COMBINED FIXED POINT AND POLICY ITERATION FOR HJB EQUATIONS IN FINANCE

Implicit methods for Hamilton Jacobi Bellman (HJB) partial differential equations give rise to highly nonlinear discretized algebraic equations. The classic policy iteration approach may not be efficient in many circumstances. In this article, we derive sufficient conditions to ensure convergence of a combined fixed point-policy iteration scheme for solution of the discretized equations. Numerical examples are included for a singular stochastic control problem arising in insurance (a Guaranteed Minimum Withdrawal Benefit) where the underlying risky asset follows a jump diffusion, and an American option assuming a regime switching process.

[1]  Huyen Pham,et al.  Continuous-time stochastic control and optimization with financial applications / Huyen Pham , 2009 .

[2]  Peter A. Forsyth,et al.  Implications of a regime-switching model on natural gas storage valuation and optimal operation , 2010 .

[3]  Peter A. Forsyth,et al.  Analysis of a penalty method for pricing a guaranteed minimum withdrawal benefit (GMWB) , 2012 .

[4]  Hasnaa Zidani,et al.  Some Convergence Results for Howard's Algorithm , 2009, SIAM J. Numer. Anal..

[5]  S. Clift,et al.  Linear and non-linear monotone methods for valuing financial options under two-factor, jump-diffusion models , 2007 .

[6]  M. Davison,et al.  Natural gas storage valuation and optimization: A real options application , 2009 .

[7]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[8]  Min Dai,et al.  GUARANTEED MINIMUM WITHDRAWAL BENEFIT IN VARIABLE ANNUITIES , 2007 .

[9]  Mary R. Hardy,et al.  A Regime-Switching Model of Long-Term Stock Returns , 2001 .

[10]  Julian Lorenz,et al.  Adaptive Arrival Price , 2007 .

[11]  H. Kushner.  Numerical Methods for Stochastic Control Problems in Continuous Time , 2000 .

[12]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[13]  H. Pham.  On some recent aspects of stochastic control and their applications , 2005, math/0509711.

[14]  P. Forsyth,et al.  Robust numerical methods for contingent claims under jump diffusion processes , 2005 .

[15]  George Labahn,et al.  A penalty method for American options with jump diffusion processes , 2004, Numerische Mathematik.

[16]  J. S. Kennedy,et al.  Hedging Contingent Claims in Markets with Jumps , 2007 .

[17]  Peter A. Forsyth,et al.  QUADRATIC CONVERGENCE OF A PENALTY METHOD FOR VALUING AMERICAN OPTIONS , 2000 .

[18]  A. Tourin,et al.  Viscosity Solutions and Numerical Schemes for Investment / Consumption Models with Transaction Costs , 1997 .

[19]  P. Forsyth,et al.  Numerical solution of the Hamilton-Jacobi-Bellman formulation for continuous time mean variance asset allocation , 2010 .

[20]  R. Schilling.  Financial Modelling with Jump Processes , 2005 .

[21]  Thomas S. Salisbury,et al.  Financial valuation of guaranteed minimum withdrawal benefits , 2006 .

[22]  Zhuliang Chen,et al.  A numerical scheme for the impulse control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit (GMWB) , 2008, Numerische Mathematik.

[23]  Daniel Bauer,et al.  A Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities , 2008 .

[24]  Abdul Q. M. Khaliq,et al.  New Numerical Scheme for Pricing American Option with Regime-Switching , 2009 .

[25]  Julian Lorenz,et al.  Adaptive Arrival Price; ; Trading; Algorithmic Trading III. Precision control, execution , 2007 .

[26]  P. Forsyth,et al.  Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance , 2007 .

[27]  Alexander Fadeev,et al.  Optimal execution for portfolio transactions , 2006 .

[28]  J. Wang,et al.  Maximal Use of Central Differencing for Hamilton-Jacobi-Bellman PDEs in Finance , 2008, SIAM J. Numer. Anal..

[29]  Peter A. Forsyth,et al.  The effect of modelling parameters on the value of GMWB guarantees , 2008 .

[30]  Iris R. Wang,et al.  Robust numerical valuation of European and American options under the CGMY process , 2007 .