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Computing stable numerical solutions for multidimensional American option pricing problems: a semi-discretization approach
TL;DR: In this article, the stability of the numerical solution of a multi-asset American option pricing problem is analyzed for the first time and sufficient stability conditions on step sizes, that also guarantee positivity and boundedness of the solution, are found.
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Abstract: The matter of the stability for multi-asset American option pricing problems is a present remaining challenge. In this paper a general transformation of variables allows to remove cross derivative terms reducing the stencil of the proposed numerical scheme and underlying computational cost. Solution of a such problem is constructed by starting with a semi-discretization approach followed by a full discretization using exponential time differencing and matrix quadrature rules. To the best of our knowledge the stability of the numerical solution is treated in this paper for the first time. Analysis of the time variation of the numerical solution with respect to previous time level together with the use of logarithmic norm of matrices are the basis of the stability result. Sufficient stability conditions on step sizes, that also guarantee positivity and boundedness of the solution, are found. Numerical examples for two and three asset problems justify the stability conditions and prove its competitiveness with other relevant methods.
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Figures
Table 3: Option price for the parameters (68). 
Figure 2: Wrong basket option price of Example 1 at τ = T with broken stability conditions. 
Figure 1: Reliable basket option price of Example 1 at τ = T . ![Table 1: CPU-time (in sec.) of the proposed method itself and matrix exponential by using algorithms of [39] and [40] for Example 1.](/figures/table1-1-1ipir2d3v3zx.png)
Table 1: CPU-time (in sec.) of the proposed method itself and matrix exponential by using algorithms of [39] and [40] for Example 1. 
Figure 5: The seven-point stencil for the 3D case. 
Table 5: Option price on an equidistant grid of n× n× n nodes.
![Table 1: CPU-time (in sec.) of the proposed method itself and matrix exponential by using algorithms of [39] and [40] for Example 1.](/figures/table1-1-1ipir2d3v3zx.webp)
Citations
Solving high-dimensional optimal stopping problems using deep learning
TL;DR: An algorithm is proposed for solving high-dimensional optimal stopping problems, which is based on deep learning and computes, in the context of early exercise option pricing, both approximations of an optimal exercise strategy and the price of the considered option.
Solving high-dimensional optimal stopping problems using deep learning
TL;DR: In this article, the authors propose an algorithm for solving high-dimensional optimal stopping problems, which is based on deep learning and computes both approximations of an optimal exercise strategy and the price of the considered option.
A general continuous time Markov chain approximation for multi-asset option pricing with systems of correlated diffusions
TL;DR: A general methodology for modeling and pricing financial derivatives which depend on systems of stochastic diffusion processes is developed with a general decorrelation procedure, which enables simple and efficient approximation of the driving processes by univariate CTMC approximations.
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