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Primal-Dual Active Set Method for American Lookback Put Option Pricing

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Partial differential equations

Published online by Cambridge University Press:  07 September 2017

Haiming Song ,
Xiaoshen Wang ,
Kai Zhang  and
Qi Zhang
Haiming Song
Affiliation:
Department of Mathematics, Jilin University Changchun, Jilin, 130012, China
Xiaoshen Wang
Affiliation:
Department of Mathematics and Statistics, University of Arkansas at Little Rock, Little Rock, Arkansas, 72204, USA
Kai Zhang*
Affiliation:
Department of Mathematics, Jilin University Changchun, Jilin, 130012, China
Qi Zhang
Affiliation:
School of Science, Shenyang University of Technology, Shenyang, Liaoning, 110870, China
*
*Corresponding author. Email address:kzhang@jlu.edu.cn (K. Zhang)
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Abstract

The pricing model for American lookback options can be characterised as a two-dimensional free boundary problem. The main challenge in this problem is the free boundary, which is also the main concern for financial investors. We use a standard technique to reduce the pricing model to a one-dimensional linear complementarity problem on a bounded domain and obtain a corresponding variational inequality. The inequality is discretised by finite differences and finite elements in the temporal and spatial directions, respectively. By enforcing inequality constraints related to the options using Lagrange multipliers, the discretised variational inequality is reformulated as a set of semi-smooth equations, which are solved by a primal-dual active set method. One of the major advantages of our algorithm is that we can obtain the option values and the free boundary simultaneously, and numerical simulations show that our approach is as efficient as some other methods.

Keywords

American lookback optionlinear complementarity problemvariational inequalityfinite element methodprimal-dual active set method

MSC classification

Secondary: 35A35: Theoretical approximation to solutions 65K10: Optimization and variational techniques 65M12: Stability and convergence of numerical methods 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 3 , August 2017 , pp. 603 - 614
Copyright
Copyright © Global-Science Press 2017 

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