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Author Contreras, G.M. pdf  doi
openurl 
  Title Stochastic volatility models at rho = +/- 1 as second class constrained Hamiltonian systems Type
  Year 2014 Publication Physica A-Statistical Mechanics And Its Applications Abbreviated Journal Physica A  
  Volume 405 Issue Pages 289-302  
  Keywords Fokker-Planck equation; Stochastic volatility models; Option pricing; Singular Lagrangian systems; Dirac's method; Constrained Hamiltonian path integrals  
  Abstract The stochastic volatility models used in the financial world are characterized, in the continuous-time case, by a set of two coupled stochastic differential equations for the underlying asset price S and volatility sigma. In addition, the correlations of the two Brownian movements that drive the stochastic dynamics are measured by the correlation parameter rho (-1 <= rho <= 1). This stochastic system is equivalent to the Fokker-Planck equation for the transition probability density of the random variables S and sigma. Solutions for the transition probability density of the Heston stochastic volatility model (Heston, 1993) were explored in Dragulescu and Yakovenko (2002), where the fundamental quantities such as the transition density itself, depend on rho in such a manner that these are divergent for the extreme limit rho = +/- 1. The same divergent behavior appears in Hagan et al. (2002), where the probability density of the SABR model was analyzed. In an option pricing context, the propagator of the bi-dimensional Black-Scholes equation was obtained in Lemmens et al. (2008) in terms of the path integrals, and in this case, the propagator diverges again for the extreme values rho = +/- 1. This paper shows that these similar divergent behaviors are due to a universal property of the stochastic volatility models in the continuum: all of them are second class constrained systems for the most extreme correlated limit rho = +/- 1. In this way, the stochastic dynamics of the rho = +/- 1 cases are different of the rho (1 <= rho <= 1) case, and it cannot be obtained as a continuous limit from the rho not equal +/- 1 regimen. This conclusion is achieved by considering the Fokker-Planck equation or the bi-dimensional Black-Scholes equation as a Euclidean quantum Schrodinger equation. Then, the analysis of the underlying classical mechanics of the quantum model, implies that stochastic volatility models at rho = +/- 1 correspond to a constrained system. To study the dynamics in an appropriate form, Dirac's method for constrained systems (Dirac, 1958, 1967) must be employed, and Dirac's analysis reveals that the constraints are second class. In order to obtain the transition probability density or the option price correctly, one must evaluate the propagator as a constrained Hamiltonian path-integral (Henneaux and Teitelboim, 1992), in a similar way to the high energy gauge theory models. In fact, for all stochastic volatility models, after integrating over momentum variables, one obtains an effective Euclidean Lagrangian path integral over the volatility alone. The role of the second class constraints is determining the underlying asset price S completely in terms of volatility, so it plays no role in the path integral. In order to examine the effect of the constraints on the dynamics for both extreme limits, the probability density function is evaluated by using semi-classical arguments, in an analogous manner to that developed in Hagan et al. (2002), for the SABR model. (C) 2014 Elsevier B.V. All rights reserved.  
  Address Univ Adolfo Ibanez, Fac Ingn & Ciencias, Santiago, Chile, Email: mauricio.contreras@uai.cl  
  Corporate Author Thesis  
  Publisher Elsevier Science Bv Place of Publication Editor  
  Language English Summary Language Original Title  
  Series Editor Series Title Abbreviated Series Title  
  Series Volume Series Issue Edition  
  ISSN 0378-4371 ISBN Medium  
  Area Expedition Conference  
  Notes WOS:000337854200030 Approved  
  Call Number UAI @ eduardo.moreno @ Serial 382  
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Author Contreras, M.; Hojman, S.A. pdf  doi
openurl 
  Title Option pricing, stochastic volatility, singular dynamics and constrained path integrals Type
  Year 2014 Publication Physica A-Statistical Mechanics And Its Applications Abbreviated Journal Physica A  
  Volume 393 Issue Pages 391-403  
  Keywords Option pricing; Stochastic volatility; Quantum mechanics; Singular Lagrangian systems; Dirac's method; Constrained Hamiltonian path integrals  
  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 p 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 rho = +/- 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 rho = +/- 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. (C) 2013 Elsevier B.V. All rights reserved.  
  Address [Contreras, Mauricio] Univ Adolfo Ibanez, Fac Ingn & Ciencias, Santiago, Chile, Email: mauricio.contreras@uai.cl  
  Corporate Author Thesis  
  Publisher Elsevier Science Bv Place of Publication Editor  
  Language English Summary Language Original Title  
  Series Editor Series Title Abbreviated Series Title  
  Series Volume Series Issue Edition  
  ISSN 0378-4371 ISBN Medium  
  Area Expedition Conference  
  Notes WOS:000328179200034 Approved  
  Call Number UAI @ eduardo.moreno @ Serial 335  
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