Abstract
Several examples of nonlinear Hamilton Jacobi Bellman (HJB) partial differential equations are given which arise in financial applications. The concept of a visocisity solution is introduced. Sufficient conditions which ensure that a numerical scheme converges to the viscosity solution are discussed. Numerical examples based on an uncertain volatility model are presented which show that seemingly reasonable discretization methods (which do not satisfy the sufficient conditions for convergence) fail to converge to the viscosity solution.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Andersen, L., Andreasen, J., & Brotherton-Ratcliffe, R. (1998). The passport option. Journal of Computational Finance, 1(3), 15–36.
Avellaneda, M., Levy, A., & Parás, A. (1995). Pricing and hedging derivative securities in markets with uncertain volatilities. Applied Mathematical Finance, 2, 73–88.
Barles, G. (1997). Convergence of numerical schemes for degenerate parabolic equations arising in finance. In L. C. G. Rogers & D. Talay (Eds.), Numerical methods in finance (pp. 1–21). Cambridge: Cambridge University Press.
Barles, G., & Burdeau, J. (1995). The Dirichlet problem for semilinear second-order degenerate elliptic equations and applications to stochastic exit time control problems. Communications in Partial Differential Equations, 20, 129–178.
Barles, G., & Rouy, E. (1998). A strong comparison result for the Bellman equation arising in stochastic exit time control problems and applications. Communications in Partial Differential Equations, 23, 1995–2033.
Barles, G., & Souganidis, P. E. (1991). Convergence of approximation schemes for fully nonlinear equations. Asymptotic Analysis, 4, 271–283.
Barles, G., Daher, C. H., & Romano, M. (1995). Convergence of numerical shemes for parabolic eqations arising in finance theory. Mathematical Models and Methods in Applied Sciences, 5, 125–143.
Basak, S., & Chabakauri, G. (2007). Dynamic mean-variance asset allocation. Working Paper, London Business School.
Bergman, Y. (1995). Option pricing with differential interest rates. Review of Financial Studies, 8, 475–500.
Chaumont, S. (2004). A strong comparison result for viscosity solutions to Hamilton-Jacobi-Bellman equations with Dirichlet conditions on a non-smooth boundary. Working paper, Institute Elie Cartan, Université Nancy I.
Chen, Z., & Forsyth, P. A. (2008). A numerical scheme for the impulse control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit (GMWB). Numerische Mathematik, 109, 535–569.
Crandall, M. G., Ishii, H., & Lions, P. L. (1992). User’s guide to viscosity solutions of second order partial differential equations. Bulletin of the American Mathematical Society, 27, 1–67.
Dai, M., Kwok, Y. K., & Zong, J. (2008). Guaranteed minimum withdrawal benefit in variable annuities. Mathematical Finance, 18, 595–611.
Derman, E., & Kani, I. (1996). The ins and outs of barrier options: Part 1. Derivatives Quarterly, 3(Winter), 55–67.
Forsyth, P. A., & Labahn, G. (2008). Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance. Journal of Computational Finance, 11, 1–44.
Kushner, H. J., & Dupuis, P. G. (1991). Numerical methods for stochastic control problems in continuous time. New York: Springer.
Leland, H. E. (1985). Option pricing and replication with transaction costs. Journal of Finance, 40, 1283–1301.
Li, D., & Ng, W.-L. (2000). Optimal dynamic portfolio selection: Multiperiod mean variance formulation. Mathematical Finance, 10, 387–406.
Lorenz, J. (2008). Optimal trading algorithms: Portfolio transactions, mulitperiod portfolio selection, and competitive online search. PhD Thesis, ETH Zurich.
Lorenz, J., & Almgren, R. (2007). Adaptive arrival price. In B. R. Bruce (Ed.), Algorithmic trading III: Precision, control, execution. New York: Institutional Investor Journals.
Lyons, T. (1995). Uncertain volatility and the risk free synthesis of derivatives. Applied Mathematical Finance, 2, 117–133.
Milevsky, M. A., & Salisbury, T. S. (2006). Financial valuation of guaranteed minimum withdrawal benefits. Insurance: Mathematics and Economics, 38, 21–38.
Mnif, M., & Sulem, A. (2001). Optimal risk control under excess of loss reinsurance. Working paper, Université Paris 6.
Pham, H. (2005). On some recent aspects of stochastic control and their applications. Probability Surveys, 2, 506–549.
Pooley, D. M., Forsyth, P. A., & Vetzal, K. R. (2003). Numerical convergence properties of option pricing PDEs with uncertain volatility. IMA Journal of Numerical Analysis, 23, 241–267.
Shreve, S., & Vecer, J. (2000). Options on a traded account: Vacation calls, vacation puts, and passport options. Finance and Stochastics, 4, 255–274.
Wang, J., & Forsyth, P. A. (2007). Maximal use of central differencing for Hamilton-Jacobi-Bellman PDEs in finance. SIAM Journal on Numerical Analysis, 46, 1580–1601.
Zhou, X. Y., & Li, D. (2000). Continuous time mean variance portfolio selection: A stochastic LQ framework. Applied Mathematics and Optimization, 42, 19–33.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Forsyth, P.A., Vetzal, K.R. (2012). Numerical Methods for Nonlinear PDEs in Finance. In: Duan, JC., Härdle, W., Gentle, J. (eds) Handbook of Computational Finance. Springer Handbooks of Computational Statistics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17254-0_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-17254-0_18
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-17253-3
Online ISBN: 978-3-642-17254-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)