Skip to main content Accessibility help
×
Home

Optimal Portfolios for Financial Markets with Wishart Volatility

  • Nicole Bäuerle (a1) and Zejing Li (a1)

Abstract

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Optimal Portfolios for Financial Markets with Wishart Volatility
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Optimal Portfolios for Financial Markets with Wishart Volatility
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Optimal Portfolios for Financial Markets with Wishart Volatility
      Available formats
      ×

Copyright

Corresponding author

Postal address: Institute for Stochastics, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany.
∗∗ Email address: nicole.baeuerle@kit.edu
∗∗∗ Email address: zejingli@stoch.uni-karlsruhe.de

References

Hide All
Bru, M.-F. (1991). Wishart processes. J. Theoret. Prob. 4, 725751.
Buraschi, A., Porchia, P. and Trojani, F. (2010). Correlation risk and optimal portfolio choice. J. Finance 65, 393420.
Cont, R. and Da Fonseca, J. (2002). Dynamics of implied volatility surfaces. Quant. Finance 2, 4560.
Da Fonseca, J., Grasselli, M. and Tebaldi, C. (2007). Option pricing when correlations are stochastic: An analytical framework. Rev. Derivatives Res. 10, 151180.
Da Fonseca, J., Grasselli, M. and Tebaldi, C. (2008). A multifactor volatility Heston model. Quant. Finance 8, 591604.
Gnoatto, A. and Grasselli, M. (2014). The explicit Laplace transform for the Wishart process. To appear in J. Appl. Prob.
Gourieroux, C. and Sufana, R. (2003). Wishart quadratic term structure models. SSRN E-library 135.
Gourieroux, C. and Sufana, R. (2004). Derivative pricing with Wishart multivariate stochastic volatility: Application to credit risk. SSRN E-library 144.
Hata, H. and Sekine, J. (2013). Risk-sensitive asset management under a Wishart Autoregressive factor model. J. Math. Finance 3, 222229.
Heston, S. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Studies 6, 327343.
Kallsen, J. and Muhle-Karbe, J. (2010). Utility maximization in affine stochastic volatility models. Internat. J. Theor. Appl. Finance 13, 459477.
Korn, R. and Kraft, H. (2004). On the stability of continuous-time portfolio problems with stochastic opportunity set. Math. Finance 14, 403414.
Kraft, H. (2005). Optimal portfolios and Heston's stochastic volatility model: an explicit solution for power utility. Quant. Finance 5, 303313.
Liptser, R. S. and Shiryaev, A. N. (2001). Statistics of random processes: I, (Appl. Math. 5). Springer, Berlin.
Liu, J. (2007). Portfolio selection in stochastic environments. Rev. Financial Studies 20, 139.
Matsumoto, S. (2012). General moments of the inverse real Wishart distribution and orthogonal Weingarten functions. J. Theoret. Prob. 25, 798822.
Mayerhofer, E., Pfaffel, O. and Stelzer, R. (2011). On strong solutions for positive definite Jump diffusions. Stoch. Process. Appl. 121, 20722086.
Muhle-Karbe, J., Pfaffel, O. and Stelzer, R. (2012). Option pricing in multivariate stochastic volatility models of OU type. SIAM J. Financial Math. 3, 6694.
Rellich, F. (1969). Perturbation Theory of Eigenvalue Problems. Gordon and Breach Science Publishers, New York.
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.
Rieder, U. and Bäuerle, N. (2005). Portfolio optimization with unobservable Markov-modulated drift process. J. Appl. Prob. 42, 362378.
Sultan, S. A. and Tracy, D. S. (1996). Moments of Wishart distribution. Stoch. Anal. Appl. 14, 237243.
Zariphopoulou, T. (2001). A solution approach to valuation with unhedgeable risks. Finance Stoch. 5, 6182.

Keywords

MSC classification

Optimal Portfolios for Financial Markets with Wishart Volatility

  • Nicole Bäuerle (a1) and Zejing Li (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed