*5.1. Data and Calibration*

The dataset contains monthly implied volatility surfaces of the S&P-500 European put and call options from January 2006 to February 2017. Each implied volatility surface contains five different strike levels (80%, 90%, 100%, 110% and 120% of *S*0) and maturities (0.25, 0.5, 1, 1.5 and 2 years). We split the dataset into a training set and a test set. The training set consists of implied volatility surfaces from January 2006 to February 2014 and is only used to identify the optimal regression components. The test set contains monthly implied volatility surfaces from March 2014 to February 2017 and is used to assess the accuracy of the VIX Heston model.

Furthermore, to assess the robustness of the VIX Heston model, we apply it to monthly implied volatility surfaces of the FTSE-100 (United Kingdom) and STOXX-50 (Europe) as well. The training set includes data from October 2010 to June 2015 and the test set contains data from July 2015 to February 2017.

To assess the accuracy of the VIX Heston model, one must first calibrate the model according to Equation (17). Using the US training set described above, we obtain

$$\Omega\_t^{\text{Histor}}(\mathbf{X}) = \begin{cases} \kappa\_t &= 1.0, \\ \upsilon\_{0t} &= \left(0.0140 + 0.0090 \cdot \text{VIX}\_t\right)^2, \\ \upsilon\_t &= \left(0.0957 + 0.0087 \cdot \text{VIX}\_{\text{filter}\_t}\right)^2, \\ \gamma\_t &= 9.6479 \cdot 10^{-5} + 0.0270 \cdot \text{VIX}\_t, \\ \rho\_t &= -0.7294. \end{cases} \tag{39}$$

The calibrated parameters of the UK and Europe datasets can be found in Appendix C. The accuracy is assessed by comparing the predicted to the observed implied volatility surfaces of the test set, according to the following error measures,

$$\begin{cases} \text{SSE} = \sum\_{t,K,T} \left( \sigma^{\text{Market}}(t,K,T) - \sigma^{\text{Histor}}(t,\Omega\_t^{\text{Histor}},K,T) \right)^2, \\ \quad \text{MAE} = \frac{1}{N\_{\mathcal{V}}} \sum\_{t,K,T} \left| \sigma^{\text{Market}}(t,K,T) - \sigma^{\text{Histor}}(t,\Omega\_t^{\text{Histor}},K,T) \right|, \\ \quad R^2 = 1 - \frac{\text{SSE}}{\sum\_{t} \left( \sigma^{\text{Market}}(t,K,T) - \mathfrak{v}^{\text{Market}} \right)^2}, \\ \quad R\_{\text{Min}}^2 = \min\_t \left\{ R\_t^2 : t \in \left[ t\_{\text{min}}, t\_{\text{max}} \right] \right\}, \end{cases} \tag{40}$$

where ΩHeston *t* is defined as the predicted parameter set, *Nσ* as the total number of observed implied volatilities and *σ*Market as the average of all observed implied volatilities. The corresponding results are displayed in Table 2. The predicted paths for the Heston parameters of the US dataset can be found in Figure 3. The predicted paths of the Heston parameters of the UK and Europe dataset are graphically presented in Appendix C.

**Table 2.** Out-of-sample accuracy of the regression models according to the error measures defined in Equation (40).


**Figure 3.** Prediction results Heston parameters of the US dataset.

The regression model loses some accuracy compared to the unrestricted model. On average, there is an error of 0.003 between the implied volatility and the unrestricted Heston model in the US dataset, which is approximately equal to an error of 2.2%. The VIX Heston model has an average absolute error of 0.012, which corresponds to a 7.7% error. Thus, by implementing the regression models, we introduce an additional error of 5.5%, on average. The accuracy of the VIX Heston model in the UK and Europe datasets is even higher, where the accuracy loss is equal to 2.9% and 1.2%, respectively.

Finally, we discuss the predictions obtained in dependence of parameter *γ*. In the US dataset, the prediction of *γ* is relatively inaccurate (see Figure 3), as the out-of-sample correlation to the VIX index is much lower than the in-sample correlation (0.86 in-sample versus 0.32 out-of-sample). In the UK and Europe datasets, this phenomenon does not seem to be present and consequently the predictions of *γ* are much more accurate (see Figures A1 and A2 in Appendix C). This also explains why the implied volatility surface predictions in the UK and Europe datasets are more accurate than the US predictions, as can be seen in Table 2. Thus, there appears to be another, ye<sup>t</sup> unknown, factor driving *γ* in the US dataset, which is absent in the UK and Europe datasets. Analysis of the cause of this phenomenon might be a topic for future study. However, in this research, we assume that the VIX Heston model is sufficiently accurate to describe the dynamic behaviour of the Heston parameters, as *γ* only has a minor effect on the implied volatility surface.
