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 φ(ST ) at expiry T if neither of the continuous time-dependent barriers b± : [0, T ]→R+ 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± and linear diffusions (St )t∈[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 Δ± : [0, T ] → R 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.