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A note on first-passage times of continuously time-changed Brownian motion

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  • Hieber, Peter
  • Scherer, Matthias

Abstract

The probability of a Brownian motion with drift to remain between two constant barriers (for some period of time) is known explicitly. In mathematical finance, this and related results are required, for example, for the pricing of single-barrier and double-barrier options in a Black–Scholes framework. One popular possibility to generalize the Black–Scholes model is to introduce a stochastic time scale. This equips the modelled returns with desirable stylized facts such as volatility clusters and jumps. For continuous time transformations, independent of the Brownian motion, we show that analytical results for the double-barrier problem can be obtained via the Laplace transform of the time change. The result is a very efficient power series representation for the resulting exit probabilities. We discuss possible specifications of the time change based on integrated intensities of shot-noise type and of basic affine process type.

Suggested Citation

  • Hieber, Peter & Scherer, Matthias, 2012. "A note on first-passage times of continuously time-changed Brownian motion," Statistics & Probability Letters, Elsevier, vol. 82(1), pages 165-172.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:1:p:165-172
    DOI: 10.1016/j.spl.2011.09.018
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    References listed on IDEAS

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    Cited by:

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    6. Deelstra, Griselda & Hieber, Peter, 2023. "Randomization and the valuation of guaranteed minimum death benefits," European Journal of Operational Research, Elsevier, vol. 309(3), pages 1218-1236.
    7. Qing-Qing Yang & Wai-Ki Ching & Jia-Wen Gu & Tak Kwong Wong, 2017. "Optimal Liquidation Problems in a Randomly-Terminated Horizon," Papers 1709.05837, arXiv.org.
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