**6. Conclusions**

The research in this paper was motivated by the open question of how to value future guarantees that are issued by insurance companies. The future value of these guarantees is essential for regulatory and Asset Liability Management purposes. The complexity of the valuation is found in the fact that, first, these guarantees involve optionalities and thus need to be valued using the risk-neutral measure; and, second, whereas this measure is well-defined at *t* = 0, the future risk-neutral measure, at future time *t* = 1, is debatable.

For a large part, the liabilities evolve according to real-world models and, therefore, the future values of these guarantees need to be computed conditionally on the real-world scenarios. In this paper, we demonstrate the benefits of option valuation under a new, so-called P Q measure in Asset Liability Management. This is done by modelling the Heston model parameters, which form the parameterization of the implied volatility surface, conditional on the real-world scenarios.

Basically, we advocate the use of dynamic risk-neutral parameters in the cases in which we need to evaluate asset prices under the P measure, before an option value is required at a future time point. It means that the development of the real-world asset paths in the future are taken into account in the option valuation.

A hedge test was implemented for an academic test case, where the dynamic strategy outperformed the strategy with static parameters. Importantly, the results from this hedge test case were confirmed by a hedge test based on 12 years of empirical, historical data. Several conclusions have already been given after each of the structured test experiments is presented.

The results obtained by the strategy for the Solvency Capital Requirement of the variable annuities exhibited differences of even 50%, as compared to the conventional risk-neutral pricing of these annuities. Next to that, we saw that the SCR was significantly less pro-cyclical under the new approach, which is a highly desired feature.

**Author Contributions:** M.T.P.D., Data Curation; C.S.L.G., Formal Analysis; C.W.O., Methodology.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors would like to thank Pieter Kloek for helpful discussions on option valuation in the context of the SCR.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **Appendix A. Least-Squares Monte Carlo Method**

In this section, we briefly describe the numerical techniques employed, which are based on the well-known least-squares Monte Carlo method.

The Least-Squares Monte Carlo method was first proposed by Longstaff and Schwartz (2001) for the valuation of American options. However, Bauer et al. (2010) were the first to implement it in an SCR context, to the best of our knowledge. The main purpose of the Least-Squares Monte Carlo algorithm is to reduce the number of inner simulations, possibly even to one path. In the first phase, a regression function is constructed using these inner estimates. The accuracy of the inner estimates is drastically reduced by reducing the number of inner simulations, but, by combining the results of all outer simulations, the inner errors cancel out. In the second phase of the algorithm, this regression function is used to evaluate the conditional expectation at *t* = 1, without the need for inner simulations. For more details regarding the Least-Squares Monte Carlo algorithm, we refer the reader to Bauer et al. (2010).

### **Appendix B. Dynamic Heston Model**

Based on the methodologies of Alexander et al. (2009), we assume *v*¯ and *γ* in Equation (13) to be stochastic in the hedge test, i.e.,

$$\begin{cases} \mathbf{d}\boldsymbol{\sigma}\_{t} = \kappa\_{\boldsymbol{\sigma}} (\boldsymbol{\sigma}\_{\text{Mean}} - \boldsymbol{\sigma}\_{t}) \mathbf{d}\mathbf{t} + a\_{\boldsymbol{\sigma}} \boldsymbol{\sigma}\_{t} \left(\rho\_{\boldsymbol{\sigma}} \mathbf{d}\mathcal{W}\_{t}^{2} + \sqrt{1 - \rho\_{\boldsymbol{\sigma}}^{2}} \mathbf{d}\mathcal{W}\_{t}^{\boldsymbol{\overline{\upsilon}}}\right), \\\ \mathbf{d}\boldsymbol{\gamma}\_{t} = \kappa\_{\boldsymbol{\gamma}} (\gamma\_{\text{Mean}} - \gamma\_{t}) \mathbf{d}\mathbf{t} + a\_{\boldsymbol{\gamma}} \gamma\_{t} \left(\rho\_{\boldsymbol{\gamma}} \mathbf{d}\mathcal{W}\_{t}^{2} + \sqrt{1 - \rho\_{\boldsymbol{\gamma}}^{2}} \mathbf{d}\mathcal{W}\_{t}^{\boldsymbol{\overline{\upsilon}}}\right), \end{cases} \tag{A1}$$

with the speed of mean reversion parameters *κv*¯ and *κγ*, long-run averages *v*¯Mean and *γ*Mean, volatilities *av*¯ and *<sup>a</sup>γ* and correlations *ρv*¯ and *ργ*. Moreover, *Wv* ¯ *t* and *<sup>W</sup>γt* are defined as independent Brownian motions. Note that, with the parameters *ρv*¯ and *ργ* close to 1, a high correlation with the volatility process is indicated, which is expected based on historical data. Under these assumptions, the option price is driven by the changes in *v*¯ and *γ* as well, giving

$$\mathbb{C}\_t^{\text{Dynamic}} \equiv \mathbb{C}\left(t, \mathbb{S}\_{t\prime} \upsilon\_{t\prime} r\_{\prime} \vartheta\_{t\prime} \gamma\_{t\prime} \kappa\_{\prime} \rho\_{\prime} \mathcal{K}\_{\prime} T\right). \tag{A2}$$

Next, we give details about the discretization of the dynamic Heston model. We can rewrite the dynamic Heston model with time-dependent *v*¯ and *γ* as follows

$$\begin{cases} \begin{aligned} \mathbf{d}\mathbf{X}\_{l} &= (r - \frac{1}{2}v\_{l})\mathbf{d}t + \sqrt{\nu\_{l}} \left(\rho \mathbf{d}\mathcal{W}\_{l}^{\overline{v}} + \sqrt{1 - \rho^{2}} \mathbf{d}\mathcal{W}\_{l}^{\bar{S}}\right), \\ \mathbf{d}\boldsymbol{v}\_{l} &= \kappa (\boldsymbol{v}\_{l} - \boldsymbol{v}\_{l}) \mathbf{d}t + \gamma\_{l} \sqrt{\boldsymbol{v}\_{l}} \mathbf{d}\mathcal{W}\_{l}^{\overline{v}}, \\ \mathbf{d}\boldsymbol{v}\_{l} &= \kappa\_{\mathcal{I}} (\boldsymbol{v}\_{\text{Mean}} - \boldsymbol{v}\_{l}) \mathbf{d}t + a\_{\mathcal{O}} \boldsymbol{v}\_{l} \left(\rho\_{\mathcal{O}} \mathbf{d}\mathcal{W}\_{l}^{\overline{v}} + \sqrt{1 - \rho\_{\mathcal{O}}^{2}} \mathbf{d}\mathcal{W}\_{l}^{\overline{v}}\right), \\ \mathbf{d}\boldsymbol{\gamma}\_{l} &= \kappa\_{\mathcal{I}} (\gamma\_{\text{Mean}} - \gamma\_{l}) \mathbf{d}t + a\_{\mathcal{I}} \gamma\_{l} \left(\rho\_{\mathcal{I}} \mathbf{d}\mathcal{W}\_{l}^{\overline{v}} + \sqrt{1 - \rho\_{\mathcal{I}}^{2}} \mathbf{d}\mathcal{W}\_{l}^{\overline{v}}\right), \end{aligned} \tag{A3}$$

with *Xt* = log(*St*). We can simulate *vt* and *St* with the Quadratic Exponential scheme proposed in Andersen (2008), with as minor difference that *v*¯ and *γ* are different in each time-step. The next step is to simulate *v*¯*t* and *γt*, such that they are correlated to *vt*. First, we discretize the processes

$$\begin{cases} \begin{aligned} v\_{t+\Delta t} &\approx v\_t + \kappa \left(\overline{v}\_t - \frac{v\_t + v\_{t\Delta t}}{2}\right) \Delta t + \gamma\_t \sqrt{\frac{v\_t + v\_{t\Delta t}}{2}} \Delta t \overline{Z}^v, \\ \overline{v}\_{t+\Delta t} &\approx \overline{v}\_t + \kappa (\overline{v}\_{\text{Mean}} - \overline{v}\_t) \Delta t + a\_{\overline{v}} \rho\_{\overline{v}} \overline{v}\_t \sqrt{\Delta t} \overline{Z}^v + a\_{\overline{v}} \overline{v}\_t \sqrt{\Delta t (1 - \rho\_{\overline{v}}^2)} \overline{Z}^\theta \\ &\quad \quad \quad \quad \quad \quad \quad \quad \quad + \frac{1}{2} a\_{\overline{v}}^2 \overline{v}\_t \left( (\underline{Z}^v)^2 - \text{d}t \right), \\ \gamma\_{t+\Delta t} &\approx \gamma\_t + \kappa (\gamma\_{\text{Mean}} - \gamma\_t) \Delta t + a\_{\gamma} \rho\_{\gamma} \gamma\_t \sqrt{\Delta t} \overline{Z}^v + a\_{\gamma} \gamma\_t \sqrt{\Delta t (1 - \rho\_{\gamma}^2)} \overline{Z}^\gamma \\ &\quad \quad \quad \quad \quad \quad \quad + \frac{1}{2} a\_{\gamma}^2 \gamma\_t \left( (\underline{Z}^\gamma)^2 - \text{d}t \right), \end{aligned} \tag{A4}$$

where *Zv*, *Zv* ¯ and *Zγ* are independent standard normal distributed random variables. Now, we are able to derive an approximation for *Zv*, given *vt*+Δ*t*,

$$\sqrt{\Delta t}Z^{\upsilon} \approx \frac{1}{\gamma\_t \sqrt{\frac{v\_l + v\_{l+\Delta t}}{2}}} \left(v\_{l+\Delta t} - v\_l - \kappa \left(\bar{v}\_l - \frac{v\_l + v\_{l+\Delta t}}{2}\right) \Delta t\right). \tag{A5}$$

This approximation can be substituted into Equation (A4), which ensures the correlation between *vt*, *v* ¯ *t* and *γt*.

### **Appendix C. VIX Heston: UK and Europe**

*Appendix C.1. Calibrated Parameters*

Appendix C.1.1. Parameters obtained from UK data

$$\Omega\_t^{\text{Histor}}(X) = \begin{cases} \kappa\_t &= 1.0, \\ \upsilon\_{0,t} &= \left(-0.0014 + 0.0096 \cdot \text{VFTSE}\_t\right)^2, \\ \upsilon\_t &= \left(0.0590 + 0.0110 \cdot \text{VFTSE}\_{\text{filller}\_t}\right)^2, \\ \gamma\_t &= 0.2556 + 0.0206 \cdot \text{VFTSE}\_t, \\ \rho\_t &= -0.6858. \end{cases} \tag{A6}$$

Appendix C.1.2. Parameters obtained from Europe data

$$\Omega\_t^{\text{Histor}}(X) = \begin{cases} \kappa\_t &= 1.0, \\ \upsilon\_{0t} &= \left(0.0013 + 0.0094 \cdot \text{VIX}\_t\right)^2, \\ \upsilon\_t &= \left(0.0518 + 0.0100 \cdot \text{VIX}\_{\text{filter}\_t}\right)^2, \\ \gamma\_t &= 0.0571 + 0.0252 \cdot \text{VIX}\_{t\prime} \\ \rho\_t &= -0.6471. \end{cases} \tag{A7}$$

*Appendix C.2. Predicted Parameters*

Appendix C.2.1. Parameters predicted for UK

**Figure A1.** Prediction results Heston parameters of the UK dataset.

Appendix C.2.2. Parameters predicted for Europe

**Figure A2.** Prediction results Heston parameters of the Europe dataset.
