An efficient numerical method for the valuation of American multi-asset options
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An efficient numerical method for the valuation of American multi-asset options Qi Zhang1 · Haiming Song2
· Chengbo Yang2 · Fangfang Wu1
Received: 18 July 2018 / Revised: 3 October 2019 / Accepted: 3 August 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, a fast and efficient numerical method which relies on the far field truncation technique, the finite element discretization, and the projection contraction method (PCM) is proposed for pricing American multi-asset options. It is well known that American multiasset option satisfies a linear complementarity problem (LCP), which is a multi-dimensional variable coefficient parabolic model on an unbounded domain. First, we transform it into a constant coefficient parabolic LCP on a bounded domain by some skillful transformations and far-field boundary estimate. Then, the variational inequality (VI) corresponding to the truncated LCP is obtained. Further, it is discretized by the finite element method and the implicit difference method in spatial and temporal directions, respectively. Based on the symmetric positive definiteness of the full-discrete matrix, the discretized VI is solved by the PCM. Finally, numerical simulations are provided to verify the efficiency of the proposed method. Keywords American multi-asset options · Linear complementarity problem · Far-field boundary estimate · Finite element method · Projection and contraction method Mathematics Subject Classification 35A35 · 90A09 · 65M12 · 65M60
1 Introduction Multi-asset options are popular trading instruments in financial markets, whose payoff functions depend on more than one underlying asset (Bensoussan 1984; Jiang 2005). Based on the traditional Black–Scholes model (Black and Scholes 1973), the multi-asset option of European-style satisfies a parabolic partial differential equation (PDE), which only could be exercised at the maturity date. While, the American-style option fulfils a linear complementarity problem (LCP), which could be exercised at any time prior to the maturity date. The difference on these two kinds of options leads to a fact that the fair price of American
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Haiming Song [email protected]
1
School of Science, Shenyang University of Technology, Shenyang 110807, China
2
Department of Mathematics, Jilin University, Changchun 130012, China 0123456789().: V,-vol
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option is always at least as much as the price of its European counterpart. And the possible difference is termed as the early-exercise premium (Hull 2006; Wilmott et al. 1997). Generally speaking, the European multi-asset option always has closed-form solution. Whereas, for American one, due to the existence of the optimal exercise boundary, it not only leads to the lack of the analytical closed-form solution, but also makes the pricing problem to be a nontrivial and demanding job. Therefore, efficient numerical algorithms are essential for pricing American multi-asset options accurately. Recently, PDE discretization
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