Convergence remedies for non-smooth payoffs in option pricing

Pooley, David M.; Vetzal, Kenneth R.; Forsyth, Peter

doi:10.21314/jcf.2003.101

Cited by 131 publications

(81 citation statements)

References 11 publications

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“…The most common way to obtain the price of an American option is to formulate a linear complementarity problem and then solve it numerically; see, for example, [19,20,23,30,31]. In the following, the price of an underlying asset is modeled in a more realistic way than only applying an asset price model with a log-normal return distribution.…”

Section: Model For American Optionsmentioning

confidence: 99%

Efficient numerical methods for pricing American options under stochastic volatility

Ikonen

Toivanen

2007

Numerical Methods Partial

143

132

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Five numerical methods for pricing American put options under Heston's stochastic volatility model are described and compared. The option prices are obtained as the solution of a two-dimensional parabolic partial differential inequality. A finite difference discretization on nonuniform grids leading to linear complementarity problems with M-matrices is proposed. The projected SOR, a projected multigrid method, an operator splitting method, a penalty method, and a componentwise splitting method are considered. The last one is a direct method while all other methods are iterative. The resulting systems of linear equations in the operator splitting method and in the penalty method are solved using a multigrid method. The projected multigrid method and the componentwise splitting method lead to a sequence of linear complementarity problems with one-dimensional differential operators that are solved using the Brennan and Schwartz algorithm. The numerical experiments compare the accuracy and speed of the considered methods. The accuracies of all methods appear to be similar. Thus, the additional approximations made in the operator splitting method, in the penalty method, and in the componentwise splitting method do not increase the error essentially. The componentwise splitting method is the fastest one. All multigrid-based methods have similar rapid grid independent convergence rates. They are about two or three times slower that the componentwise splitting method. On the coarsest grid the speed of the projected SOR is comparable with the multigrid methods while on finer grids it is several times slower.

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Section: Model For American Optionsmentioning

confidence: 99%

Efficient numerical methods for pricing American options under stochastic volatility

Ikonen

Toivanen

2007

Numerical Methods Partial

143

132

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“…to smooth the initial conditions (Pooley et al, 2001). The most theoretically sound method for doing this is projecting the initial conditions onto the set of basis functions used to discretize the equations (Wahlbin, 1980).…”

Section: Digital Call Optionsmentioning

confidence: 99%

Numerical convergence properties of option pricing PDEs with uncertain volatility

Pooley

Forsyth

Vetzal

2003

IMA Journal of Numerical Analysis

125

112

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The pricing equations derived from uncertain volatility models in finance are often cast in the form of nonlinear partial differential equations. Implicit timestepping leads to a set of nonlinear algebraic equations which must be solved at each timestep. To solve these equations, an iterative approach is employed. In this paper, we prove the convergence of a particular iterative scheme for one factor uncertain volatility models. We also demonstrate how non-monotone discretization schemes (such as standard Crank-Nicolson timestepping) can converge to incorrect solutions, or lead to instability. Numerical examples are provided.

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“…The stability of GF G and F GF is evaluated by freezing the coefficients in (9), (10), (4), and (14) at s and assuming a periodic, smooth solution u (4), (9), (10), and (14) to obtain the polynomial fulfilled by z. With…”

Section: Accuracy and Stabilitymentioning

confidence: 99%

“…For this reason the so called Rannacher scheme [11] is often employed [3], [10]. In this method a number of initial time-steps with the backward Euler method are preceding the Crank-Nicolson method in order to damp the oscillations occurring from the discontinuities in the pay-off function.…”

Section: Introductionmentioning

confidence: 99%