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Weak approximation of averaged diffusion processes

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  • Gobet, Emmanuel
  • Miri, Mohammed

Abstract

We derive expansion results in order to approximate the law of the average of the marginal of diffusion processes. The average is computed w.r.t. a general parameter that is involved in the diffusion dynamics. Our approximation is based on the use of proxys with normal distribution or log-normal distribution, so that the expansion terms are explicit. We provide non asymptotic error bounds, which justifies the expansion accuracy as the time or the diffusion coefficients are small in a suitable sense.

Suggested Citation

  • Gobet, Emmanuel & Miri, Mohammed, 2014. "Weak approximation of averaged diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 475-504.
    • Handle: RePEc:eee:spapps:v:124:y:2014:i:1:p:475-504
    DOI: 10.1016/j.spa.2013.08.007
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    References listed on IDEAS

    as
    1. Pierre Etoré & Emmanuel Gobet, 2012. "Stochastic expansion for the pricing of call options with discrete dividends," Post-Print hal-00507787, HAL.
    2. Romain Bompis & Emmanuel Gobet, 2012. "Asymptotic and non asymptotic approximations for option valuation," Post-Print hal-00720650, HAL.
    3. E. Benhamou & E. Gobet & M. Miri, 2012. "Analytical formulas for a local volatility model with stochastic rates," Quantitative Finance, Taylor & Francis Journals, vol. 12(2), pages 185-198, September.
    4. Hélyette Geman & Marc Yor, 1993. "Bessel Processes, Asian Options, And Perpetuities," Mathematical Finance, Wiley Blackwell, vol. 3(4), pages 349-375, October.
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    6. E. Benhamou & E. Gobet & M. Miri, 2009. "Smart expansion and fast calibration for jump diffusions," Finance and Stochastics, Springer, vol. 13(4), pages 563-589, September.
    7. Turnbull, Stuart M. & Wakeman, Lee Macdonald, 1991. "A Quick Algorithm for Pricing European Average Options," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 26(3), pages 377-389, September.
    8. Paolo Foschi & Stefano Pagliarani & Andrea Pascucci, 2011. "Black-Scholes formulae for Asian options in local volatility models," Quaderni di Dipartimento 7, Department of Statistics, University of Bologna.
    9. E. Benhamou & E. Gobet & M. Miri, 2010. "Expansion Formulas For European Options In A Local Volatility Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 13(04), pages 603-634.
    10. Rainer Avikainen, 2009. "On irregular functionals of SDEs and the Euler scheme," Finance and Stochastics, Springer, vol. 13(3), pages 381-401, September.
    11. Pierre Étor� & Emmanuel Gobet, 2012. "Stochastic Expansion for the Pricing of Call Options with Discrete Dividends," Applied Mathematical Finance, Taylor & Francis Journals, vol. 19(3), pages 233-264, August.
    12. Naoto Kunitomo & Akihiko Takahashi, 2001. "The Asymptotic Expansion Approach to the Valuation of Interest Rate Contingent Claims," Mathematical Finance, Wiley Blackwell, vol. 11(1), pages 117-151, January.
    13. Bernard Lapeyre & Emmanuel Temam, 2001. "Competitive Monte Carlo methods for the pricing of Asian options," Post-Print hal-01667057, HAL.
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    Citations

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    Cited by:

    1. Florian Bourgey & Stefano De Marco & Emmanuel Gobet, 2022. "Weak approximations and VIX option price expansions in forward variance curve models," Papers 2202.10413, arXiv.org, revised May 2022.
    2. Julien Hok & Philip Ngare & Antonis Papapantoleon, 2018. "Expansion formulas for European quanto options in a local volatility FX-LIBOR model," Papers 1801.01205, arXiv.org, revised Apr 2018.
    3. Pagliarani, S. & Pascucci, A. & Pignotti, M., 2017. "Intrinsic expansions for averaged diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2560-2585.
    4. Romain Bompis, 2017. "Weak approximations for arithmetic means of geometric Brownian motions and applications to Basket options," Working Papers hal-01502886, HAL.
    5. Julien Hok & Philip Ngare & Antonis Papapantoleon, 2018. "Expansion Formulas For European Quanto Options In A Local Volatility Fx-Libor Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(02), pages 1-43, March.
    6. Hyungbin Park & Jonghwa Park, 2019. "Pricing and hedging short-maturity Asian options in local volatility models," Papers 1911.12944, arXiv.org.
    7. Cai, Ning & Li, Chenxu & Shi, Chao, 2021. "Pricing discretely monitored barrier options: When Malliavin calculus expansions meet Hilbert transforms," Journal of Economic Dynamics and Control, Elsevier, vol. 127(C).
    8. Dan Pirjol & Lingjiong Zhu, 2016. "Short Maturity Asian Options in Local Volatility Models," Papers 1609.07559, arXiv.org.
    9. Dan Pirjol & Lingjiong Zhu, 2017. "Short Maturity Asian Options for the CEV Model," Papers 1702.03382, arXiv.org.

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