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Pricing multi-asset option problems: a Chebyshev pseudo-spectral method

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Abstract

The aim of this paper is to contribute a new second-order pseudo-spectral method via a non-uniform distribution of the computational nodes for solving multi-asset option pricing problems. In such problems, the prices are required to be as accurately as possible around the strike price. Accordingly, the proposed modification of the Chebyshev–Gauss–Lobatto points would concentrate on this area. This adaptation is also fruitful for the non-smooth payoffs which cause discontinuities in the strike price. The proposed scheme competes well with the existing methods such as finite difference, meshfree, and adaptive finite difference methods on several benchmark problems.

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Acknowledgements

The author is grateful to two anonymous referees and the handling editor for several corrections and comments, which directly contribute to the readability and reliability of the current paper.

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Authors and Affiliations

  1. Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, 45137-66731, Iran

    Fazlollah Soleymani

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  1. Fazlollah Soleymani
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Correspondence to Fazlollah Soleymani.

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Communicated by Elisabeth Larsson.

Appendix

Appendix

We here provide the PDE formulation in Option Problem 4 with the initial and boundary conditions given in Sect. 2 for interested readers as follows.

$$\begin{aligned} \frac{\partial \mathcal {V}}{\partial \tau }&= \frac{1}{2}\sigma _1^2 S_1^2 \frac{\partial ^2 \mathcal {V}}{\partial S_1^2} +\frac{1}{2}\sigma _2^2 S_2^2 \frac{\partial ^2 \mathcal {V}}{\partial S_2^2} +\frac{1}{2}\sigma _3^2 S_3^2 \frac{\partial ^2 \mathcal {V}}{\partial S_3^2} +\frac{1}{2}\sigma _4^2 S_4^2 \frac{\partial ^2 \mathcal {V}}{\partial S_4^2} +\frac{1}{2}\sigma _5^2 S_5^2 \frac{\partial ^2 \mathcal {V}}{\partial S_5^2}\nonumber \\&\quad +\sigma _1 \sigma _2 \rho _{1,2} S_1 S_2 \frac{\partial ^2 \mathcal {V}}{\partial S_1\, \partial S_2} +\sigma _1 \sigma _3 \rho _{1,3} S_1 S_3 \frac{\partial ^2 \mathcal {V}}{\partial S_1\, \partial S_3}\nonumber \\&\quad +\sigma _1 \sigma _4 \rho _{1,4} S_1 S_4 \frac{\partial ^2 \mathcal {V}}{\partial S_1\, \partial S_4} +\sigma _1 \sigma _5 \rho _{1,5} S_1 S_5 \frac{\partial ^2 \mathcal {V}}{\partial S_1\, \partial S_5}\nonumber \\&\quad +\sigma _2 \sigma _3 \rho _{2,3} S_2 S_3 \frac{\partial ^2 \mathcal {V}}{\partial S_2\, \partial S_3} +\sigma _2 \sigma _4 \rho _{2,4} S_2 S_4 \frac{\partial ^2 \mathcal {V}}{\partial S_2\, \partial S_4}\nonumber \\&\quad +\sigma _2 \sigma _5 \rho _{2,5} S_2 S_5 \frac{\partial ^2 \mathcal {V}}{\partial S_2\, \partial S_5} +\sigma _3 \sigma _4 \rho _{3,4} S_3 S_4 \frac{\partial ^2 \mathcal {V}}{\partial S_3\, \partial S_4}\nonumber \\&\quad +\sigma _3 \sigma _5 \rho _{3,5} S_3 S_5 \frac{\partial ^2 \mathcal {V}}{\partial S_3\, \partial S_5} +\sigma _4 \sigma _5 \rho _{4,5} S_4 S_5 \frac{\partial ^2 \mathcal {V}}{\partial S_4\, \partial S_5}\nonumber \\&\quad +(r-q_1) S_1 \frac{\partial \mathcal {V}}{\partial S_1} +(r-q_2) S_2 \frac{\partial \mathcal {V}}{\partial S_2} +(r-q_3) S_3 \frac{\partial \mathcal {V}}{\partial S_3}\nonumber \\&\quad +(r-q_4) S_4 \frac{\partial \mathcal {V}}{\partial S_4} +(r-q_5) S_5 \frac{\partial \mathcal {V}}{\partial S_5} -r \mathcal {V}. \end{aligned}$$
(68)

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Soleymani, F. Pricing multi-asset option problems: a Chebyshev pseudo-spectral method. Bit Numer Math 59, 243–270 (2019). https://doi.org/10.1007/s10543-018-0722-0

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