International portfolio choice under multi-factor stochastic volatility

Abstract

In this article, we develop an identifiable multi-factor stochastic volatility model for international portfolio choice problems in complete and incomplete markets. Allowing for stochastic covariance between financial asset returns and foreign exchange rates, optimal investment strategies are derived in closed form and welfare losses arising from suboptimal investment strategies are analysed. Moreover, we provide a two-step procedure for estimating as well as calibrating the model parameters and use this ansatz to illustrate optimal investment decisions for the S&P 500, the German blue chip index DAX, and the USD/EUR foreign exchange rate. We find, both theoretically and empirically, that the model satisfies various well-known stylized facts of equity and foreign exchange rate markets and that investors who invest myopically or ignore derivative assets can incur substantial welfare losses implying strong evidence for significant welfare benefits from international diversification across different asset classes.

Keywords:

• International portfolio choice
• Portfolio choice with stochastic covariance
• Foreign exchange markets
• Welfare loss analysis

JEL classifications:

• F31
• G11
• G15

Data availability statement

The data that support the findings of this study are available from the corresponding author, Christoph Gschnaidtner (CG), upon reasonable request.

Disclosure statement

The authors declare that they have no conflict of interest.

Supplemental data

Supplemental data for this article can be accessed at http://dx.doi.org/10.1080/14697688.2021.2019820.

Notes

1 For example, this is one of the reasons Da Fonseca and Grasselli 2011 calibrate the Wishart model to a fixed day rather than a time series of option prices.

2 Buraschi et al. 2010 and Da Fonseca et al. 2014 do not evaluate the welfare losses from myopic investment strategies. Moreover, the analysis in Da Fonseca et al. 2014 is only for a specific type of derivatives (swaps), whereas our results apply to a much wider class of derivative securities.

3 In an accompanying online appendix, we expand the empirical analysis with estimations of the PCSV model to intraday price data from January, 2009 to January, 2021.

4 The parameter of the constant relative risk aversion (CRRA) utility function is equal to 4. See equation (13(13) $J(t,x,\mathit{v})=\underset{\mathit{\varphi }\in \mathcal{U}[t,T]}{sup}E[\frac{1}{1-\gamma }{\left(X_T\right)}^{1-\gamma }|X_t=x,{\mathit{v}}_t=\mathit{v}],$(13) ).

5 This property of stochastic volatility models does not hold in general. For example, the well-known Stochastic Alpha Beta Rho (SABR) stochastic volatility model does not satisfy this property.

6 It is worth mentioning that De Col et al. 2013 do not make this assumption because they only calibrate the model to the observed option prices and thus, do not estimate the model parameters per se.

7 The assumption (8(8) ${\mathit{\lambda }}_t^P={\mathit{V}}_t^{1/2}{\mathit{A}}^{\mathsf{T}}{\mathit{\lambda }}^P,$(8) ) implies constant market price of variance–covariance risk. The same assumption is also made in the univariate models in Heston 1993 and Liu 2007 as well as in the multivariate models in Buraschi et al. 2010 and Bäuerle and Li 2013.

8 Following Björk 2009 the market prices of risk for the factors $B_1,B_2,B_3$ are relative to the specified numéraire and depend indeed on the perspective of the investor (domestic versus foreign) – somewhat comparable to risk neutrality being a property that holds only relative to the respective numéraire leading to the difference between the domestic and the foreign risk-neutral measure. At first one would naturally guess that the market prices of risk are identical for the domestic and foreign investor. Yet, unless the exchange rate is deterministic, this is not the case. Indeed, the specific way of quoting exchange rates (units of domestic currency per unit of foreign currency) gives rise to the asymmetry between the domestic and the foreign perspective. Hence, depending on the perspective of the investor we refer to the market prices of risk either as ‘domestic’ or ‘foreign’ and thus, attach an origin to the market prices of risk.

9 Similar equations can also be derived for the domestic (${\mu }_D=r^D+{\mathit{\sigma }}_D^{\mathsf{T}}{\mathit{V}}_t{\mathit{A}}^{\mathsf{T}}{\mathit{\lambda }}^P$) and the foreign (${\mu }_F=r^F+{\mathit{\sigma }}_F^{\mathsf{T}}{\mathit{V}}_t({\mathit{A}}^{\mathsf{T}}{\mathit{\lambda }}^P-e)$) stock drifts.

10 The dependence of $X_t$ on ϕ is omitted from the notation.

11 This property is contrary to the result obtained in Larsen 2010: The increase in risk aversion results in larger investment in the hedge portfolio. The difference is due to the fact that the investment opportunity set in Larsen 2010 is stochastic as a result of the stochastic interest rates, whereas in our model this is due to the covariance between the asset returns.

12 To distinguish from the solution J to the HJB equation (14(14) $\begin{array}{rl}& J_t+\underset{\mathit{\varphi }\in {\mathbb{R}}^3}{sup}\{\frac{1}{2}{x}^2{\mathit{\varphi }}^{\mathsf{T}}\mathbf{\Sigma }{\mathbf{\Sigma }}^{\mathsf{T}}\mathit{\varphi }{J}_{xx}+x\left({\mathit{\varphi }}^{\mathsf{T}}\right(\mathit{\mu }-\mathit{r})+r^D)J_x\\ & +x{\mathit{\varphi }}^{\mathsf{T}}\mathbf{\Sigma }{\mathit{R}}^{\mathsf{T}}{\mathbf{\Sigma }}^{v\mathsf{T}}{\mathit{J}}_{xv}+\frac{1}{2}\mathrm{trace}\left({\mathbf{\Sigma }}^v{\mathbf{\Sigma }}^{v\mathsf{T}}{\mathit{J}}_{vv}\right)+{\mathit{\mu }}^{v\mathsf{T}}{\mathit{J}}_v\}=0,\end{array}$(14) ) for incomplete market, we write $\overline{J}$ as the solution to the HJB equation (35(35) $\begin{array}{rl}& {\overline{J}}_t+\underset{\mathit{\eta }\in {\mathbb{R}}^6}{sup}\{\frac{1}{2}{x}^2{\mathit{\eta }}^{\mathsf{T}}\overline{\mathit{V}}\mathit{\eta }{\overline{J}}_{xx}+x(r^D+{\mathit{\eta }}^{\mathsf{T}}\overline{\mathit{V}}\overline{\mathit{\lambda }}){\overline{J}}_x\\ & +x{\mathit{\eta }}^{\mathsf{T}}{\overline{\mathit{V}}}^{1/2}{\overline{\mathit{R}}}^{\mathsf{T}}{\mathbf{\Sigma }}^{v\mathsf{T}}{\overline{\mathit{J}}}_{xv}+\frac{1}{2}\mathrm{trace}\left({\mathbf{\Sigma }}^v{\mathbf{\Sigma }}^{v\mathsf{T}}{\overline{\mathit{J}}}_{vv}\right)+{\mathit{\mu }}^{v\mathsf{T}}{\overline{\mathit{J}}}_v\}\\ & =0,\end{array}$(35) ) for complete market.

13 To be more precise, this component arises due to the correlation between assets and eigenvalues (i.e. due to the leverage effect) and allows the investor to take advantage of the eigenvalue risk premium. Thus, even if the expression might suggest otherwise, the second term of equation (47(47) $\begin{array}{rl}{\overline{\mathit{\varphi }}}_t^S& =\frac{1}{\gamma }{\mathit{\lambda }}^P-\frac{1}{\gamma }\mathit{A}\mathit{R}(\mathit{I}-{\mathit{R}}^2{)}^{-1/2}{\mathit{\lambda }}^v-D[\mathit{S}\left]{\mathit{H}}_p^{\mathsf{T}}{D}^{-1}\right[\mathit{O}]{\overline{\mathit{\varphi }}}_t^O,\end{array}$(47) ) does not represent a compensation from the Brownian risk factors $B_4,B_5,B_6$ but rather from $B_1,B_2,B_3$ which are shared between assets and eigenvalues.

14 See Larsen and Munk 2012.

15 We would like to point out that without this assumption the strategy of the home-biased investor is no longer independent of the state variables ${\mathit{v}}_t$. Furthermore, the corresponding HJB equation does not seem to have a closed-form solution.

16 In the accompanying online appendix, we also provide the results of the estimation of the PCSV model to time series price data at the intraday level (down to 1-minute intervals) of the S&P 500 index, the DAX index, and the USD/EUR exchange rate.

17 Adjusting for the number of calibration instruments allows for comparing the errors over time when the number of available calibration instruments varies. Here, the same number of calibration instruments are used for all calibrations. Nonetheless, we report the MRSVE as this is the standard in the literature, see e.g. Christoffersen et al. 2009, Da Fonseca and Grasselli 2011, or Escobar and Gschnaidtner 2018.

18 While calibration to foreign exchange rate options are usually conducted using error measures based on Black–Scholes implied volatilities, calibrations to index or equity options minimize the difference between market and model option prices. Based on the findings of Escobar and Gschnaidtner 2016, we decided to use the MRSVE rather than the MRSPE for the calibration. This decision is in line with the arguments put forward by Christoffersen et al. 2009.

19 Notice that, following Bessembinder 1994, the USD/EUR exchange rate is given in American terms; $E_t$ constitutes the units of USD necessary to buy one unit of EUR, i.e. 1 EUR = $E_t$ USD

20 As an approximation for the interest rates we use the averages over the entire estimation period of the respective 3 months swap rates which were found to be 0.4% for the US and 0.32% for Germany. For simplicity, we set both interest rates equal to 0.35%, leading to an interest rate differential of $r^D-r^F=0$.

21 Here, $\mathrm{\%}$-moneyness is defined as the option's strike divided by the current spot price of the underlying.

22 Notice that the matrices $\mathit{R}$ and ${\mathbf{\Sigma }}^{v^{\prime }}$ are diagonal.

23 Liu and Pan 2003 mentioned several values for the market price of volatility risk (one-dimensional model of Heston 1993), ranging from 0 to $-23$ (including $-6$, $-10$ and $-12$, see Liu and Pan 2003, page 418 and footnote 14). A simple moment matching approach on the stationary distributions between the model of Heston 1993 and our model 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 ${\lambda }_4$, ${\lambda }_5$, and ${\lambda }_6$. A similar range is also confirmed by our calibration exercise whose results are given in table 2.

24 For example, an increase in volatility of U.S. equity returns is followed by an increase in volatility of German equity returns.

Additional information

Funding

This work was supported by Natural Sciences and Engineering Research Council of Canada [CRD grant, Discovery grant].