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Option pricing, stochastic volatility, singular dynamics and constrained path integrals

Author

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  • Contreras, Mauricio
  • Hojman, Sergio A.

Abstract

Stochastic volatility models have been widely studied and used in the financial world. The Heston model (Heston, 1993) [7] is one of the best known models to deal with this issue. These stochastic volatility models are characterized by the fact that they explicitly depend on a correlation parameter ρ which relates the two Brownian motions that drive the stochastic dynamics associated to the volatility and the underlying asset. Solutions to the Heston model in the context of option pricing, using a path integral approach, are found in Lemmens et al. (2008) [21] while in Baaquie (2007,1997) [12,13] propagators for different stochastic volatility models are constructed. In all previous cases, the propagator is not defined for extreme cases ρ=±1. It is therefore necessary to obtain a solution for these extreme cases and also to understand the origin of the divergence of the propagator. In this paper we study in detail a general class of stochastic volatility models for extreme values ρ=±1 and show that in these two cases, the associated classical dynamics corresponds to a system with second class constraints, which must be dealt with using Dirac’s method for constrained systems (Dirac, 1958,1967) [22,23] in order to properly obtain the propagator in the form of a Euclidean Hamiltonian path integral (Henneaux and Teitelboim, 1992) [25]. After integrating over momenta, one gets an Euclidean Lagrangian path integral without constraints, which in the case of the Heston model corresponds to a path integral of a repulsive radial harmonic oscillator. In all the cases studied, the price of the underlying asset is completely determined by one of the second class constraints in terms of volatility and plays no active role in the path integral.

Suggested Citation

  • Contreras, Mauricio & Hojman, Sergio A., 2014. "Option pricing, stochastic volatility, singular dynamics and constrained path integrals," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 393(C), pages 391-403.
    • Handle: RePEc:eee:phsmap:v:393:y:2014:i:c:p:391-403
    DOI: 10.1016/j.physa.2013.08.057
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    References listed on IDEAS

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    1. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    2. 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.
    3. 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.
    4. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    5. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
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    1. Contreras G., Mauricio, 2014. "Stochastic volatility models at ρ=±1 as second class constrained Hamiltonian systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 405(C), pages 289-302.
    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, 2017. "Dynamic optimization and its relation to classical and quantum constrained systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 479(C), pages 12-25.
    4. G., Mauricio Contreras & Peña, Juan Pablo, 2019. "The quantum dark side of the optimal control theory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 515(C), pages 450-473.
    5. Shafi, Khuram & Latif, Natasha & Shad, Shafqat Ali & Idrees, Zahra & Gulzar, Saqib, 2018. "Estimating option greeks under the stochastic volatility using simulation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 503(C), pages 1288-1296.
    6. Mauricio Contreras & Alejandro Llanquihu'en & Marcelo Villena, 2015. "On the Solution of the Multi-asset Black-Scholes model: Correlations, Eigenvalues and Geometry," Papers 1510.02768, arXiv.org.

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