Optimal investment under multi-factor stochastic volatility

Abstract

We consider a model for multivariate intertemporal portfolio choice in complete and incomplete markets with a multi-factor stochastic covariance matrix of asset returns. The optimal investment strategies are derived in closed form. We estimate the model parameters and illustrate the optimal investment based on two stock indices: S&P500 and DAX. It is also shown that the model satisfies several stylized facts well known in the literature. We analyse the welfare losses due to suboptimal investment strategies and we find that investors who invest myopically, ignore derivative assets, model volatility by one factor and ignore stochastic covariance between asset returns can incur significant welfare losses.

Keywords:

• Multivariate portfolio choice
• Stochastic covariance
• Welfare loss

JEL Classification:

• G11

Notes

No potential conflict of interest was reported by the authors.

1 For example, this optimisation difficulty is one of the reasons that Da Fonseca and Grasselli (2011) calibrate their Wishart model based on a set of options quoted at a fixed day instead of a time series of option prices.

2 For example, an increase in volatility of US equity returns is followed by an increase in volatility of German equity returns.

3 The dependence of on is omitted from the notation.

4 A similar assumption is also made in univariate models of Heston (1993), Liu (2007), and multivariate models of Buraschi et al. (2010), and Bäuerle and Li (2013).

5 The fact that follows form the initial condition and from the fact that in (A6) the derivative whenever .

6 This specification of the market prices of risks is consistent with the two-factor model () of option pricing considered in Christoffersen et al. (2009).

7 This interpretation of is similar to that of in Liu and Pan (2003).

8 To distinguish the solution J to the HJB equation (4.5(4.5) ) for the incomplete markets, we write as the solution to the HJB equation (4.24(4.24) ) for the complete markets.

9 We omit the dependency of the suboptimal strategies and on in the notation.

10 See Larsen and Munk (2012).

11 We choose a uniform density function in a bounded interval in order to obtain better estimates.

12 The proof is similar to the one given in appendix 2.

13 It is not difficult to establish this result rigorously from equation E6.

14 Extending the range of and gives the same lower bounds for the loss.

15 Liu and Pan (2003) mentioned several values for the market price of volatility risk, ranging from 0 to (including , and , see page 418 and footnote 14). A simple moments matching approach on the stationary distributions between the model of Heston (1993) and the two-factor model of Christoffersen et al. (2009) would allow for extrapolating Liu and Pan (2003) market price of volatility risk estimates to our market price of eigenvalues risk values. This exercise led to a similar range of values for and .