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Volume 27, Issue 4
Finite Difference Approximation for Pricing the American Lookback Option

Tie Zhang, Shuhua Zhang & Danmei Zhu

DOI: 10.4208/jcm.2009.27.4.015

J. Comp. Math., 27 (2009), pp. 484-494.

Published online: 2009-08

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  • Abstract

In this paper we are concerned with the pricing of lookback options with American type constrains. Based on the differential linear complementary formula associated with the pricing problem, an implicit difference scheme is constructed and analyzed. We show that there exists a unique difference solution which is unconditionally stable. Using the notion of viscosity solutions, we also prove that the finite difference solution converges uniformly to the viscosity solution of the continuous problem. Furthermore, by means of the variational inequality analysis method, the $\mathcal{O}(Δt+Δx^2)$-order error estimate is derived in the discrete $L_2$-norm provided that the continuous problem is sufficiently regular. In addition, a numerical example is provided to illustrate the theoretical results.

  • Keywords

American lookback options, Finite difference approximation, Stability and convergence, Error estimates.

  • AMS Subject Headings

65M12, 65M06, 91B28.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-27-484, author = {Tie Zhang, Shuhua Zhang and Danmei Zhu}, title = {Finite Difference Approximation for Pricing the American Lookback Option}, journal = {Journal of Computational Mathematics}, year = {2009}, volume = {27}, number = {4}, pages = {484--494}, abstract = {

In this paper we are concerned with the pricing of lookback options with American type constrains. Based on the differential linear complementary formula associated with the pricing problem, an implicit difference scheme is constructed and analyzed. We show that there exists a unique difference solution which is unconditionally stable. Using the notion of viscosity solutions, we also prove that the finite difference solution converges uniformly to the viscosity solution of the continuous problem. Furthermore, by means of the variational inequality analysis method, the $\mathcal{O}(Δt+Δx^2)$-order error estimate is derived in the discrete $L_2$-norm provided that the continuous problem is sufficiently regular. In addition, a numerical example is provided to illustrate the theoretical results.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2009.27.4.015}, url = {http://global-sci.org/intro/article_detail/jcm/8585.html} }
TY - JOUR T1 - Finite Difference Approximation for Pricing the American Lookback Option AU - Tie Zhang, Shuhua Zhang & Danmei Zhu JO - Journal of Computational Mathematics VL - 4 SP - 484 EP - 494 PY - 2009 DA - 2009/08 SN - 27 DO - http://doi.org/10.4208/jcm.2009.27.4.015 UR - https://global-sci.org/intro/article_detail/jcm/8585.html KW - American lookback options, Finite difference approximation, Stability and convergence, Error estimates. AB -

In this paper we are concerned with the pricing of lookback options with American type constrains. Based on the differential linear complementary formula associated with the pricing problem, an implicit difference scheme is constructed and analyzed. We show that there exists a unique difference solution which is unconditionally stable. Using the notion of viscosity solutions, we also prove that the finite difference solution converges uniformly to the viscosity solution of the continuous problem. Furthermore, by means of the variational inequality analysis method, the $\mathcal{O}(Δt+Δx^2)$-order error estimate is derived in the discrete $L_2$-norm provided that the continuous problem is sufficiently regular. In addition, a numerical example is provided to illustrate the theoretical results.

Tie Zhang, Shuhua Zhang and Danmei Zhu. (2009). Finite Difference Approximation for Pricing the American Lookback Option. Journal of Computational Mathematics. 27 (4). 484-494. doi:10.4208/jcm.2009.27.4.015
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