Quantum physics methods in share option valuation

Marcin Wróblewski


This paper deals with European share option pricing using quantum physics methods. These contingent claims are usually priced using the Black-Scholes equation. This nonlinear parabolic equation is based on geometric Brownian motion model of the stock price stochastic process. Similar processes also appear among quantum particles and are described by the time-dependent Schrödinger equation. In this paper, the option pricing based on the Schrödinger equation approach is proposed. Using Wick transformation, the Black-Scholes equation is transformed into the equivalent Schrödinger equation. The Fourier separation method is used to find analytical solutions to this equation. The last square method is used to calibrate the Schrödinger model based on real market data. Numerical results are provided and discussed.

Słowa kluczowe: option pricing, econophysics, quantum physics methods

Schrödinger E., Quantisierung als Eigenwertproblem (Erste Mitteilung), Ann. Phys. 79, 1926, 361-376.

Black F., Scholes M., The pricing of options and corporate liabilities, Journal of Political Economy, 81 (3), 637-654.

Merton R., Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, Vol. 3, Issue 1–2, 1976, 125-144.

Kishimoto M., On the Black-Scholes Equation: Various Derivations, MSE 408 Term Paper, 2-3.

Papoulis A., Wiener-Lévy Process, §15-3 in Probability, Random Variables, and Stochastic Processes, 2nd ed. McGraw-Hill, New York 1984, 292-293.

Honarkhah M., Caers J., Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling, Mathematical Geosciences, 42, 2010, 487-517.

Dzhrbashyan M.M., On integral representation and expansion in generalized Taylor series of entire functions of several complex variables, Mat. Sb. (N.S.), 41(83), 3, 1957, 257-276.

Contreras M., Pellicer R., Villena M., Ruiz A., A quantum model of option pricing: When Black Scholes meets Schrodinger and its semi-classical limit, Physica A 389, 2010, 5447-5459.

Hazewinkel M., ed., Fourier method, Encyclopedia of Mathematics, Springer, 2001.

Peleg Y., Pnini R., Zaarur E., Hecht E., Quantum mechanics. Schuam’s outlines, (2nd ed.), McGraw Hill., 2010, 68-69.

Shleifer A., Vishny R., The limits of arbitrage, Journal of Finance, 52, 1997, 35-55.

Bohr N., Discussion with Einstein, [in:] Schilpp P.A., ed., Albert Einstein: Philosopher- Scientist, 235.

Challet D., Marsili M., Zhang Y.C., Modeling Market Mechanism with Minority Game, Physica A-276, 2000, 284-315.

McCrary S.A., Chapter 1: Introduction to Hedge Funds. How to Create and Manage a Hedge Fund: A Professional’s Guide, John Wiley & Sons, 2002, 7-8.

Cherpakov P.V., Periodic solutions of the heat equation, Izv. Vyssh. Uchebn. Zaved. Mat., No. 2, 1959, 247-251.