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arXiv · 0809.1747

Local time and the pricing of time-dependent barrier options

Aleksandar Mijatovic
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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 $b_\pm:[0,T]\to \RR_+$ 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 $b_\pm$ and linear diffusions $(S_t)_{t\in[0,T]}$. We show that the barrier premium can be expressed as a sum of integrals along the barriers $b_\pm$ of the option's deltas $Δ_\pm:[0,T]\to\RR$ 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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Aleksandar Mijatovic. 2008-09-10. Local time and the pricing of time-dependent barrier options. https://arxiv.org/abs/0809.1747

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