Abstract
Discretized singular control problems in finance result in highly nonlinear algebraic equations which must be solved at each timestep. We consider a singular stochastic control problem arising in pricing a guaranteed minimum withdrawal benefit (GMWB), where the underlying asset is assumed to follow a jump diffusion process. We use a scaled direct control formulation of the singular control problem and examine the conditions required to ensure that a fast fixed point policy iteration scheme converges. Our methods take advantage of the special structure of the GMWB problem in order to obtain a rapidly convergent iteration. The direct control method has a scaling parameter which must be set by the user. We give estimates for bounds on the scaling parameter so that convergence can be expected in the presence of round-off error. Example computations verify that these estimates are of the correct order. Finally, we compare the scaled direct control formulation to a formulation based on a block version of the penalty method (Huang and Forsyth in IMA J Numer Anal 32:320–351, 2012). We show that the scaled direct control method has some advantages over the penalty method.
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References
Andersen L., Andreasen J.: Jump-diffusion processes: volatility smile fitting and numerical methods for option pricing. Rev. Derivatives Res. 4, 231–262 (2000)
Barles G.: Convergence of numerical schemes for degenerate parabolic equations arising in finance. In: Rogers, L.C.G., Talay, D. (eds) Numerical Methods in Finance, pp. 1–21. Cambridge University Press, Cambridge (1997)
Barles G., Souganidis P.E.: Convergence of approximation schemes for fully nonlinear equations. Asymptotic Anal. 4, 271–283 (1991)
Bertsekas D.P., Tsitsiklis J.: Neuro-Dynamic Programming. Athena, Massachusetts (1996)
Bokanowski O., Maroso S., Zidani H.: Some convergence results for Howard’s algorithm. SIAM J. Numer. Anal. 47, 3001–3026 (2009)
Budhriaja A., Ross K.: Convergent numerical scheme for singular stochastic control with state constraints in a portfolio selection problem. SIAM J. Control Optim. 45, 2169–2206 (2007)
Chen Z., Forsyth P.A.: A numerical scheme for the impulse control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit (GMWB). Numer. Math. 109, 535–569 (2008)
Chen Z., Vetzal K., Forsyth P.A.: The effect of modelling parameters on the value of GMWB guarantees. Insur. Math. Econ. 43, 165–173 (2008)
Cont R., Tankov P.: Financial Modelling with Jump Processes. Chapman and Hall, London (2004)
Dai M., Kwok Y.K., Zong J.: Guaranteed minimum withdrawal benefit in variable annuities. Math. Finance 18, 595–611 (2008)
d’Halluin Y., Forsyth P.A., Vetzal K.R.: Robust numerical methods for contingent claims under jump diffusion processes. IMA J. Numer. Anal. 25, 87–112 (2005)
Forsyth P.A., Labahn G.: Numerical methods for controlled Hamilton–Jacobi–Bellman PDEs in finance. J. Comput. Finance 11(Winter), 1–44 (2008)
Hindy A., Huang C., Zhu S.: Numerical analysis of a free boundary singular control problem in financial economics. J. Econ. Dyn. Control 21, 297–327 (1997)
Huang, Y.: Numerical methods for pricing a guaranteed minimum withdrawal benefit (GMWB) as a singular control problem. PhD thesis, School of Computer Science, University of Waterloo (2011)
Huang Y., Forsyth P.A.: Analysis of a penalty method for pricing a guaranteed minimum withdrawal benefit (GMWB). IMA J. Numer. Anal. 32, 320–351 (2012)
Huang, Y., Forsyth, P.A., Labahn, G.: Combined fixed point policy iteration for HJB equations in finance. Working paper, University of Waterloo (submitted to SIAM J. Numer. Anal.) (2010)
Kennedy J.S., Forsyth P.A., Vetzal K.R.: Dynamic hedging under jump diffusion with transaction costs. Oper. Res. 57, 541–559 (2009)
Kumar S., Mithiraman K.: A numerical method for solving singular stochastic control problems. Oper. Res. 52, 563–582 (2004)
Kushner H.J., Dupuis P.G: Numerical Methods for Stochastic Control Problems in Continuous Time. Springer, New York (1991)
Merton R.C.: Option pricing when underlying stock returns are discontinuous. J. Financial Econ. 3, 125–144 (1976)
Milevsky M.A., Salisbury T.S.: Financial valuation of guaranteed minimum withdrawal benefits. Insur. Math. Econ. 38, 21–38 (2006)
Munos, R.: Error bounds for approximate policy iteration. In: Proceedings of the 20th international congress on machine learning, pp. 560–567. Washington (2003)
Pham H.: On some recent aspects of stochastic control and their applications. Probability Surv. 2, 506–549 (2005)
Pooley D.M., Forsyth P.A., Vetzal K.R.: Numerical convergence properties of option pricing PDEs with uncertain volatility. IMA J. Numer. Anal. 23, 241–267 (2003)
Stoer J., Bulirsch R.: Introduction to Numerical Analysis. 2nd edn. Springer, Berlin (1993)
Tourin A., Zariphopoulou T. : Viscosity solutions and numerical schemes for investment/consumption models with transaction costs. In: Rogers, L.C.G., Talay, D. (eds) Numerical Methods in Finance, Cambridge University Press, Cambridge (1997)
Varga R.: Matrix Iterative Analysis. Prentice Hall, New Jersey (1961)
Wang I.R., Wan J.W.I., Forsyth P.A.: Robust numerical valuation of European and American options under the CGMY proces. J. Comput. Finance 10(4 Summer), 86–115 (2007)
Wang J., Forsyth P.A.: Maximal use of central differencing for Hamilton–Jacobi–Bellman PDEs in finance. SIAM J. Numer. Anal. 46, 1580–1601 (2008)
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This work was supported by Credit Suisse, New York and the Natural Sciences and Engineering Research Council of Canada.
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Huang, Y., Forsyth, P.A. & Labahn, G. Iterative methods for the solution of a singular control formulation of a GMWB pricing problem. Numer. Math. 122, 133–167 (2012). https://doi.org/10.1007/s00211-012-0455-y
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DOI: https://doi.org/10.1007/s00211-012-0455-y