Proximity of Bachelier and Samuelson Models for Different Metrics
https://doi.org/10.26794/2308-944X-2021-9-3-52-76
Abstract
This paper proposes a method of comparing the prices of European options, based on the use of probabilistic metrics, with respect to two models of price dynamics: Bachelier and Samuelson. In contrast to other studies on the subject, we consider two classes of options: European options with a Lipschitz continuous payout function and European options with a bounded payout function. For these classes, the following suitable probability metrics are chosen: the Fortet-Maurier metric, the total variation metric, and the Kolmogorov metric. It is proved that their computation can be reduced to computation of the Lambert in case of the Fortet-Mourier metric, and to the solution of a nonlinear equation in other cases. A statistical estimation of the model parameters in the modern oil market gives the order of magnitude of the error, including the magnitude of sensitivity of the option price, to the change in the volatility.
Keywords
Supporting agencies: The authors express their gratitude to I. Vorobieva for valuable comments and advice in the design of this article, as well as to the reviewer for useful comments, which helped to improve the presentation.
About the Authors
S. Smirnov
Lomonosov Moscow State University
Russian Federation
Russian Federation
Sergey Smirnov — Cand. Sci. of Physical and Mathematical Sciences since 1982, Faculty of Computational Mathematics and Cybernetics, Department of System Analysis, Associate Professor since April 1, 2017. S. N. Smirnov also is Professor at National Research University Higher School of Economics (HSE). Scopus Author ID: 57211114685
D. Sotnikov
Lomonosov Moscow State University
Russian Federation
Russian Federation
Dmitry Sotnikov — student, Faculty of Computational Mathematics and Cybernetics
References
1. Bachelier, L. (1900). Théorie de la speculation. Annales Scientifiques de l’École Normale Supérieure, 3(17), 21–86.
2. Black, F. (1976). The pricing of commodity contracts. Journal of Financial Economics, 3, 167–179.
3. Black, F., & Scholes, M. S. (1973). The pricing of options and corporate liabilities, Journal of Political Economy, 81, 637–654.
4. Bogachev, V. I. (2007). Measure theory. Vol. I, II. Springer-Verlag, Berlin.
5. Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J., & Knuth, D. E. (1996). On the Lambert W function. Advances in Computational Mathematics, 5(1), 329–359.
6. Glazyrina, A., & Melnikov A. (2020). Bachelier model with stopping time and its insurance application. Insurance: Mathematics and Economics, 93, 156–167.
7. DeGroot, M. H., & Schervish, M. J. (2011). Probability and Statistics. 4th Ed. In Pearson, Delbaen, F., Schachermayer, W. The Mathematics of Arbitrage., Berlin Heidelberg: Springer-Verlag.
8. Gouriéroux, C. (1997). ARCH Models and Financial Applications. Springer Series in Statistics.
9. Grunspan, C. (2011). A note on the equivalence between the normal and the lognormal implied volatility: A model free approach. Retrieved from https://arxiv.org/pdf/1112.1782.pdf.
10. Hull, J. (2012). Options, Futures, and Other Derivatives. Boston: Prentice Hall.
11. Kolmogorov, A. (1933). Sulla determinazione empirica di una legge di distribuzione. G.Ist. Ital. Attuari., 4, 83–91.
12. Melnikov, A., Wan, H. (2021). On modifications of the Bachelier model. Annals of Finance, 17(2), 187–214.
13. Merton, R.C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(1), 141–183.
14. Øksendal, B. (1991). Stochastic Differential Equations. New York: Springer.
15. Rachev, S.T., Klebanov, L.B., Stoyanov, S.V., & Fabozzi, F.J. (2013). The Methods of Distances in the Theory of Probability and Statistics. New York: Springer.
16. Samuelson, P.A. (1965). Rational theory of warrant pricing. Industrial Management Review, 6(2), 13–39.
17. Schachermayer, W., & Teichmann, J. (2005). How close are the option pricing formulas of Bachelier and Black-Merton-Scholes. Mathematical Finance, 18(1), 55–76.
18. Thomson, I.A. (2016). Option Pricing Model: Comparing Louis Bachelier with Black-Scholes Merton. Available at SSRN 2782719.
19. Versluis, C. (2006). Option pricing: back to the thinking of Bachelier , Appl. Financ. Econ. Lett., 2(3), 205–209.
20. Melnikov, A. V., Volkov, S. V., & Nechaev, M. L., (2001). Mathematics of Financial Liabilities. Moscow: Publishing house of State University — Higher School of Economics. 260 p. (In Russian)
21. Samarsky, A. A., & Gulin, A. V. (1989). Numerical methods. Moscow: Nauka. 432 p. (In Russian)
22. Sverchkov, M. Y., & Smirnov S. N. (1990). Maximum coupling for processes from D[0, ∞]. Doklady. AN SSSR, 311, 5, 1059–1061; Dokl. Math., 41, 2, 352–354. (In Russian)
23. Shiryaev A. N. (1998). Fundamentals of Stochastic Financial Mathematics. Vol. 2. Theory Moscow: PHASIS. 512 p. (In Russian)
Review
For citations:
Smirnov S., Sotnikov D. Proximity of Bachelier and Samuelson Models for Different Metrics. Review of Business and Economics Studies. 2021;9(3):52-76. https://doi.org/10.26794/2308-944X-2021-9-3-52-76
Email this article 