General information
Latest issue
Impact factor
Guidelines for authors
Submit a manuscript

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Teor. Veroyatnost. i Primenen.:

Personal entry:
Save password
Forgotten password?

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.


Full text: PDF file (3954 kB)
References: PDF file   HTML file

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

Bibliographic databases:

Received: 11.12.2002

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
\by P.~Carr, M.~Schr\"oder
\paper Bessel processes, the integral of geometric Brownian motion, and Asian options
\jour Teor. Veroyatnost. i Primenen.
\yr 2003
\vol 48
\issue 3
\pages 503--533
\jour Theory Probab. Appl.
\yr 2004
\vol 48
\issue 3
\pages 400--425

Linking options:

    SHARE: FaceBook Twitter Livejournal

    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Schröder M., “Laguerre series in contingent claim valuation, with applications to Asian options”, Math. Finance, 15:3 (2005), 491–531  crossref  mathscinet  zmath  isi  scopus
    2. Schröder M., “On ladder height densities and Laguerre series in the study of stochastic functionals. II. Exponential functionals of Brownian motion and Asian option values”, Adv. in Appl. Probab., 38:4 (2006), 995–1027  crossref  mathscinet  zmath  isi  scopus
    3. Peskir G., “On the fundamental solution of the Kolmogorov-Shiryaev equation”, From stochastic calculus to mathematical finance: The Shiryaev Festschift, Springer, Berlin, 2006, 535–546  crossref  mathscinet  isi
    4. Schröder M., “On constructive complex analysis in finance: explicit formulas for Asian options”, Quart. Appl. Math., 66:4 (2008), 633–658  crossref  mathscinet  zmath  isi  scopus
    5. Cruz-Báez D.I., González-Rodríguez J.M., “A different approach for pricing Asian options”, Appl. Math. Lett., 21:3 (2008), 303–306  crossref  mathscinet  zmath  isi  scopus
    6. Carr P., Ewald Ch.-O., Xiao Ya., “On the qualitative effect of volatility and duration on prices of Asian options”, Finance Research Letters, 5:3 (2008), 162–171  crossref  isi  scopus
    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
    8. Pintoux C., Privault N., “A direct solution to the Fokker-Planck equation for exponential Brownian functionals”, Anal. Appl. (Singap.), 8:3 (2010), 287–304  crossref  mathscinet  zmath  isi  scopus
    9. Gulisashvili A., Stein E.M., “Asymptotic behavior of distribution densities in models with stochastic volatility. I”, Math. Finance, 20:3 (2010), 447–477  crossref  mathscinet  zmath  isi  scopus
    10. Pintoux C., Privault N., “The Dothan pricing model revisited”, Math. Finance, 21:2 (2011), 355–363  mathscinet  zmath  isi
    11. Yang Zh., Ewald Ch.-O., Menkens O., “Pricing and hedging of Asian options: quasi-explicit solutions via Malliavin calculus”, Math. Methods Oper. Res., 74:1 (2011), 93–120  crossref  mathscinet  zmath  isi  scopus
    12. V. R. Fatalov, “Integral Functionals for the Exponential of the Wiener Process and the Brownian Bridge: Exact Asymptotics and Legendre Functions”, Math. Notes, 92:1 (2012), 79–98  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    13. Feng R., Volkmer H.W., “Analytical Calculation of Risk Measures for Variable Annuity Guaranteed Benefits”, Insur. Math. Econ., 51:3 (2012), 636–648  crossref  mathscinet  zmath  isi  scopus
    14. Cai N., Kou S., “Pricing Asian Options Under a Hyper-Exponential Jump Diffusion Model”, Oper. Res., 60:1 (2012), 64–77  crossref  mathscinet  zmath  isi  scopus
    15. Schroeder M., “On Arithmetic-Average Asian Power Options: Closed Forms and Explicit Methods for Valuation”, Q. J. Mech. Appl. Math., 66:1 (2013), 1–27  crossref  mathscinet  zmath  isi  scopus
    16. Privault N., Uy W.I., “Monte Carlo Computation of the Laplace Transform of Exponential Brownian Functionals”, Methodol. Comput. Appl. Probab., 15:3 (2013), 511–524  crossref  mathscinet  zmath  isi  scopus
    17. Patie P., “Asian Options Under One-Sided Levy Models”, J. Appl. Probab., 50:2 (2013), 359–373  crossref  mathscinet  zmath  isi  scopus
    18. Metzler A., “The Laplace Transform of Hitting Times of Integrated Geometric Brownian Motion”, J. Appl. Probab., 50:1 (2013), 295–299  crossref  mathscinet  zmath  isi  scopus
    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
    20. Urban R., “An Explicit Formula For the Heat Kernel of a Left Invariant Operator”, ALEA-Latin Am. J. Probab. Math. Stat., 11:1 (2014), 299–306  mathscinet  zmath  isi
    21. Shi Q., Yang X., “Pricing Asian Options in a Stochastic Volatility Model With Jumps”, Appl. Math. Comput., 228 (2014), 411–422  crossref  mathscinet  zmath  isi  scopus
    22. Feng R., Volkmer H.W., “Conditional Asian Options”, Int. J. Theor. Appl. Financ., 18:6 (2015), 1550040  crossref  mathscinet  zmath  isi  scopus
    23. Pirjol D., Zhu L., “Discrete sums of geometric Brownian motions, annuities and Asian options”, Insur. Math. Econ., 70 (2016), 19–37  crossref  mathscinet  zmath  isi  elib  scopus
    24. Vrins F., “Characteristic function of time-inhomogeneous Lévy-driven Ornstein–Uhlenbeck processes”, Stat. Probab. Lett., 116 (2016), 55–61  crossref  mathscinet  zmath  isi  scopus
    25. Privault N., Yu J., “Stratified approximations for the pricing of options on average”, J. Comput. Financ., 19:4 (2016), 95–113  crossref  isi  scopus
    26. Penney R., Urban R., “Poisson kernels on nilpotent, 3-meta-abelian groups”, Ann. Mat. Pura Appl., 195:2 (2016), 293–307  crossref  mathscinet  zmath  isi  scopus
    27. Feng R., Volkmer H.W., “An identity of hitting times and its application to the valuation of guaranteed minimum withdrawal benefit”, Math. Financ. Econ., 10:2 (2016), 127–149  crossref  mathscinet  zmath  isi  scopus
    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
    29. Belius D., Kistler N., “The subleading order of two dimensional cover times”, Probab. Theory Relat. Field, 167:1-2 (2017), 461–552  crossref  mathscinet  zmath  isi  scopus
    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
    31. Privault N., Wei X., “Fast Computation of Risk Measures For Variable Annuities With Additional Earnings By Conditional Moment Matching”, Astin Bull., 48:1 (2018), 171–196  crossref  mathscinet  zmath  isi  scopus
    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
    33. Levendorskii S., “Pricing Arithmetic Asian Options Under Levy Models By Backward Induction in the Dual Space”, SIAM J. Financ. Math., 9:1 (2018), 1–27  crossref  mathscinet  zmath  isi  scopus
    34. Pigato P., “Tube Estimates For Diffusion Processes Under a Weak Hormander Condition”, Ann. Inst. Henri Poincare-Probab. Stat., 54:1 (2018), 299–342  crossref  mathscinet  zmath  isi  scopus
    35. Ocejo A., “Asian Option as a Fixed-Point”, J. Fixed Point Theory Appl., 20:2 (2018), UNSP 93  crossref  mathscinet  isi  scopus
    36. Pirjol D., Zhu L., “Short Maturity Asian Options For the Cev Model”, Probab. Eng. Inform. Sci., 33:2 (2019), 258–290  crossref  mathscinet  isi  scopus
  • Теория вероятностей и ее применения Theory of Probability and its Applications
    Number of views:
    This page:379
    Full text:81

    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019