*3.1. Heston Model*

We assume the Heston model as a benchmark. The Heston stochastic volatility model (Heston 1993) assumes the volatility of the stock price process to be driven by a CIR model, i.e., under the Q measure, 

$$\begin{cases} \mathbf{d}S\_{l} = rS\_{l}\mathbf{d}t + \sqrt{\upsilon\_{l}}\mathbf{S}\_{l}\mathbf{d}\mathcal{W}\_{t}^{1} & \mathcal{S}(0) = \mathcal{S}\_{0\prime} \\ \mathbf{d}\upsilon\_{l} = \kappa(\upsilon - \upsilon\_{l}) + \gamma\sqrt{\upsilon\_{l}}\mathbf{d}\mathcal{W}\_{t}^{2} & \upsilon(0) = \upsilon\_{0} \\ \langle \mathbf{d}\mathcal{W}\_{t}^{1}, \mathbf{d}\mathcal{W}\_{t}^{2} \rangle = \rho\mathbf{d}t. \end{cases} \tag{13}$$

where *r* denotes the risk-free rate, *κ* the speed of mean-reversion, *v*¯ the long-term variance, *γ* the volatility of variance and *ρ* the correlation between asset price and variance. The risk-free rate is assumed to be constant throughout this research. To calibrate these parameters according to the market's expectations, one wishes to minimize the distance between the model's and market's implied volatilities. For the Heston model, consider the following search space for the parameters

$$\begin{split} \Omega^{\text{Search}} &= D\_{\text{X}} \times D\_{\text{U}\_{0}} \times D\_{\text{J}} \times D\_{\text{Y}} \times D\_{\text{\textdegree P}} \\ &= [0, 10] \times [0, 1] \times [0, 1] \times [0, 2] \times [-1, 1], \end{split} \tag{14}$$

where *Dp* denotes the search domain for parameter *p*. Using this search space, one is able to find the calibrated parameters at time *t* by minimizing the sum of squared errors:

$$\Omega\_t^{\text{Hesson}} = \underset{\Omega \in \Omega^{\text{Sowch}}}{\text{arg min}} \left( \sum\_{K} \sum\_{T} \left( \sigma^{\text{Market}}(t, K, T) - \sigma^{\text{Hesson}}(t, \Omega, K, T) \right)^2 \right). \tag{15}$$

The set of parameters ΩHeston *t* = {*<sup>κ</sup>*, *v*0, *v*¯, *γ*, *ρ*} minimizing this expression is considered risk-neutral and reflects the market's expectations.

When the market is subject to changes, its expectations will change accordingly. Hence, the implied volatility surface will evolve dynamically over time. Consequently, the Heston parameters may change over time. Figure 1 shows the monthly evolution of the Heston parameters from January 2006 to February 2017.

**Figure 1.** Evolution of the Heston parameters over time.

During this calibration procedure, we assumed *κ* to be constant; an unrestricted *κ* led to unstable results and did not significantly improve the accuracy. The other parameters, however, do not appear constant over time. This may give rise to issues in risk-management applications, where one simulates many real-world paths to assess the sensitivity to the market of a portfolio, balance sheet, etc. In many cases, a risk-neutral valuation is required, which is nested inside the real-world simulation, for example, when the portfolio or balance sheet contains options. Consequently, the future implied volatility surface for each trajectory needs to be known, such that the Heston parameters can be calibrated accordingly. Modelling the implied volatility surface over time is a challenging task (see, e.g., Cont et al. (2002); Mixon (2002); Audrino and Colangelo (2010)), as it quantifies the market's expectations that depend on many factors. Moreover, even if the implied volatility surface is modelled, one would still need to perform a costly calibration procedure. Performing this calibration for each of the simulated trajectories would require a significant computational effort. As the Heston model is a parameterization of the implied volatility surface, an attractive alternative is to simulate the Heston parameters directly.

Figure 1 clearly shows that the parameters are time-dependent, which is in contrast with the assumptions of the plain Heston model. In this research we first search for relations between the risk-neutral parameters and observe real-world market indices, such as the VIX index. When a relation is found, the future risk-neutral parameters can be extracted from a real-world simulation. In this way, by performing a real-world simulation, we can directly forecast the set of Heston parameters within each simulated trajectory. The risk-neutral measure is then conditioned on the simulated state of the market, without the need of a simulated implied volatility surface.
