Abstract
In this chapter, different Newton-based solvers are introduced to solve fully nonlinear PDEs generated from financial problems. The first one concentrates on solving the root-finding problem from the nonlinear system after applying the standard finite difference method with implicit scheme. The second one addresses to solve the deferred correction problem which is transformed from the original PDE. Different numerical experiments in terms of accuracy and efficiency are compared and some improvements using Newton-like methods are also discussed.
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Barles, G., Soner, H.M.: Option pricing with transaction costs and a nonlinear Black-Scholes equation. Finance Stochast. 2, 369–397 (1998). https://people.math.ethz.ch/~hmsoner/pdfs/38-Soner-FS-98.pdf
Black, F., Scholes, M.S.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973)
Bordag, L.A., Frey, R.: Pricing options in illiquid markets: symmetry reductions and exact solutions, in Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing, pp. 103–130. Nova Science, New York (2008)
Company, R., Jódar, L., Pintos, J.: A consistent stable numerical scheme for a nonlinear option pricing model in illiquid markets. Math. Comput. Simul. 82(10), 1972–1985 (2012)
Dembo, R.S., Eisenstat, S.C., Steihaug,T.: Inexact Newton methods. SIAM J. Numer. Anal. 19(2), 400–408 (1982)
Ďuriš, K., Tan, S.-H., Lai, C.-H., Ševčovič, D.: Comparison of the analytical approximation formula and Newton’s method for solving a class of nonlinear Black-Scholes parabolic equations, Comput. Methods Appl. Math. 16(1), 35–50 (2016)
Egorova, V., Tan, S.-H., Lai, C.-H., Company, R., Jódar, L.: Moving boundary transformation for American call options with transaction cost: finite difference methods and computing. Int. J. Comput. Math. 94, 1–18 (2015). Published online: 08 Dec 2015
Ehrhardt, M. (ed.): Nonlinear Models in Mathematical Finance. Nova Science, New York (2008)
Erdman, D.J., Rose, D.J.: Newton waveform relaxation techniques for tightly coupled systems. IEEE Trans. Comput. Aided Des. 11(5), 598–606 (1992)
Frey, R.: Market illiquidity as a source of model risk in dynamic hedging. In: Gilbson, E. (ed.) Model Risk. RISK Publication, London (2000)
Frey, R., Patie, P.: Risk management for derivatives in illiquid markets: a simulation study. In: Sandmann, K., Schönbucher, P.J. (eds.) Advances in Finance and Stochastics, pp. 137–160. Springer, Berlin (2002). http://www.math.uni-leipzig.de/~frey/frey-patie-simulation-study.pdf
Griewank, A.: Broyden updating, the good and the bad. Documenta Mathematica - Extra Volume ISMP 2012, pp. 301–315. http://www.math.uiuc.edu/documenta/vol-ismp/45_griewank-andreas-broyden.pdf.
Guyon, J., Henry-Labordère, P.: Nonlinear Option Pricing. CRC Press, Boca Raton (2013)
Heider, P.: Numerical methods for nonlinear Black-Scholes equations. Appl. Math. Finance 17, 59–81 (2010)
Jandačka, M., Ševčovič, D.: On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile. J. Appl. Math. 3, 235–258 (2005)
Kelley, C.T.: Solving Nonlinear Equation with Newton’s Method. SIAM, Philadelphia (1987)
Kevenaar, T.A.M.: Periodic steady state analysis using shooting and waveform-Newton. Int. J. Circuit Theory Appl. 22, 51–60 (1994)
Knoll, D.A., Keyes, D.E.: Jacobian-free Newton-Krylov methods: a survey of approaches and applications, J. Comput. Phys. 193, 357–397 (2004)
Martínez, J.M.: Practical quasi-Newton methods for solving nonlinear systems, J. Comput. Appl. Math. 124(1), 97–121 (2000)
Rheinboldt,W.C.: Quasi-Newton methods. Lecture Notes TU Munich (2000). https://www-m2.ma.tum.de/foswiki/pub/M2/Allgemeines/SemWs09/quasi-newt.pdf
Saleh, R.A., White,J.: Accelerating relaxation algorithms for circuit simulation using waveform-Newton and step-size refinement. IEEE Trans. Comput. Aided Des. 9(9), 951–958 (1990)
Schubert, L.K.: Modification of a quasi-Newton method for nonlinear equations with a sparse Jacobian. J. Math. Comput. 24, 27–23 (1970)
Ševčovič, D., Stehlíková, B., Mikula,K.: Analytical and Numerical Methods for Pricing Financial Derivatives. Nova Science, New York (2011)
Acknowledgements
The authors thank Prof. Matthias Ehrhardt, Dr. E. Jan W. ter Maten from Bergische Universität Wuppertal, Prof. Daniel Ševčovič from Comenius University Bratislava, Prof. Kevin Parrott and Dr. André Ribeiro from University of Greenwich, for the fruitful discussions.
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Tan, SH., Lai, CH. (2017). Newton-Based Solvers for Nonlinear PDEs in Finance. In: Ehrhardt, M., Günther, M., ter Maten, E. (eds) Novel Methods in Computational Finance. Mathematics in Industry(), vol 25. Springer, Cham. https://doi.org/10.1007/978-3-319-61282-9_12
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DOI: https://doi.org/10.1007/978-3-319-61282-9_12
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