Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-13T00:50:09.724Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal Portfolios for Financial Markets with Wishart Volatility

Part of: Stochastic systems and control Mathematical finance

Published online by Cambridge University Press:  30 January 2018

Nicole Bäuerle  and
Zejing Li
Nicole Bäuerle*
Affiliation:
Karlsruhe Institute of Technology
Zejing Li*
Affiliation:
Karlsruhe Institute of Technology
*
∗ Postal address: Institute for Stochastics, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany.
∗ Postal address: Institute for Stochastics, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a multi asset financial market with stochastic volatility modeled by a Wishart process. This is an extension of the one-dimensional Heston model. Within this framework we study the problem of maximizing the expected utility of terminal wealth for power and logarithmic utility. We apply the usual stochastic control approach and obtain, explicitly, the optimal portfolio strategy and the value function in some parameter settings. In particular, we do this when the drift of the assets is a linear function of the volatility matrix. In this case the affine structure of the model can be exploited. In some cases we obtain a Feynman-Kac representation of the candidate value function. Though the approach we use is quite standard, the hard part is to identify when the solution of the Hamilton-Jacobi-Bellman equation is finite. This involves a couple of matrix analytic arguments. In a numerical study we discuss the influence of the investors' risk aversion on the hedging demand.

Keywords

Wishart processportfolio problemCRRA utilitystochastic controlHamilton-Jacobi-Bellman equationmatrix exponential

MSC classification

Primary: 93E20: Optimal stochastic control 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems) 91G10: Portfolio theory
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 4 , December 2013 , pp. 1025 - 1043
Copyright
© Applied Probability Trust 

References

Bru, M.-F. (1991). Wishart processes. J. Theoret. Prob. 4, 725–751.CrossRefGoogle Scholar
Buraschi, A., Porchia, P. and Trojani, F. (2010). Correlation risk and optimal portfolio choice. J. Finance 65, 393–420.Google Scholar
Cont, R. and Da Fonseca, J. (2002). Dynamics of implied volatility surfaces. Quant. Finance 2, 45–60.CrossRefGoogle Scholar
Da Fonseca, J., Grasselli, M. and Tebaldi, C. (2007). Option pricing when correlations are stochastic: An analytical framework. Rev. Derivatives Res. 10, 151–180.CrossRefGoogle Scholar
Da Fonseca, J., Grasselli, M. and Tebaldi, C. (2008). A multifactor volatility Heston model. Quant. Finance 8, 591–604.Google Scholar
Gnoatto, A. and Grasselli, M. (2014). The explicit Laplace transform for the Wishart process. To appear in J. Appl. Prob. CrossRefGoogle Scholar
Gourieroux, C. and Sufana, R. (2003). Wishart quadratic term structure models. SSRN E-library 1–35.Google Scholar
Gourieroux, C. and Sufana, R. (2004). Derivative pricing with Wishart multivariate stochastic volatility: Application to credit risk. SSRN E-library 1–44.Google Scholar
Hata, H. and Sekine, J. (2013). Risk-sensitive asset management under a Wishart Autoregressive factor model. J. Math. Finance 3, 222–229.CrossRefGoogle Scholar
Heston, S. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Studies 6, 327–343.CrossRefGoogle Scholar
Kallsen, J. and Muhle-Karbe, J. (2010). Utility maximization in affine stochastic volatility models. Internat. J. Theor. Appl. Finance 13, 459–477.CrossRefGoogle Scholar
Korn, R. and Kraft, H. (2004). On the stability of continuous-time portfolio problems with stochastic opportunity set. Math. Finance 14, 403–414.Google Scholar
Kraft, H. (2005). Optimal portfolios and Heston's stochastic volatility model: an explicit solution for power utility. Quant. Finance 5, 303–313.CrossRefGoogle Scholar
Liptser, R. S. and Shiryaev, A. N. (2001). Statistics of random processes: I, (Appl. Math. 5). Springer, Berlin.Google Scholar
Liu, J. (2007). Portfolio selection in stochastic environments. Rev. Financial Studies 20, 1–39.CrossRefGoogle Scholar
Matsumoto, S. (2012). General moments of the inverse real Wishart distribution and orthogonal Weingarten functions. J. Theoret. Prob. 25, 798–822.Google Scholar
Mayerhofer, E., Pfaffel, O. and Stelzer, R. (2011). On strong solutions for positive definite Jump diffusions. Stoch. Process. Appl. 121, 2072–2086.Google Scholar
Muhle-Karbe, J., Pfaffel, O. and Stelzer, R. (2012). Option pricing in multivariate stochastic volatility models of OU type. SIAM J. Financial Math. 3, 66–94.Google Scholar
Rellich, F. (1969). Perturbation Theory of Eigenvalue Problems. Gordon and Breach Science Publishers, New York.Google Scholar
Richter, A. (2012). Explicit solutions to quadratic BSDEs and applications to utility maximization in multivariate affine stochastic volatility models. Preprint. Available at http://arxiv.org/abs/1201.2877v1.Google Scholar
Rieder, U. and Bäuerle, N. (2005). Portfolio optimization with unobservable Markov-modulated drift process. J. Appl. Prob. 42, 362–378.CrossRefGoogle Scholar
Sultan, S. A. and Tracy, D. S. (1996). Moments of Wishart distribution. Stoch. Anal. Appl. 14, 237–243.CrossRefGoogle Scholar
Zariphopoulou, T. (2001). A solution approach to valuation with unhedgeable risks. Finance Stoch. 5, 61–82.CrossRefGoogle Scholar