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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bachelier, L.: Théorie de la spéculation. Ann Sci l’École Norm Supér 3e série, 17, 21–86 (1900)
Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Boca Raton: Chapman Hall (2004)
Föllmer, H., Leukert, P.: Quantile hedging. Finance Stoch 3(3), 251–273 (1999)
Föllmer, H., Leukert, P.: Efficient hedging: cost versus shortfall risk. Finance Stoch 4(2), 117–146 (2000)
Gall, J.F.L.: Brownian Motion, Martingales, and Stochastic Calculus. Cham: Springer (2016)
Glazyrina, A., Melnikov, A.: Bachelier model with stopping time and its insurance application. Insur Math Econ 93, 156–167 (2020)
Goldenberg, D.H.: A unified method for pricing options on diffusion processes. J Financ Econ 29(1), 3–34 (1991)
Kolmogorov, A.: On the analytic methods of probability theory. Math Ann 104(1), 415–458 (1931)
Kramkov, D.: Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Prob Theory Relat Fields 105, 459–479 (1996)
Melnikov, A., Nosrati, A.: Equity-Linked Life Insurance Partial Hedging Methods. Boca Raton: Chapman and Hall (2017)
Melnikov, A., Smirnov, I.: Dynamic hedging of conditional value-at-risk. Insur Math Econ 51(1), 182–190 (2012)
Pilipenko, A.: An Introduction to Stochastic Differential Equations with Reflection. Potsdam: Universitätsverlag Potsdam (2014)
Privault, N.: Stochastic Finance: An Introduction with Market Examples. Boca Raton: Chapman and Hall (2014)
Rockafellar, R., Uryasev, S.: Conditional value-at-risk for general loss distributions. J Bank Finance 26, 1443–1471 (2002)
Samuelson, P.: Rational theory of warrant pricing. Ind Manag Rev 6, 13–39 (1965)
Schachermayer, W., Teichmann, J.: How close are the option pricing formulas of Bachelier and Black–Merton–Scholes? Math Finance 18(1), 155–170 (2008)
Skorokhod, A.: Stochastic equations for diffusion processes in a bounded region. Theory Prob Appl 6(3), 264–274 (1961)
Taqqu, M.S.: Bachelier and his times: a conversation with Bernard Bru. Finance Stoch 5, 3–32 (2001)
Acknowledgements
The authors are grateful to anonymous reviewers and the editor for fruitful suggestions to improve the paper. The research is supported by Natural Sciences and Engineering Research Council of Canada. Discovery Grant RES0043487
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Melnikov, A., Wan, H. On modifications of the Bachelier model. Ann Finance 17, 187–214 (2021). https://doi.org/10.1007/s10436-020-00381-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10436-020-00381-1