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Teor. Veroyatnost. i Primenen., 2003, Volume 48, Issue 3, Pages 503–533 (Mi tvp268)  

This article is cited in 36 scientific papers (total in 36 papers)

Bessel processes, the integral of geometric Brownian motion, and Asian options

P. Carr, M. Schröder

New York University

Abstract: This paper is motivated by questions about averages of stochastic processes which originate in mathematical finance, originally in connection with valuing the so-called Asian options. Starting with [M. Yor, Adv. Appl. Probab., 24 (1992), pp. 509–531], these questions about exponential functionals of Brownian motion have been studied in terms of Bessel processes using the Hartman–Watson theory of [M. Yor, Z. Wahrsch. Verw. Gebiete, 53 (1980), pp. 71–95]. Consequences of this approach for valuing Asian options proper have been spelled out in [H. Geman and M. Yor, Math. Finance, 3 (1993), pp. 349–375] whose Laplace transform results were in fact regarded as a significant advance. Unfortunately, a number of difficulties with the key results of this last paper have surfaced which are now addressed in this paper. One of them in particular is of a principal nature and originates with the Hartman–Watson approach itself: this approach is in general applicable without modifications only if it does not involve Bessel processes of negative indices. The main mathematical contribution of this paper is the development of three principal ways to overcome these restrictions, in particular by merging stochastics and complex analysis in what seems a novel way, and the discussion of their consequences for the valuation of Asian options proper.

Keywords: Asian options, integral of geometric Brownian motion, Bessel processes, Laplace transform, complex analytic methods in stochastics.

DOI: https://doi.org/10.4213/tvp268

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Theory of Probability and its Applications, 2004, 48:3, 400–425

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Received: 11.12.2002
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Citation: P. Carr, M. Schröder, “Bessel processes, the integral of geometric Brownian motion, and Asian options”, Teor. Veroyatnost. i Primenen., 48:3 (2003), 503–533; Theory Probab. Appl., 48:3 (2004), 400–425

Citation in format AMSBIB
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    This publication is cited in the following articles:
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    7. Gulisashvili A., Stein E.M., “Asymptotic behavior of the stock price distribution density and implied volatility in stochastic volatility models”, Appl. Math. Optim., 61:3 (2010), 287–315  crossref  mathscinet  zmath  isi  scopus
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    19. Yakubovich S., “On the Yor Integral and a System of Polynomials Related to the Kontorovich-Lebedev Transform”, Integral Transform. Spec. Funct., 24:8 (2013), 672–683  crossref  mathscinet  zmath  isi  scopus
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    28. Pirjol D., Zhu L., “Short Maturity Asian Options in Local Volatility Models”, SIAM J. Financ. Math., 7:1 (2016), 947–992  crossref  mathscinet  zmath  isi  scopus
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    30. Pirjol D., Zhu L., “Asymptotics For the Discrete-Time Average of the Geometric Brownian Motion and Asian Options”, Adv. Appl. Probab., 49:2 (2017), 446–480  crossref  mathscinet  isi  scopus
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    32. Pirjol D., Zhu L., “Sensitivities of Asian Options in the Black-Scholes Model”, Int. J. Theor. Appl. Financ., 21:1 (2018), 1850008  crossref  mathscinet  zmath  isi  scopus
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