summations of an infinite number of normal distribution functions for standard double barrier options and in many non-trivial cases the solution consists of only finite terms. Double barrier options with rebates and exponentially curved barriers are also considered. Numerical approximations are provided as a verification of the closed-form solutions.

Keywords: Double Barrier Options, Reflection Principal of Brownian Motion,Change of Measures

JEL Classification: G13