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Endogenous stochastic arbitrage bubbles and the Black–Scholes model

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  • Contreras G., Mauricio

Abstract

This paper develops a model that incorporates the presence of stochastic arbitrage explicitly in the Black–Scholes equation. Here, the arbitrage is generated by a stochastic bubble, which generalizes the deterministic arbitrage model obtained in the literature (Contreras et al., 2010). It is considered to be a generic stochastic dynamic for the arbitrage bubble, and a generalized Black–Scholes equation is then derived. The resulting equation is similar to that of the stochastic volatility models, but there are no undetermined parameters as the market price of risk.

Suggested Citation

  • Contreras G., Mauricio, 2021. "Endogenous stochastic arbitrage bubbles and the Black–Scholes model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 583(C).
    • Handle: RePEc:eee:phsmap:v:583:y:2021:i:c:s0378437121005963
    DOI: 10.1016/j.physa.2021.126323
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    References listed on IDEAS

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    1. Kirill Ilinski, 1999. "How to account for virtual arbitrage in the standard derivative pricing," Papers cond-mat/9902047, arXiv.org.
    2. Bustamante, M. & Contreras, M., 2016. "Multi-asset Black–Scholes model as a variable second class constrained dynamical system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 457(C), pages 540-572.
    3. Contreras, Mauricio & Pellicer, Rely & Villena, Marcelo & Ruiz, Aaron, 2010. "A quantum model of option pricing: When Black–Scholes meets Schrödinger and its semi-classical limit," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(23), pages 5447-5459.
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    5. Contreras, M. & Echeverría, J. & Peña, J.P. & Villena, M., 2020. "Resonance phenomena in option pricing with arbitrage," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
    6. Kirill Ilinski & Alexander Stepanenko, 1999. "Derivative pricing with virtual arbitrage," Papers cond-mat/9902046, arXiv.org.
    7. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
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    9. 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.
    10. Contreras, Mauricio & Montalva, Rodrigo & Pellicer, Rely & Villena, Marcelo, 2010. "Dynamic option pricing with endogenous stochastic arbitrage," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(17), pages 3552-3564.
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