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Valuation Equations for Stochastic Volatility Models

SIAM J. Finan. Math., 3(1), 351–373, 2012.

25 Pages Posted: 28 May 2016  

Erhan Bayraktar

University of Michigan at Ann Arbor - Department of Mathematics

Constantinos Kardaras

London School of Economics & Political Science (LSE)

Hao Xing

London School of Economics & Political Science (LSE)

Date Written: January 30, 2012

Abstract

We analyze the valuation partial differential equation for European contingent claims in a general framework of stochastic volatility models where the diffusion coefficients may grow faster than linearly and degenerate on the boundaries of the state space. We allow for various types of model behavior: the volatility process in our model can potentially reach zero and either stay there or instantaneously reflect, and the asset-price process may be a strict local martingale. Our main result is a necessary and sufficient condition on the uniqueness of classical solutions to the valuation equation: the value function is the unique nonnegative classical solution to the valuation equation among functions with at most linear growth if and only if the asset-price is a martingale.

Keywords: stochastic volatility models, valuation equations, Feynman–Kac theorem, strict local martingales, necessary and sufficient conditions for uniqueness

Suggested Citation

Bayraktar, Erhan and Kardaras, Constantinos and Xing, Hao, Valuation Equations for Stochastic Volatility Models (January 30, 2012). SIAM J. Finan. Math., 3(1), 351–373, 2012.. Available at SSRN: https://ssrn.com/abstract=2785690

Erhan Bayraktar (Contact Author)

University of Michigan at Ann Arbor - Department of Mathematics ( email )

2074 East Hall
530 Church Street
Ann Arbor, MI 48109-1043
United States

Constantinos Kardaras

London School of Economics & Political Science (LSE) ( email )

Houghton Street
London, WC2A 2AE
United Kingdom

Hao Xing

London School of Economics & Political Science (LSE) ( email )

Houghton Street
London, WC2A 2AE
United Kingdom

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