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A spectral element approximation to price European options with one asset and stochastic volatility. (English) Zbl 1203.91307
Summary: We develop a Legendre quadrilateral spectral element approximation for the Black-Scholes equation to price European options with one underlying asset and stochastic volatility. A weak formulation of the equations imposes the boundary conditions naturally along the boundaries where the equation becomes singular, and in particular, we use an energy method to derive boundary conditions at outer boundaries for which the problem is well-posed on a finite domain. Using Heston’s analytical solution as a benchmark, we show that the spectral element approximation along with the proposed boundary conditions gives exponential convergence in the solution and the Greeks to the level of time and boundary errors in a domain of financial interest.
91G60Numerical methods in mathematical finance
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
65M70Spectral, collocation and related methods (IVP of PDE)
91B70Stochastic models in economics
91G20Derivative securities
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