October  2014, 10(4): 1129-1146. doi: 10.3934/jimo.2014.10.1129

Dynamic optimization models in finance: Some extensions to the framework, models, and computation

1. 

Department of Mathematics & Statistics, University of Melbourne, Victoria 3010, Australia

2. 

Victoria University, P.O. Box 14428, Melbourne, Vic. 6001, Australia

Received  December 2012 Revised  January 2014 Published  February 2014

Both mathematical characteristics and computational aspects of dynamic optimization in finance have potential for extensions. Various proposed extensions are presented in this paper for dynamic optimization modelling in finance, adapted from developments in other areas of economics and mathematics. They show the need and potential for further areas of study and extensions in financial modelling. The extensions discussed and made concern (a) incorporation of the elements of a dynamic optimization model, (b) an improved model including physical capital, (c) some computational experiments. These extensions make dynamic financial optimisation relatively more organized, coherent and coordinated. These extensions are relevant for applications of financial models to academic and practical exercises. This paper reports initial efforts in providing some useful extensions; further work is necessary to complete the research agenda.
Citation: Bruce D. Craven, Sardar M. N. Islam. Dynamic optimization models in finance: Some extensions to the framework, models, and computation. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1129-1146. doi: 10.3934/jimo.2014.10.1129
References:
[1]

P. Brandimarte, Numerical Methods in Finance: A MATLAB-based introduction, Second edition. Statistics in Practice. Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2006. doi: 10.1002/0470080493.

[2]

W. Brock, Sensitivity of optimal growth paths with respect to a change in target stocks, Zeitchrift für National Ökonomie, Supplementum, 1 (1971), 73-89.

[3]

B. D. Craven, Convergence of discrete approximations for constrained minimizatiion, Journal of the Australian Mathematical Society, Series B, 36 (1994), 50-59. doi: 10.1017/S0334270000010237.

[4]

B. D. Craven, Control and Optimization, Chapman & Hall, London, 1995.

[5]

B. D. Craven, Optimal control and invexity, Computers and Mathematics with Applications, 35 (1998), 17-25. doi: 10.1016/S0898-1221(98)00002-9.

[6]

B. D. Craven, Optimal control of an economic model with a small stochastic term, Pacific Journal of Optimization, 1 (2005), 233-241.

[7]

B. D. Craven, K. de Haas and J. Wettenhall, Computing optimal control, Dynamics of Continuous, Discrete and Impulsive Systems, 4 (1988), 601-615.

[8]

B. D. Craven and S. M. N. Islam, Computing optimal control on MATLAB: The scom package and economic growth models, in Optimisation and Related Topics, (Ballarat/Melbourne, 1999), 61-70, Appl. Optim., Volume 47 in the Series Applied Optimization, Kluwer Academic Publishers, Dordrecht, 2001.

[9]

B. D. Craven and S. M. N. Islam, Optimization in Economics and Finance, Springer, Dordrecht, 2005.

[10]

K. Cuthbertson, Quantitative Financial Economics: Stocks, Bonds, and Foreign Exchange, John Wiley, Chichester, England, 1996.

[11]

B. Davis and D. Elzinga, The solution of an optimal control problem in financial modeling, Operations Research, 19 (1971), 1419-1433. doi: 10.1287/opre.19.6.1419.

[12]

W. Diewert, Generalized Concavity and Economics, in Generalized Concavity in Optimization and Economics, S. Schaible and W. T. Ziemba (eds.), Academic Press, New York, 1981.

[13]

P. Dutta, On specifying the parameters of a development plan, in Capital, Investment and Development, K. Basu, M. Majumdar and T. Mitra (eds.), Blackwell, Oxford, 1993, 75-98.

[14]

K. Fox, J. K. Sengupta and E. Thorbecke, The Theory of Quantitative Economic Policy with Applications to Economic Growth, Stabilization and Planning, North-Holland, Amsterdam, 1973.

[15]

C. Goh and K. L. Teo, MISER: A FORTRAN program for solving optimal control problems, Advances in Engineering Software, 10 (1988), 90-99. doi: 10.1016/0141-1195(88)90005-8.

[16]

C. Gourieroux and J. Janiak, Financial Econometrics, Princeton University Press, Princeton, 2001. doi: 10.7202/010560ar.

[17]

N. Hakansson, Optimal investment and consumption strategies under risk for a class of utility functions, Econometrica, 38 (1970), 587-607. doi: 10.2307/1912196.

[18]

G. Heal, Valuing the Future: Economic Theory and Sustainability, Columbia University Press, New York, 1998.

[19]

S. M. N. Islam and B. D. Craven, Computation of non-linear continuous capital growth models: Experiments with optimal control algorithms and computer programs, Economic Modelling: The International Journal of Theoretical and Applied Papers on Economic Modelling 18 (2001), 551-586.

[20]

S. M. N. Islam and B. D. Craven, Measuring Sustainable Growth, in Governance and Social Responsibility, J. Batten, (ed.), ELsevier-North-Holland, Amsterdam, 2002.

[21]

K. Judd, Numerical Methods in Economics, MIT Press, Cambridge, 1998.

[22]

D. Leonard D. and N. V. Long, Optimal Control Theory and Static Optimization in Economics, Cambridge University Press, Cambridge, U.K, 1992.

[23]

R. Lucas, Asset prices in an exchange economy, Econometrica, 46 (1978), 1429-1445. doi: 10.2307/1913837.

[24]

A. Malliaris and W. Brock, Stochastic Methods in Economics and Finance, Elsevier Science, Amsterdam, 1982.

[25]

T. Mitra, Sensitivity of optimal programmes with respect to changes in target stocks: The case of irreversible investment, Journal of Economic Theory, 29 (1983), 172-184. doi: 10.1016/0022-0531(83)90128-X.

[26]

T. Mitra and D. Ray, Dynamic optimization on a non-convex feasible set: Some general resulots for non-emooth technologies, Zeitchrift für National Ökonomie, 44 (1984), 151-175. doi: 10.1007/BF01289475.

[27]

M. Ramon and A. Scott, (Ed.), Computational Methods for the Study of Dynamic Economies, Oxford University Press, Oxford, 1999.

[28]

A. L. Schwartz, Theory and implementation of numerical methods based on Runge-Kutta integration for solving optimal control problems, Dissertation, University of California at Berkeley, 1989.

[29]

J. K. Sengupta and P. Fanchon, Control Theory Methods in Economics, Kluwer Academic, Boston, 1997. doi: 10.1007/978-1-4615-6285-6.

[30]

C. Tapiero, Applied Stochastic Models and Control for Insurance and Finance, Kluwer Academic, London, 1998. doi: 10.1007/978-1-4615-5823-1.

[31]

K. L. Teo, C. Goh and K. Wong, A Unified Computational Approach for Optimal Control Problems, Pitman Monographs and Surveys in Pure and Applied Mathematics, 55. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991.

[32]

G. Thompson and S. Thore, Computational Economics: Economic Modeling with Optimization Software, (Chapters 21 to 22 and Appendices A&B), Scientific Press, 1993.

[33]

R. Vickson and W. Ziemba, Stochastic Optimisation Models in Finance, Academic Press, New York, 1975.

[34]

S. Zenios, (Ed.), Financial Optimization, Cambridge University Press, Cambridge, 1993.

show all references

References:
[1]

P. Brandimarte, Numerical Methods in Finance: A MATLAB-based introduction, Second edition. Statistics in Practice. Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2006. doi: 10.1002/0470080493.

[2]

W. Brock, Sensitivity of optimal growth paths with respect to a change in target stocks, Zeitchrift für National Ökonomie, Supplementum, 1 (1971), 73-89.

[3]

B. D. Craven, Convergence of discrete approximations for constrained minimizatiion, Journal of the Australian Mathematical Society, Series B, 36 (1994), 50-59. doi: 10.1017/S0334270000010237.

[4]

B. D. Craven, Control and Optimization, Chapman & Hall, London, 1995.

[5]

B. D. Craven, Optimal control and invexity, Computers and Mathematics with Applications, 35 (1998), 17-25. doi: 10.1016/S0898-1221(98)00002-9.

[6]

B. D. Craven, Optimal control of an economic model with a small stochastic term, Pacific Journal of Optimization, 1 (2005), 233-241.

[7]

B. D. Craven, K. de Haas and J. Wettenhall, Computing optimal control, Dynamics of Continuous, Discrete and Impulsive Systems, 4 (1988), 601-615.

[8]

B. D. Craven and S. M. N. Islam, Computing optimal control on MATLAB: The scom package and economic growth models, in Optimisation and Related Topics, (Ballarat/Melbourne, 1999), 61-70, Appl. Optim., Volume 47 in the Series Applied Optimization, Kluwer Academic Publishers, Dordrecht, 2001.

[9]

B. D. Craven and S. M. N. Islam, Optimization in Economics and Finance, Springer, Dordrecht, 2005.

[10]

K. Cuthbertson, Quantitative Financial Economics: Stocks, Bonds, and Foreign Exchange, John Wiley, Chichester, England, 1996.

[11]

B. Davis and D. Elzinga, The solution of an optimal control problem in financial modeling, Operations Research, 19 (1971), 1419-1433. doi: 10.1287/opre.19.6.1419.

[12]

W. Diewert, Generalized Concavity and Economics, in Generalized Concavity in Optimization and Economics, S. Schaible and W. T. Ziemba (eds.), Academic Press, New York, 1981.

[13]

P. Dutta, On specifying the parameters of a development plan, in Capital, Investment and Development, K. Basu, M. Majumdar and T. Mitra (eds.), Blackwell, Oxford, 1993, 75-98.

[14]

K. Fox, J. K. Sengupta and E. Thorbecke, The Theory of Quantitative Economic Policy with Applications to Economic Growth, Stabilization and Planning, North-Holland, Amsterdam, 1973.

[15]

C. Goh and K. L. Teo, MISER: A FORTRAN program for solving optimal control problems, Advances in Engineering Software, 10 (1988), 90-99. doi: 10.1016/0141-1195(88)90005-8.

[16]

C. Gourieroux and J. Janiak, Financial Econometrics, Princeton University Press, Princeton, 2001. doi: 10.7202/010560ar.

[17]

N. Hakansson, Optimal investment and consumption strategies under risk for a class of utility functions, Econometrica, 38 (1970), 587-607. doi: 10.2307/1912196.

[18]

G. Heal, Valuing the Future: Economic Theory and Sustainability, Columbia University Press, New York, 1998.

[19]

S. M. N. Islam and B. D. Craven, Computation of non-linear continuous capital growth models: Experiments with optimal control algorithms and computer programs, Economic Modelling: The International Journal of Theoretical and Applied Papers on Economic Modelling 18 (2001), 551-586.

[20]

S. M. N. Islam and B. D. Craven, Measuring Sustainable Growth, in Governance and Social Responsibility, J. Batten, (ed.), ELsevier-North-Holland, Amsterdam, 2002.

[21]

K. Judd, Numerical Methods in Economics, MIT Press, Cambridge, 1998.

[22]

D. Leonard D. and N. V. Long, Optimal Control Theory and Static Optimization in Economics, Cambridge University Press, Cambridge, U.K, 1992.

[23]

R. Lucas, Asset prices in an exchange economy, Econometrica, 46 (1978), 1429-1445. doi: 10.2307/1913837.

[24]

A. Malliaris and W. Brock, Stochastic Methods in Economics and Finance, Elsevier Science, Amsterdam, 1982.

[25]

T. Mitra, Sensitivity of optimal programmes with respect to changes in target stocks: The case of irreversible investment, Journal of Economic Theory, 29 (1983), 172-184. doi: 10.1016/0022-0531(83)90128-X.

[26]

T. Mitra and D. Ray, Dynamic optimization on a non-convex feasible set: Some general resulots for non-emooth technologies, Zeitchrift für National Ökonomie, 44 (1984), 151-175. doi: 10.1007/BF01289475.

[27]

M. Ramon and A. Scott, (Ed.), Computational Methods for the Study of Dynamic Economies, Oxford University Press, Oxford, 1999.

[28]

A. L. Schwartz, Theory and implementation of numerical methods based on Runge-Kutta integration for solving optimal control problems, Dissertation, University of California at Berkeley, 1989.

[29]

J. K. Sengupta and P. Fanchon, Control Theory Methods in Economics, Kluwer Academic, Boston, 1997. doi: 10.1007/978-1-4615-6285-6.

[30]

C. Tapiero, Applied Stochastic Models and Control for Insurance and Finance, Kluwer Academic, London, 1998. doi: 10.1007/978-1-4615-5823-1.

[31]

K. L. Teo, C. Goh and K. Wong, A Unified Computational Approach for Optimal Control Problems, Pitman Monographs and Surveys in Pure and Applied Mathematics, 55. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991.

[32]

G. Thompson and S. Thore, Computational Economics: Economic Modeling with Optimization Software, (Chapters 21 to 22 and Appendices A&B), Scientific Press, 1993.

[33]

R. Vickson and W. Ziemba, Stochastic Optimisation Models in Finance, Academic Press, New York, 1975.

[34]

S. Zenios, (Ed.), Financial Optimization, Cambridge University Press, Cambridge, 1993.

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