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On modifications of the Bachelier model

Author

Listed:
  • Alexander Melnikov

    (University of Alberta)

  • Hongxi Wan

    (University of Alberta)

Abstract

Mathematically, stock prices described by a classical Bachelier model are sums of a Brownian motion and an absolute continuous drift. Hence, stock prices can take negative values, and financially, it is not appropriate. This drawback is overcome by Samuelson who has proposed the exponential transformation and provided the so-called Geometrical Brownian motion. In this paper, we introduce two additional modifications which are based on SDEs with absorption and reflection. We show that the model with reflection may admit arbitrage, but the model with an appropriate absorption leads to a better model. Comparisons regarding option pricing among the standard Bachelier model, the Black–Scholes model and the modified Bachelier model with absorption at zero are executed. Moreover, our main findings are also devoted to the Conditional Value-at-Risk based partial hedging in the framework of these models. Illustrative numerical examples are provided.

Suggested Citation

  • Alexander Melnikov & Hongxi Wan, 2021. "On modifications of the Bachelier model," Annals of Finance, Springer, vol. 17(2), pages 187-214, June.
    • Handle: RePEc:kap:annfin:v:17:y:2021:i:2:d:10.1007_s10436-020-00381-1
    DOI: 10.1007/s10436-020-00381-1
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    References listed on IDEAS

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    1. Walter Schachermayer & Josef Teichmann, 2008. "How Close Are The Option Pricing Formulas Of Bachelier And Black–Merton–Scholes?," Mathematical Finance, Wiley Blackwell, vol. 18(1), pages 155-170, January.
    2. Hans FÃllmer & Peter Leukert, 2000. "Efficient hedging: Cost versus shortfall risk," Finance and Stochastics, Springer, vol. 4(2), pages 117-146.
    3. Glazyrina, Anna & Melnikov, Alexander, 2020. "Bachelier model with stopping time and its insurance application," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 156-167.
    4. Murad S. Taqqu, 2001. "Bachelier and his times: A conversation with Bernard Bru," Finance and Stochastics, Springer, vol. 5(1), pages 3-32.
    5. Goldenberg, David H., 1991. "A unified method for pricing options on diffusion processes," Journal of Financial Economics, Elsevier, vol. 29(1), pages 3-34, March.
    6. Melnikov, Alexander & Smirnov, Ivan, 2012. "Dynamic hedging of conditional value-at-risk," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 182-190.
    7. Rockafellar, R. Tyrrell & Uryasev, Stanislav, 2002. "Conditional value-at-risk for general loss distributions," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1443-1471, July.
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    Citations

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    Cited by:

    1. Dean Buckner & Kevin Dowd & Hardy Hulley, 2024. "Arbitrage problems with reflected geometric Brownian motion," Finance and Stochastics, Springer, vol. 28(1), pages 1-26, January.
    2. Svetlozar Rachev & Nancy Asare Nyarko & Blessing Omotade & Peter Yegon, 2023. "Bachelier's Market Model for ESG Asset Pricing," Papers 2306.04158, arXiv.org.
    3. Sergey Smirnov & Dimitri Sotnikov & Andrey Zanochkin, 2024. "Approximation and asymptotics in the superhedging problem for binary options," Annals of Finance, Springer, vol. 20(4), pages 421-458, December.
    4. Alexis Anagnostakis & David Criens & Mikhail Urusov, 2025. "On weak notions of no-arbitrage in a 1D general diffusion market with interest rates," Papers 2503.14078, arXiv.org.

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    Keywords

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    JEL classification:

    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • G1 - Financial Economics - - General Financial Markets
    • G2 - Financial Economics - - Financial Institutions and Services
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies

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