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Stochastic volatility models, singular dynamics and constrained path integrals

Mauricio Contreras and Sergio Hojman

posted on 15 May 2016

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Stochastic volatility models have been widely studied and used in the nancial world. Heston model [1] 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 which relates the two Brownian motions that drive the stochastic dynamics associated to the volatility and the underlying asset. Solutions to the Heston model, using a path integral approach, are found in [15] while in [16], [17] propagators for dierent stochastic volatility models are constructed. In all previous cases, the propagator is not dened 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 the 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 [18], [19] in order to properly obtain the propagator in the form of a Euclidean Hamiltonian path integral [21]. After integrating over momenta, one gets a 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 this case, 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.

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It is the first paper on the applications of constrained systems ideas in option pricing.

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