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On Optimal Cash Management under a Stochastic Volatility Model

Published online by Cambridge University Press:  28 May 2015

Na Song ,
Wai-Ki Ching ,
Tak-Kuen Siu  and
Cedric Ka-Fai Yiu
Na Song*
Affiliation:
School of Management and Economics, University of Electronic Science and Technology, Chengdu, China
Wai-Ki Ching*
Affiliation:
Advanced Modeling and Applied Computing Laboratory, Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Tak-Kuen Siu*
Affiliation:
Cass Business School, City University London, 106 Bunhill Row, London, ECY1 8TZ, United Kingdom and Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, Macquarie University, Sydney, NSW 2109, Australia
Cedric Ka-Fai Yiu*
Affiliation:
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong
*
Corresponding author. Email: smynmath@163.com
Corresponding author. Email: wching@hku.hk
Corresponding author. Email: Ken.Siu.1@city.ac.uk
Corresponding author. Email: macyiu@polyu.edu.hk
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Abstract

We discuss a mathematical model for optimal cash management. A firm wishes to manage cash to meet demands for daily operations, and to maximize terminal wealth via bank deposits and stock investments that pay dividends and have uncertain capital gains. A Stochastic Volatility (SV) model is adopted for the capital gains rate of a stock, providing a more realistic way to describe its price dynamics. The cash management problem is formulated as a stochastic optimal control problem, and solved numerically using dynamic programming. We analyze the implications of the heteroscedasticity described by the SV model for evaluating risk, by comparing the terminal wealth arising from the SV model to that obtained from a Constant Volatility (CV) model.

Keywords

60K2568M2091A80Optimal cash managementstochastic volatilitydynamic programmingHJB equations
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 3 , Issue 2 , May 2013 , pp. 81 - 92
Copyright
Copyright © Global-Science Press 2013

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References

[1]Bensoussan, A, Chutani, A and Sethi, S.P. (2009) Optimal cash management under uncertainty. Operations Research Letters, 37:425–429.Google Scholar
[2]Chang, M.H. and Krishna, K (1986) A successive approximation algorithm for stochastic control problems. Applied Mathematics and Computation, 18(2):155–165.Google Scholar
[3]Chen, L (1996) Stochastic mean and stochastic volatility – A three-factor model of the term structure of interest rates and its application to the pricing of interest rate derivatives. Financial Markets, Institutions, and Instruments, 5:1–88.Google Scholar
[4]Ghysels, E, Harvey, AC and Renault, E (1996) Stochastic Volatility. Handbook of Statistics, 14 North Holland, Amsterdam, Holland.Google Scholar
[5]Herzog, F, Peyrl, H, Geering, H.P. (2005) Proof of the convergence of the successive approximation algorithm for numerically solving the Hamilton-Jacobi-Bellman equation. WSEAS Transactions on Systems, 12:2238–2245.Google Scholar
[6]Hull, J and White, A (1987) The pricing of options on assets with stochastic volatilities. The Journal of Finance, 42(2):281–300.Google Scholar
[7]Kilin, F (2006) Accelerating the calibration of stochastic volatility models. Munich Personal RePEc Archive, 3.Google Scholar
[8]Peyrl, H, Herzog, F and Geering, HP (2005) Numerical solution ofthe Hamilton-Jacobi-Bellman equation for stochastic optimal control problems. 2005 WSEAS Int. Conference on Dynamical Systems and Control, 489–497.Google Scholar
[9]Pillay, E, O'Hara, JG (2011) FFT based option pricing under a mean reverting process with stochastic volatility and jumps. Journal of Computational and Applied Mathematics, 235:3378–3384.Google Scholar
[10]Sethi, SP and Thompson, GL (1970) Application of mathematical control theory to finance. Journal of Finance and Quantitative Analysis, 5:381–394.Google Scholar
[11]Shephard, N (2004) Stochastic Volatility: Selected Readings. Oxford University Press, Oxford.Google Scholar
[12]Strikwerda, JC (1989) Finite Differential Schemes and Partial Differential Equations, Wadsworth & Brooks, California.Google Scholar
[13]Taylor, SJ (1986) Modeling Financial Time Series. Wiley, Chichester.Google Scholar
[14]Taylor, SJ (2006) Asset Price Dynamics and Prediction. Princeton University Press, Princeton.Google Scholar
[15]Wiggins, JB (1987) Option values under stochastic volatilities. Journal of Financial Economics, 19:351–372.Google Scholar
[16]Yong, J and Zhou, X (1999) Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York.CrossRefGoogle Scholar
[17]Zhang, X, Zhang, K (2009) Using genetic algorithm to solve a new multi-period stochastic optimization mode. Journal of Computational and Applied Mathematics, 231:114–123.CrossRefGoogle Scholar
[18]Zvan, R, Forsyth, PA, Vetzal, KR (1998) Penalty methods for American options with stochastic volatility. Journal of Computational and Applied Mathematics, 91:199–218.Google Scholar