Local time and the pricing of time-dependent barrier options

TL;DR

This paper develops a probabilistic method to price time-dependent double-barrier options by decomposing their value into European options minus a barrier premium, which is characterized through integral equations and semi-analytic solutions.

Contribution

It introduces a novel probabilistic decomposition of barrier option prices and derives integral equations for barrier deltas, providing new semi-analytic and numerical methods for time-dependent barriers.

Findings

Barrier premium expressed as integrals of deltas along barriers

Deltas solve a system of Volterra integral equations

Semi-analytic solution for constant barriers

Abstract

A time-dependent double-barrier option is a derivative security that delivers the terminal value $\phi(S_T)$ at expiry $T$ if neither of the continuous time-dependent barriers $b_\pm:[0,T]\to \RR_+$ have been hit during the time interval $[0,T]$. Using a probabilistic approach we obtain a decomposition of the barrier option price into the corresponding European option price minus the barrier premium for a wide class of payoff functions $\phi$, barrier functions $b_\pm$ and linear diffusions $(S_t)_{t\in[0,T]}$. We show that the barrier premium can be expressed as a sum of integrals along the barriers $b_\pm$ of the option's deltas $\Delta_\pm:[0,T]\to\RR$ at the barriers and that the pair of functions $(\Delta_+,\Delta_-)$ solves a system of Volterra integral equations of the first kind. We find a semi-analytic solution for this system in the case of constant double barriers and briefly discus a numerical algorithm for the time-dependent case.