*3.2. VIX Heston Model*

We have already described the difficulties of modelling the implied volatility surface, or equivalently, the option prices. A different approach is therefore required to calibrate the Heston parameters in simulated markets. The simulated parameter sets should accurately reflect the expectations of the simulated market, for example by linking the dynamics of the Heston parameters to the dynamics of the market. In Singor et al. (2017), an approach is considered which is based on the assumptions of a linear relationship between the VIX index and the Heston parameters. After analysis, it was concluded that:


To this end, the following restrictions are imposed on the Heston model parameters:

$$\Omega\_t^{\text{Histor}}(X) = \begin{cases} \kappa\_t &= \kappa\_\prime & \kappa \in \mathbb{R}\_+, \\ \upsilon\_{0,t} &= \left(a\_{\upsilon\_0} \cdot \text{VIX}\_t + b\_{\upsilon\_0}\right)^2, & a\_{\upsilon\_0}, b\_{\upsilon\_0} \in \mathbb{R}\_\prime \\ \upsilon\_t &= \left(a\_\sigma \cdot \text{VIX}\_{\text{filter}\_l} + b\_\sigma\right)^2, & a\_\sigma, b\_\sigma \in \mathbb{R}\_\prime \\ \gamma\_t &= a\_\gamma \cdot \text{VIX}\_t + b\_{\gamma\_t}, & a\_{\gamma\_t}, b\_\gamma \in \mathbb{R}\_\prime \\ \rho\_t &= \rho\_\prime & \rho \in [-1, 1]. \end{cases} \tag{16}$$

where both the speed of mean reversion *κt* and the correlation coefficient *ρt* are assumed to be constant over time. The constant *ρ* assumption is justified by the fact that *ρ* displays a mean reverting pattern and it can therefore be approximated by its long-term mean. The constant *κ* assumption is justified by observations in Gauthier and Rivaille (2009). They argued that the effect on the implied volatility surface of increasing *κ* is similar to decreasing *γ*. Thus, allowing *κ* to change over time unnecessarily overcomplicates the model. Moreover, numerical experiments show that an unrestricted *κ* sometimes leads to unstable results.

The purpose of the restrictions is to accurately reflect the market's expectations. To this end, we wish to minimize the distance between the observed and predicted implied volatility surfaces. Therefore, we calibrate the constraint parameters with a procedure similar to Equation (15). By changing the parameter set from ΩHeston *t* to *X* = {*<sup>κ</sup>*, *<sup>α</sup>v*0 , *bv*0 , *av*¯, *bv*¯, *<sup>a</sup>γ*, *bγ*, *ρ*} and summing over all points in time one obtains

$$X = \underset{X^{\mathrm{S}} \in X^{\mathrm{Ssearch}}}{\arg\min} \left( \sum\_{t, K, T} \left( \sigma^{\mathrm{Market}}(t, K, T) - \sigma^{\mathrm{Histor}}(t, \Omega\_t^{\mathrm{Histor}}(X^{\mathrm{S}}), K, T) \right)^2 \right), \tag{17}$$

with,

$$X^{\text{Ssearch}} = D\_{\mathbf{x}} \times D\_{\mathbf{a}\_{\text{V}\_0}} \times D\_{\mathbf{b}\_{\text{V}\_0}} \times D\_{\mathbf{a}\_{\text{\theta}}} \times D\_{\mathbf{b}\_{\text{\theta}}} \times D\_{\mathbf{a}\_{\text{\gamma}}} \times D\_{\mathbf{b}\_{\gamma}} \times D\_{\mathbf{b}\_{\gamma}}$$

$$= \{1\} \times \mathbb{R}^6 \times [-1, 1]. \tag{18}$$

By including the VIX-index in the real-world simulation, one is able to efficiently evaluate the set of Heston parameters in line with the simulated state of the market. For more information regarding the derivation and properties of the VIX Heston model, we refer the reader to Singor et al. (2017).

It is, however, important to stress the different assumptions in the real-world and risk-neutral markets. Risk-neutral valuations are performed under the Heston model, which assumes constant parameters. However, in real-world simulations, we assume the Heston parameters to *change over time*, according to the simulated state of the market, similar to Figure 1. One could argue that this approach is invalid, since we are violating the assumptions of the risk-neutral market. To this end, we discuss a justification of this approach, by means of a hedge test. Moreover, to assess the impact of time-dependent Heston parameters, we implement the VIX Heston model as proposed in Singor et al. (2017) in a risk-management application: the Solvency Capital Requirement.
