Abstract
In this paper, we consider the nonlinear complementary problem (NCP) arising from the pricing American options in a liquidity switching market. The NCP arises from discretising a coupled system of Hamilton-Bellman-Jacobi (HJB) equations whose solutions are the American option buyer indifference prices. In order to price American options, we derive a complementary problem. Due to the form of liquidity assumptions, the system of (HJB) equations are nonlinear which when discretised give rise to a NCP. We apply two Newton-like methods and perform various numerical experiments to illustrate the method.
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References
Bjerksund, P., Stensland, G.: Closed-form approximation of american options. Scand. J. Manag. 9, S88–S99 (1993)
Carmona, R.: Indifference Pricing: Theory and Applications. Princeton University Press, Princeton (2009)
De Luca, T., Facchinei, F., Kanzow, C.: A semismooth equation approach to the solution of nonlinear complementary problems. Math. Program. 75, 407–439 (1996)
De Luca, T., Facchinei, F., Kanzow, C.: A theoretic and numerical comparison of some semismooth algorithms for complementary problems. Comput. Optim. Appl. 16, 173–205 (1996)
Harker, P.T., Pang, J.S.: A damped-Newton method for the linear complementarity problem. In: Lectures in Applied Mathematics, vol. 26, pp. 265–284. American Mathematical Society (1996)
Leung, T.S.T.: A Markov-modulated stochastic control problem with optimal multiple stopping with application to finance. In: IEEE Conference on Decision and Control (CDC), pp. 559–566 (2010)
Ludkovski, M., Shen, Q.: European option pricing with liquidity shocks. Int. J. Theor. Appl. Finan. 16, 219–249 (2013)
Øksendal, B.: Stochastic Differential Equations. Springer, Heidelberg (2010)
Pang, J.S.: Newton’s method for B-differential equations. Math. Oper. Res. 15(2), 311–341 (1990)
Pang, J.S.: A B-differentiable equation-based, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational inequality problems. Math. Program. 51, 101–131 (1991)
Pang, J.S., Huang, J.: Pricing American options with transaction costs by complementarity methods. In: Avellaneda, M. (ed.) Quantitative Analysis in Financial Markets, pp. 172–198 (2002)
Seydel, R.U.: Tools for Computational Finance. Springer, Heidelberg (2006)
Sun, Z., Zheng, J.: A monotone semismooth Newton type method for a class of complementarity problems. J. Comput. Appl. Math. 235, 1261–1274 (2011)
Wilmott, P., Howison, S., Dewynne, J.: The Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press, Cambridge (1995)
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Mudzimbabwe, W., Vulkov, L. (2017). American Options in an Illiquid Market: Nonlinear Complementary Method. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Numerical Analysis and Its Applications. NAA 2016. Lecture Notes in Computer Science(), vol 10187. Springer, Cham. https://doi.org/10.1007/978-3-319-57099-0_56
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DOI: https://doi.org/10.1007/978-3-319-57099-0_56
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