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Geometric Brownian motion with affine drift and its time-integral

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  • Runhuan Feng
  • Pingping Jiang
  • Hans Volkmer

Abstract

The joint distribution of a geometric Brownian motion and its time-integral was derived in a seminal paper by Yor (1992) using Lamperti's transformation, leading to explicit solutions in terms of modified Bessel functions. In this paper, we revisit this classic result using the simple Laplace transform approach in connection to the Heun differential equation. We extend the methodology to the geometric Brownian motion with affine drift and show that the joint distribution of this process and its time-integral can be determined by a doubly-confluent Heun equation. Furthermore, the joint Laplace transform of the process and its time-integral is derived from the asymptotics of the solutions. In addition, we provide an application by using the results for the asymptotics of the double-confluent Heun equation in pricing Asian options. Numerical results show the accuracy and efficiency of this new method.

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  • Runhuan Feng & Pingping Jiang & Hans Volkmer, 2020. "Geometric Brownian motion with affine drift and its time-integral," Papers 2012.09661, arXiv.org.
    • Handle: RePEc:arx:papers:2012.09661
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    References listed on IDEAS

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    1. Feng, Runhuan & Volkmer, Hans W., 2014. "Spectral Methods For The Calculation Of Risk Measures For Variable Annuity Guaranteed Benefits," ASTIN Bulletin, Cambridge University Press, vol. 44(3), pages 653-681, September.
    2. Alan L. Lewis, 1998. "Applications of Eigenfunction Expansions in Continuous‐Time Finance," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 349-383, October.
    3. Joseph Abate & Ward Whitt, 2006. "A Unified Framework for Numerically Inverting Laplace Transforms," INFORMS Journal on Computing, INFORMS, vol. 18(4), pages 408-421, November.
    4. Feng, Runhuan & Volkmer, Hans W., 2012. "Analytical calculation of risk measures for variable annuity guaranteed benefits," Insurance: Mathematics and Economics, Elsevier, vol. 51(3), pages 636-648.
    5. Joseph Abate & Ward Whitt, 1995. "Numerical Inversion of Laplace Transforms of Probability Distributions," INFORMS Journal on Computing, INFORMS, vol. 7(1), pages 36-43, February.
    6. Vadim Linetsky, 2004. "Spectral Expansions for Asian (Average Price) Options," Operations Research, INFORMS, vol. 52(6), pages 856-867, December.
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