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On the probability flow in the Stock market I: The Black-Scholes case

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  • Ivan Arraut
  • Alan Au
  • Alan Ching-biu Tse
  • Joao Alexandre Lobo Marques

Abstract

It is known that the probability is not a conserved quantity in the stock market, given the fact that it corresponds to an open system. In this paper we analyze the flow of probability in this system by expressing the ideal Black-Scholes equation in the Hamiltonian form. We then analyze how the non-conservation of probability affects the stability of the prices of the Stocks. Finally, we find the conditions under which the probability might be conserved in the market, challenging in this way the non-Hermitian nature of the Black-Scholes Hamiltonian.

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  • Ivan Arraut & Alan Au & Alan Ching-biu Tse & Joao Alexandre Lobo Marques, 2019. "On the probability flow in the Stock market I: The Black-Scholes case," Papers 2001.00516, arXiv.org.
    • Handle: RePEc:arx:papers:2001.00516
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    References listed on IDEAS

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    1. Arraut, Ivan & Au, Alan & Tse, Alan Ching-biu & Segovia, Carlos, 2019. "The connection between multiple prices of an Option at a given time with single prices defined at different times: The concept of weak-value in quantum finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 526(C).
    2. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    3. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    4. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    5. Baaquie, Belal E. & Corianò, Claudio & Srikant, Marakani, 2004. "Hamiltonian and potentials in derivative pricing models: exact results and lattice simulations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 334(3), pages 531-557.
    6. Jones, E. Philip, 1984. "Option arbitrage and strategy with large price changes," Journal of Financial Economics, Elsevier, vol. 13(1), pages 91-113, March.
    7. Ivan Arraut & Alan Au & Alan Ching-biu Tse & Carlos Segovia, 2019. "The connection between multiple prices of an Option at a given time with single prices defined at different times: The concept of weak-value in quantum finance," Papers 1905.05813, arXiv.org.
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