Local time and the pricing of time-dependent barrier options

Abstract: A time-dependent double-barrier option is a derivative security that delivers the terminal value $φ(S_T)$ 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 $φ$, 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 $Δ\pm:[0,T]\to\RR$ at the barriers and that the pair of functions $(Δ+,Δ_-)$ 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.