Stochastic volatility models at [formula omitted] as second class constrained Hamiltonian systems

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 σ. In addition, the correlations of the two Brownian movements that drive the stochastic dynamics are measured by the correlation parameter ρ(−1≤ρ≤1). This stochastic system is equivalent to the Fokker–Planck equation for the transition probability density of the random variables S and σ. 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 ρ in such a manner that these are divergent for the extreme limit ρ=±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 ρ=±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 ρ=±1. In this way, the stochastic dynamics of the ρ=±1 cases are different of the −1<ρ<1 case, and it cannot be obtained as a continuous limit from the ρ≠±1 regimen. This conclusion is achieved by considering the Fokker–Planck equation or the bi-dimensional Black–Scholes equation as a Euclidean quantum Schrödinger equation. Then, the analysis of the underlying classical mechanics of the quantum model, implies that stochastic volatility models at ρ=±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.