Adaptive θ-methods for pricing American options

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:222

Article

Khaliq, Abdul Q. M.²  Voss, David A.¹  Kazmi, Kamran³ 

[1] Western Illinois Univ, Dept Math, Macomb, IL 61455 USA

[2] Middle Tennessee State Univ, Dept Math Sci, Murfreesboro, TN 37132 USA

[3] Univ Iowa, Dept Math, Iowa City, IA 52240 USA

关键词: Black-Scholes PDE;    American options;    theta-methods;    Method of Lines;    Locally one-dimensional exponential splitting;    Adaptive time-stepping;   

DOI  :  10.1016/j.cam.2007.10.035

来源: Elsevier

【 摘 要 】

We develop adaptive theta-methods for solving the Black-Scholes PDE for American options. By adding a small, continuous term, the Black-Scholes PDE becomes an advection-diffusion-reaction equation on a fixed spatial domain. Standard implementation of theta-methods would require a Newton-type iterative procedure at each time step thereby increasing the computational complexity of the methods. Our linearly implicit approach avoids such complications. We establish a general framework under which theta-methods satisfy a discrete version of the positivity constraint charactetistic of American options, and numerically demonstrate the sensitivity of the constraint. The positivity results are established for the single-asset and independent two-asset models. In addition, we have incorporated and analyzed an adaptive time-step control strategy to increase the computational efficiency. Numerical experiments are presented for one- and two-asset American options, using adaptive exponential splitting for two-asset problems. The approach is compared with an iterative solution of the two-asset problem in terms of computational efficiency. (C) 2007 Elsevier B.V. All rights reserved.

【 授权许可】

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