Local time and the pricing of time-dependent barrier options

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 (+/-):[0,T]-> a"e(+) 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 (+/-) and linear diffusions (S (t) ) (ta[0,T]). We show that the barrier premium can be expressed as a sum of integrals along the barriers b (+/-) of the option's deltas Delta (+/-):[0,T]-> a"e 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.