Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-08T13:24:05.921Z Has data issue: false hasContentIssue false
  • English
  • Français

A martingale approach for asset allocation with derivative security and hidden economic risk

Part of: Mathematical finance Stochastic analysis

Published online by Cambridge University Press:  01 October 2019

Tak Kuen Siu ,
Jinxia Zhu  and
Hailiang Yang
Tak Kuen Siu*
Affiliation:
Macquarie University
Jinxia Zhu*
Affiliation:
The University of New South Wales
Hailiang Yang*
Affiliation:
The University of Hong Kong
*
*Postal address: Department of Actuarial Studies and Business Analytics, Macquarie Business School, Macquarie University, Sydney, NSW 2109, Australia.
***Postal address: School of Risk and Actuarial Studies, Business School, The University of New South Wales, Sydney, Australia.
****Postal address: Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong, China.
Get access Rights & Permissions [Opens in a new window]

Abstract

Asset allocation with a derivative security is studied in a hidden, Markovian regime-switching, economy using filtering theory and the martingale approach. A generalized delta-hedged ratio and a generalized elasticity of an option are introduced to accommodate the presence of the information state process and the derivative security. Malliavin calculus is applied to derive a solution for a general utility function which includes an exponential utility, a power utility, and a logarithmic utility. A compact solution is obtained for a logarithmic utility. Some economic implications of the solutions are discussed.

Keywords

Asset allocationDerivativesFilteringMalliavin calculus

MSC classification

Primary: 91G10: Portfolio theory
Secondary: 91G20: Derivative securities 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems) 60H07: Stochastic calculus of variations and the Malliavin calculus
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 3 , September 2019 , pp. 723 - 749
Copyright
© Applied Probability Trust 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baeuerle, N. and Rieder, U. (2007). Portfolio optimization with jumps and unobservable intensity process. Math. Finance 17, 205–224.CrossRefGoogle Scholar
Barucci, E. and Marazzina, D. (2015). Risk seeking, non-convex remuneration and regime switching. Int. J. Theoret. Appl. Finance 18, 1550009.CrossRefGoogle Scholar
Capponi, A. and Figueroa-López, J. E. (2014). Dynamic portfolio optimization with a defaultable security and regime switching. Math. Finance 24, 207–249.CrossRefGoogle Scholar
Clark, J. M. C. (1978). The design of robust approximations to the stochastic differential equations for nonlinear filtering. In Communications Systems and Random Process Theory, ed. Skwirzynski, J. K.. Sijthoff and Noorhoff, Amsterdam, pp. 721–734.CrossRefGoogle Scholar
Cox, J. C. and Huang, C. F. (1989). Optimal consumption and portfolio policies when asset prices follow a diffusion process. J. Econom. Theory 49, 33–83.CrossRefGoogle Scholar
Cui, J., Oldenkamp, B. and Vellekoop, M. (2013). When do derivatives add value in pension fund asset allocation? Rotman Int. J. Pension Management 6, 46–57.Google Scholar
Cvitanic, J. and Karatzas, I. (1992). Convex duality in constrained portfolio optimization. Ann. Appl. Prob. 2, 767–818.CrossRefGoogle Scholar
Detemple, J. and Rindisbacher, M. (2005). Closed-form solutions for optimal portfolio selection with stochastic interest rate and investment constraints. Math. Finance 15, 539–568.CrossRefGoogle Scholar
Di Nunno, G., Øksendal, B. and Proske, F. (2009). Malliavin Calculus for Lévy Processes with Applications to Finance. Springer, New York.CrossRefGoogle Scholar
Elliott, R. J., Aggoun, L. and Moore, J. B. (1995). Hidden Markov Models: Estimation and Control. Springer, New York.Google Scholar
Elliott, R. J., Siu, T. K. and Badescu, A. (2010). On mean-variance portfolio selection under a hidden Markovian regime-switching model. Econom. Model. 27, 678–686.CrossRefGoogle Scholar
Elliott, R. J. and Siu, T. K. (2015). Asset pricing using trading volumes in a hidden regime switching environment. Asia-Pacific Financial Markets 22, 133–149.CrossRefGoogle Scholar
Fu, J., Wei, J. and Yang, H. (2014). Portfolio optimization in a regime-switching market with derivatives. Europ. J. Operat. Res. 233, 184–192.CrossRefGoogle Scholar
Haugh, M. B. and Lo, A. W. (2001). Asset allocation and derivatives. Quant. Finance 1, 45–72.CrossRefGoogle Scholar
Hirshleifer, J. and Hirshleifer, D. (1998). Price Theory And Applications, 6th edn. Prentice Hall, New Jersey.Google Scholar
Honda, T. (2003). Optimal portfolio choice for unobservable and regime-switching mean returns. J. Econom. Dynam. Control 28, 45–78.CrossRefGoogle Scholar
Karatzas, I., Lehoczky, J. P. and Shreve, S. E. (1987). Optimal portfolio and consumption decisions for a “small investor” on a finite horizon. SIAM J. Control Optimization 25, 1557–1586.CrossRefGoogle Scholar
Karatzas, I. (1989). Optimization problems in the theory of continuous trading. SIAM J. Control Optimization 27, 1221–1259.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus. Springer, New York.Google Scholar
Karatzas, I., Lehoczky, J., Shreve, S. E. and Xu, G. L. (1991). Martingale and duality methods for utility maximization in an incomplete market. SIAM J. Control Optimization 29, 702–730.CrossRefGoogle Scholar
Kallianpur, G. (1980). Stochastic Filtering Theory. Springer, New York.CrossRefGoogle Scholar
Korn, R. and Trautmann, S. (1999). Optimal control of option portfolios. OR Spektrum 21, 123–146.CrossRefGoogle Scholar
Korn, R. and Kraft, H. (2002). A stochastic control approach to portfolio problems with stochatic interest rates. SIAM J. Control Optimization 40, 1250–1269.CrossRefGoogle Scholar
Korn, R., Siu, T. K. and Zhang, A. (2011). Asset allocation for a DC pension fund under regime-switching environment. Europ. Actuar. J. 1, 361–377.CrossRefGoogle Scholar
Kraft, H. (2003). Elasticity approach to portfolio optimization. Math. Meth. Operat. Res. 58, 159–182.CrossRefGoogle Scholar
Lakner, P. and Nygren, L. N. (2006). Portfolio optimization with downside constraints. Math. Finance 16, 283–299.CrossRefGoogle Scholar
Lipster, R. and Shiryaev, A. N. (2003). Statistics of Random Processes. Springer, New York.Google Scholar
Liu, H. (2011). Dynamic portfolio choice under ambiguity and regime switching mean returns. J. Econom. Dynam. Control 35, 623–640.CrossRefGoogle Scholar
Liu, J. and Pan, J. (2003). Dynamic derivative strategies. J. Finan. Econom. 69, 401–430.CrossRefGoogle Scholar
Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: the continuous-time model. Rev. Econom. Statist. 51, 247–257.CrossRefGoogle Scholar
Merton, R. C. (1971). Optimum consumption and portfolio rules in a continuous-time model. J. Econom. Theory 3, 373–413.CrossRefGoogle Scholar
Nagai, H., and Runggaldier, W. J. (2008). PDE approach to utility maximization for market models with hidden Markov factors. In Seminar on Stochastic Analysis, Random Fields and Applications V. Progress in Probability, Vol. 59, eds. Dalang, R. C., Dozzi, M. , and Russo, F.. Birkhaeuser, Basel, pp. 493–506.Google Scholar
Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd edn. Springer, New York.Google Scholar
Pliska, S. (1986). A stochastic calculus model of continuous trading: optimal portfolios. Math. Operat. Res. 11, 371–384.CrossRefGoogle Scholar
Rieder, U. and Bäuerle, N. (2005). Portfolio optimization with unobservable Markov-modulated drift process. J. Appl. Prob. 42, 362–378.CrossRefGoogle Scholar
Sass, J. and Haussmann, G. (2004). Optimizing the terminal wealth under partial information: the drift process as a continuous time Markov chain. Finance Stoch. 8, 553–577.CrossRefGoogle Scholar
Shen, Y. and Siu, T. K. (2017). Consumption-portfolio optimization and filtering in a hidden-Markov-modulated asset price model. J. Indust. Manage. Optim. 13, 23–46.Google Scholar
Siu, T. K. (2011). Long-term strategic asset allocation with inflation risk and regime switching. Quant. Finance 11, 1565–1580.CrossRefGoogle Scholar
Siu, T. K. (2014). A hidden Markov-modulated jump diffusion model for European option pricing. In Hidden Markov models in Finance, Vol. 2, eds. Mamon, R. and Elliott, R. J.. Springer, New York, pp. 185–209.CrossRefGoogle Scholar
Siu, T. K. (2015). A stochastic flows approach for asset allocation with hidden economic environment. Int. J. Stoch. Anal. 2015, 462524.Google Scholar
Sotomayor, A. and Cadenillas, A. (2009). Explicit solution of consumption–investment problems in financial markets with regime switching. Math. Finance 9, 251–279.CrossRefGoogle Scholar
Zhou, X. and Yin, G. (2003). Markowitz’s mean-variance portfolio selection with regime switching: a continuous-time model. SIAM J. Control Optim. 42, 1466–1482.CrossRefGoogle Scholar