Optimal portfolio in a continuous-time self-exciting threshold model

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April 2013, 9(2): 487-504. doi: 10.3934/jimo.2013.9.487

Optimal portfolio in a continuous-time self-exciting threshold model

Hui Meng ^1, , Fei Lung Yuen ^2, , Tak Kuen Siu ^3, and Hailiang Yang ^4,

1.	China Institute for Actuarial Science, Central University of Finance and Economics, Beijing 100081, China
2.	Department of Actuarial Mathematics and Statistics, and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, United Kingdom
3.	Cass Business School, City University, 106 Bunhill Row, London EC1Y 8TZ, United Kingdom
4.	Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong, China

Received February 2011 Revised October 2012 Published February 2013

Abstract
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This paper discusses an optimal portfolio selection problem in a continuous-time economy, where the price dynamics of a risky asset are governed by a continuous-time self-exciting threshold model. This model provides a way to describe the effect of regime switching on price dynamics via the self-exciting threshold principle. Its main advantage is to incorporate the regime switching effect without introducing an additional source of uncertainty. A martingale approach is used to discuss the problem. Analytical solutions are derived in some special cases. Numerical examples are given to illustrate the regime-switching effect described by the proposed model.

Keywords: logarithmic utility., self-exciting threshold model, Portfolio selection, power utility, regime switching.

Mathematics Subject Classification: Primary: 93E2.

Citation: Hui Meng, Fei Lung Yuen, Tak Kuen Siu, Hailiang Yang. Optimal portfolio in a continuous-time self-exciting threshold model. Journal of Industrial and Management Optimization, 2013, 9 (2) : 487-504. doi: 10.3934/jimo.2013.9.487

References:

[1]	R. J. Elliott and P. E. Kopp, "Mathematics of Financial Markets," $2^{nd}$ edition, Springer Finance, Springer-Verlag, New York, 2005.
[2]	S. M. Goldfeld and R. E. Quandt, A Markov model for switching regressions, Journal of Econometrics, 1 (1973), 3-16.
[3]	J. D. Hamilton, A new approach to the economic analysis of non-stationary time series and the business cycle, Econometrica, 57 (1989), 357-384. doi: 10.2307/1912559.
[4]	B. G. Jang, H. K. Koo and M. Loewenstein, Liquidity premia and transaction costs, Journal of Finance, 62 (2007), 2329-2366.
[5]	I. Karatzas and S. E. Shreve, "Methods of Mathematical Finance," Applications of Mathematics (New York), 39, Springer-Verlag, New York, 1998.
[6]	J. Lintner, The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets, Review of Economics and Statistics, 47 (1965), 12-37.
[7]	H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91.
[8]	R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time model, Review of Economics and Statistics, 51 (1969), 247-257.
[9]	R. C. Merton, Optimal consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413.
[10]	J. Mossin, Equilibrium in a capital asset market, Econometrica, 34 (1966), 768-783.
[11]	R. E. Quandt, The estimation of parameters of linear regression system obeying two separate regimes, Journal of the American Statistical Association, 53 (1958), 873-880.
[12]	P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, Review of Economics and Statistics, 51 (1969), 239-246.
[13]	W. F. Sharpe, Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance, 19 (1964), 425-442.
[14]	H. Tong, On a threshold model, in "Pattern Recognition and Signal Processing," Sijthoff and Noordhoff, Amsterdam, (1978), 575-586.
[15]	H. Tong and K. S. Lim, Threshold autoregression, limit cycles and cyclical data (with discussion), Journal of Royal Statistical Society - B, 42 (1980), 245-292.
[16]	H. Tong, "Threshold Models in Nonlinear Time Series Analysis," Lecture Notes in Statistics, 21, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4684-7888-4.
[17]	J. L. Treynor, Toward a theory of market value of risky assets, unpublished manuscript.
[18]	G. Yin and X. Zhou, Markowitz's mean-variance portfolio selection with regime switching: From discrete-time models to their continuous-time limits, IEEE Transactions on Automatic Control, 49 (2004), 349-360. doi: 10.1109/TAC.2004.824479.
[19]	X. Zhou and G. Yin, Markowitz's mean-variance portfolio selection with regime switching: A continuous-time model, SIAM Journal on Control and Optimization, 42 (2003), 1466-1482. doi: 10.1137/S0363012902405583.

show all references

References:

[1]	R. J. Elliott and P. E. Kopp, "Mathematics of Financial Markets," $2^{nd}$ edition, Springer Finance, Springer-Verlag, New York, 2005.
[2]	S. M. Goldfeld and R. E. Quandt, A Markov model for switching regressions, Journal of Econometrics, 1 (1973), 3-16.
[3]	J. D. Hamilton, A new approach to the economic analysis of non-stationary time series and the business cycle, Econometrica, 57 (1989), 357-384. doi: 10.2307/1912559.
[4]	B. G. Jang, H. K. Koo and M. Loewenstein, Liquidity premia and transaction costs, Journal of Finance, 62 (2007), 2329-2366.
[5]	I. Karatzas and S. E. Shreve, "Methods of Mathematical Finance," Applications of Mathematics (New York), 39, Springer-Verlag, New York, 1998.
[6]	J. Lintner, The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets, Review of Economics and Statistics, 47 (1965), 12-37.
[7]	H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91.
[8]	R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time model, Review of Economics and Statistics, 51 (1969), 247-257.
[9]	R. C. Merton, Optimal consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413.
[10]	J. Mossin, Equilibrium in a capital asset market, Econometrica, 34 (1966), 768-783.
[11]	R. E. Quandt, The estimation of parameters of linear regression system obeying two separate regimes, Journal of the American Statistical Association, 53 (1958), 873-880.
[12]	P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, Review of Economics and Statistics, 51 (1969), 239-246.
[13]	W. F. Sharpe, Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance, 19 (1964), 425-442.
[14]	H. Tong, On a threshold model, in "Pattern Recognition and Signal Processing," Sijthoff and Noordhoff, Amsterdam, (1978), 575-586.
[15]	H. Tong and K. S. Lim, Threshold autoregression, limit cycles and cyclical data (with discussion), Journal of Royal Statistical Society - B, 42 (1980), 245-292.
[16]	H. Tong, "Threshold Models in Nonlinear Time Series Analysis," Lecture Notes in Statistics, 21, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4684-7888-4.
[17]	J. L. Treynor, Toward a theory of market value of risky assets, unpublished manuscript.
[18]	G. Yin and X. Zhou, Markowitz's mean-variance portfolio selection with regime switching: From discrete-time models to their continuous-time limits, IEEE Transactions on Automatic Control, 49 (2004), 349-360. doi: 10.1109/TAC.2004.824479.
[19]	X. Zhou and G. Yin, Markowitz's mean-variance portfolio selection with regime switching: A continuous-time model, SIAM Journal on Control and Optimization, 42 (2003), 1466-1482. doi: 10.1137/S0363012902405583.

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