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Penalty methods for American options with stochastic volatility. (English) Zbl 0945.65005

Summary: The American early exercise constraint can be viewed as transforming the original linear two-dimensional stochastic volatility option pricing partial differential equation (PDE) into a PDE with a nonlinear source term. Several methods are described for enforcing the early exercise constraint by using a penalty source term in the discrete equations. The resulting nonlinear algebraic equations are solved using an approximate Newton iteration. The solution of the Jacobian is obtained using an incomplete LU preconditioned conjugate gradient-like method. Some example computations are presented for option pricing problems based on a stochastic volatility model, including an exotic American chooser option written on a put and call with discrete double knockout barriers and discrete dividends.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
91B24 Microeconomic theory (price theory and economic markets)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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