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(S _T ) 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 (S _t ) _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.