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Windings of planar processes, exponential functionals and Asian options

Part of: Stochastic processes Mathematical finance Markov processes Limit theorems

Published online by Cambridge University Press:  16 November 2018

Wissem Jedidi  and
Stavros Vakeroudis
Wissem Jedidi*
Affiliation:
King Saud University and Université de Tunis El Manar
Stavros Vakeroudis*
Affiliation:
University of the Aegean
*
* Postal address: Department of Statistics & OR, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia. Email address: wissem_jedidi@yahoo.fr
** Postal address: Department of Mathematics, University of the Aegean, Vourliotis Building, Office Y5, 83200 Karlovasi, Samos, Greece. Email address: stavros.vakeroudis@gmail.com
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Abstract

Motivated by a common mathematical finance topic, we discuss the reciprocal of the exit time from a cone of planar Brownian motion which also corresponds to the exponential functional of Brownian motion in the framework of planar Brownian motion. We prove a conjecture of Vakeroudis and Yor (2012) concerning infinite divisibility properties of this random variable and present a novel simple proof of the result of DeBlassie (1987), (1988) concerning the asymptotic behavior of the distribution of the Bessel clock appearing in the skew-product representation of planar Brownian motion, as t→∞. We use the results of the windings approach in order to obtain results for quantities associated to the pricing of Asian options.

Keywords

Planar Brownian motionLévy processstable processwindingskew-product representationBessel clockBougerol's identityinfinite divisibilityBernstein functionLévy measureAsian option

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes 60G52: Stable processes 60J65: Brownian motion
Secondary: 60F05: Central limit and other weak theorems 60G44: Martingales with continuous parameter 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 3 , September 2018 , pp. 726 - 742
Copyright
Copyright © Applied Probability Trust 2018 

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