LOCAL TIME AND THE PRICING OF TIME-DEPENDENT BARRIER OPTIONS

Abstract. A time-dependent barrier option is a derivative security that delivers the terminal payoff
at expiry T if neither of the continuous time-dependent barriers b, B: [0,T] -> R (satisfying b(t) < B(t)
for all t) have been hit during the time interval [0,T]. In this paper we describe a decomposition of the
time-dependent barrier option price into the corresponding European option price minus the barrier premium
for a wide class of linear diffusions, possibly discontinuous payoff functions and twice differentiable
barrier functions b, B. We show that the barrier premium can be expressed as an integral of the option's
delta at the barriers and that the pair of functions describing the deltas at the barriers solves a system
of Volterra integral equations of the first kind.

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