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A Wiener–Hopf based approach to numerical computations in fluctuation theory for Lévy processes

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  • Peter Iseger
  • Paul Gruntjes
  • Michel Mandjes

Abstract

This paper focuses on numerical evaluation techniques related to fluctuation theory for Lévy processes; they can be applied in various domains, e.g., in finance in the pricing of so-called barrier options. More specifically, with $$\bar{X}_t:= \sup _{0\le s\le t} X_s$$ denoting the running maximum of the Lévy process $$X_t$$ , the aim is to evaluate $$\mathbb{P }(\bar{X}_t \in \mathrm{d}x)$$ for $$t,x>0$$ . The starting point is the Wiener–Hopf factorization, which yields an expression for the transform $$\mathbb E e^{-\alpha \bar{X}_{e(\vartheta )}}$$ of the running maximum at an exponential epoch (with $$\vartheta ^{-1}$$ the mean of this exponential random variable). This expression is first rewritten in a more convenient form, and then it is pointed out how to use Laplace inversion techniques to numerically evaluate $$\mathbb{P }(\bar{X}_t\in \mathrm{d}x).$$ In our experiments we rely on the efficient and accurate algorithm developed in den Iseger (Probab Eng Inf Sci 20:1–44, 2006 ). We illustrate the performance of the algorithm with various examples: Brownian motion (with drift), a compound Poisson process, and a jump diffusion process. In models with jumps, we are also able to compute the density of the first time a specific threshold is exceeded, jointly with the corresponding overshoot. The paper is concluded by pointing out how our algorithm can be used in order to analyze the Lévy process’ concave majorant. Copyright Springer-Verlag Berlin Heidelberg 2013

Suggested Citation

  • Peter Iseger & Paul Gruntjes & Michel Mandjes, 2013. "A Wiener–Hopf based approach to numerical computations in fluctuation theory for Lévy processes," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 78(1), pages 101-118, August.
  • Handle: RePEc:spr:mathme:v:78:y:2013:i:1:p:101-118
    DOI: 10.1007/s00186-013-0434-9
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    References listed on IDEAS

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    1. Chris Carolan & Richard Dykstra, 2003. "Characterization of the least concave majorant of brownian motion, conditional on a vertex point, with application to construction," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 55(3), pages 487-497, September.
    2. Soeren Asmussen & Dilip Madan & Martijn Pistorius, 2007. "Pricing Equity Default Swaps under an approximation to the CGMY L\'{e}% vy Model," Papers 0711.2807, arXiv.org.
    3. Asmussen, Søren & Avram, Florin & Pistorius, Martijn R., 2004. "Russian and American put options under exponential phase-type Lévy models," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 79-111, January.
    4. 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.
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