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Local time and the pricing of time-dependent barrier options

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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 ±:[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.

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  1. Department of Mathematics, Imperial College London, Huxley Building, 180 Queen’s Gate, London, SW7 2AZ, UK

    Aleksandar Mijatović

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  1. Aleksandar Mijatović
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Correspondence to Aleksandar Mijatović.

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I should like to thank Petros Spanoudakis for pointing out the problem and for stimulating discussion. Thanks for many useful comments are due to Dirk Becherer, Nick Bingham, Johan Tysk and Michalis Zervos.

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Mijatović, A. Local time and the pricing of time-dependent barrier options. Finance Stoch 14, 13–48 (2010). https://doi.org/10.1007/s00780-008-0077-5

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