An efficient numerical method for the valuation of American multi-asset options

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.

Lattice Boltzmann method for the linear complementarity problem arising from American option pricing

“…Observing the model (2), it is easy to see that the LCP defined on the unbounded is very difficult to solve numerically. The far field estimate method is one of the efficient methods for estimating the unbounded solution domain, which has been successfully introduced into the American option pricing problem in recent years [24,30,35]. Therefore, this approach is followed here for the truncation of the LCP (2).…”

“…Since the streaming process equation ( 25) is handled in a similar way to the assignment process, only the collision process equation (24) needs to be analyzed. We consider the stability of f + i (x, t), which is sufficient to obtain the stability of u(x, t).…”

“…Kadalbajoo et al [23] presented a new radial basis function based numerical method for European and American options, which can be combined with the operator splitting method to solve the LCP for American options. Zhang et al [24], based on the far-field truncation method, the finite element discretization and the projection contraction method, proposed an efficient method for the LCP of pricing American multi-asset options. Other efficient algorithms for the LCP of American option pricing are also commonly seen and cited in the literature [25][26][27][28][29][30].…”

A Generalized Finite Difference Method for Solving Hamilton–Jacobi–Bellman Equations in Optimal Investment

“…However, mesh methods require a regular mesh of grids and are only applicable to regular boundaries, whereas, in dynamic stochastic economic models, the natural modeling domain may have an irregular shape [27,28]. For example, in the option pricing problem, the pricing model for American backdated options can be described as a two-dimensional free boundary problem [29][30][31]. De Angelis et al [32] considered the problem of optimal entry into an irreversible investment scheme with a cost function that is non-convex for the control variables, where it is necessary to study optimal entry policies with an irregular boundary.…”