Risk-based decisions on assets structure of a bank — partially observed economic conditions

Hałaj, Grzegorz (2006): Risk-based decisions on assets structure of a bank — partially observed economic conditions.

Abstract

A model of bank’s dynamic asset management problem in case of partially observed future economic conditions and requirements concerning level of risk taken has been built. It requires solving the resulting optimal control with random terminal condition resulting from partial observation of parameter of maximized functional. Stochastic Maximum Principle reduces the problem to solving FBSDE. As optimization may usually imply dependence of forward equation on solutions of backward equation we allow the drift and diffusion of forward part to be functions of solution of backward equation. The necessary conditions for existence of solutions of FBSDE in such a form have been derived. A numerical scheme is then implemented for a particular choice of parameters of the problem.

• Item Type: MPRA Paper
• Language: English
• Keywords: Portfolio optimization; bank’s assets; partial observation; stochastic maximum principle; + FBSDEs
• Subjects: C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61 - Optimization Techniques ; Programming Models ; Dynamic Analysis; G - Financial Economics > G1 - General Financial Markets > G11 - Portfolio Choice ; Investment Decisions

References

Benes V., Karatzas I., Ocone D., Wang H. (2004) Control with Partial Observations and Explicit Solution to Mortensen’s Equation,Applied Mathematics & Optimization, 49, pp. 217–239. Basel Committee on Banking Supervision (1999) A NEW CAPITAL ADEQUACY FRAMEWORK,Consultative paper, Basel. Bensoussan A. (1992) Stochastic Control of Partially Observable Systems Cambridge University Press, Cambridge. Brendle S. (2005) Portfolio Selection under Partial Observation and Filtering, preliminary version of working paper, Prinston University. Cadenillas A., Karatzas I. (1995) The stochastic maximum principle for linear, convex optimal control with random coefficients. SIAM Journal on Control and Optimization, vol 33, number 2, pp. 590–624. Carmona R., Ludkovski M. (2004) Convenience yield model with partial observations and exponential utility, Prinston working paper. Cuoco D., Liu H. (2005) An analysis of VaR-based capital requirements Journal of Financial Intermediation (in press). Delarue F., Menozzi S. (2005) A Forward-Backward Stochastic Algorithm for Quasi-Linear PDEs, submitted to Annals of Applied Probability. Emmer S., Kluppelberg C., Korn R. (2001) Optimal portfolios with bounded capital at risk, Math. Finance 11, pp. 365–384. Lefevre D. (2001) An introduction to utility maximization with partial observation, INRIA rapport de recherche, N 4183. Karatzas I., Shreve S. E. (1997) Brownian Motion and Stochastic Calculus Springer-Verlag, New York. Kohlmann M., Zhou X. Y. (2000) Relationship between backward stochastic differential equations and stochastic controls: a linear-quadratic approach, SIAM J. Control Optim., 38, No. 5, pp. 1392–1407. Lipster R. Sh., Shiryaev A. N. (2001) Statistics of Random Processes. Springer (Berlin). Pastor L., Veronesi P. (2003) Stock Valuation and Learning about Profitability, The Journal of Finance, Vol 58, pp. 1749–1789. Peng Sh., Wu Z. (1999) Fully Coupled Forward-Backward Stochastic Differential Equations and Application to Optimal Control, SIAM J. of Control& Optimization, Vol. 37, No. 3, pp. 825–843. Riviere O. (2005) Equations differentielles stochastiques progressives retrogrades: Equations aux derivees partielles et discretisation, PhD Thesis, 1st August (in French). Santos, J. (2002) Bank capital regulation in contemporary banking theory: A review of the literature, Working Paper 90, Bank for International Settlements. Yong J., Zhou X. (1999) Stochastic Controls. Hamiltonian Systems and HJB Equations, Springer-Verlag, New York. Zhang J. (2004) A Numerical Scheme for BSDEs, The Annals of Applied Probability, Vol. 14, pp. 459–488. Zhou X. Y (1998) Stochastic Near-Optimal Controls: Necessary and Sufficient Conditions for Near-Optimality, SIAM J. of Control & Optimization, Vol. 38, pp.929–947.