Primal-dual active-set method for solving the unilateral pricing problem of American better-of options on two assets

Yiyuan Qian; Haiming Song; Xiaoshen Wang; Kai Zhang; Yiyuan Qian; Haiming Song; Xiaoshen Wang; Kai Zhang

doi:10.3934/era.2022005

Electronic Research Archive

2022, Volume 30, Issue 1: 90-115. doi: 10.3934/era.2022005

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Primal-dual active-set method for solving the unilateral pricing problem of American better-of options on two assets

1.
Department of Mathematics, Jilin University, Changchun 130012, China
2.
Department of Mathematics and Statistics, University of Arkansas at Little Rock, Arkansas 72204, USA

Received: 28 September 2021 Revised: 07 November 2021 Accepted: 21 November 2021 Published: 10 December 2021

In this paper, an efficient numerical algorithm is proposed for the valuation of unilateral American better-of options with two underlying assets. The pricing model can be described as a backward parabolic variational inequality with variable coefficients on a two-dimensional unbounded domain. It can be transformed into a one-dimensional bounded free boundary problem by some conventional transformations and the far-field truncation technique. With appropriate boundary conditions on the free boundary, a bounded linear complementary problem corresponding to the option pricing is established. Furthermore, the full discretization scheme is obtained by applying the backward Euler method and the finite element method in temporal and spatial directions, respectively. Based on the symmetric positive definite property of the discretized matrix, the value of the option and the free boundary are obtained simultaneously by the primal-dual active-set method. The error estimation is established by the variational theory. Numerical experiments are carried out to verify the efficiency of our method at the end.
- American better-of option,
- far-field truncation technique,
- linear complementary problem,
- finite element method,
- primal-dual active-set method
Citation: Yiyuan Qian, Haiming Song, Xiaoshen Wang, Kai Zhang. Primal-dual active-set method for solving the unilateral pricing problem of American better-of options on two assets[J]. Electronic Research Archive, 2022, 30(1): 90-115. doi: 10.3934/era.2022005

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Abstract

In this paper, an efficient numerical algorithm is proposed for the valuation of unilateral American better-of options with two underlying assets. The pricing model can be described as a backward parabolic variational inequality with variable coefficients on a two-dimensional unbounded domain. It can be transformed into a one-dimensional bounded free boundary problem by some conventional transformations and the far-field truncation technique. With appropriate boundary conditions on the free boundary, a bounded linear complementary problem corresponding to the option pricing is established. Furthermore, the full discretization scheme is obtained by applying the backward Euler method and the finite element method in temporal and spatial directions, respectively. Based on the symmetric positive definite property of the discretized matrix, the value of the option and the free boundary are obtained simultaneously by the primal-dual active-set method. The error estimation is established by the variational theory. Numerical experiments are carried out to verify the efficiency of our method at the end.

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