**1. Introduction**

Liabilities of insurance companies depend on the fair value of the outstanding claims that typically involve guarantees (that are also called embedded options). The market consistent value of these guarantees is defined under the risk-neutral measure Q, i.e., they are then computed with pricing formulas that agree on the current implied volatility surfaces. To hedge against the risks involved in these claims, insurers often acquire (complex) option portfolios that also require market consistent risk-neutral valuation themselves. Furthermore, on 1 January 2016 the so-called Solvency II directive came into effect which introduced the Solvency Capital Requirement (SCR). The SCR is defined as the minimum amount of capital which should be held by an insurer, such that the insurer is able to pay its claims over a one-year horizon with a 99.5% probability. The regulator demands that the insurer's available capital should be greater than, or equal to, the SCR. Because the claims typically depend on *future* market consistent valuation, computing the SCR, and, more generally performing proper Asset Liability Management (ALM), is a challenging task.

To compute these future market consistent values of the embedded options, insurers require the probability distribution of the values of these embedded options. Typically, this is done by simulating a large number of random future states of the market and, after that, the different states are valued

under the market consistent risk-neutral measure. From the simulated embedded option values, the desired statistics can then be extracted. The future states of the market can be computed by means of risk-neutral models (Q in Q), or real-world models (Q in P). Risk-neutral simulations are, for example, used to calculate the Credit Value Adjustment (CVA) (see, e.g., Pykhtin (2012)), which is a traded quantity and should therefore be computed using no-arbitrage arguments. For quantities that are not traded (or hedged), the Q in Q approach appears to be incorrect (see, e.g., Stein (2016)) and the future state of the market should be modelled using the real-world measure (Q in P). Real-world models are calibrated to the observed historical time-series and are typically used to compute non-traded quantities such as Value-at-Risk (VaR).

The risk-neutral measure at *t* = 0 is connected to the observed implied volatility surface and is therefore well-defined. However, the definition of the risk-neutral measure at future time *t* = 1 is debatable. Despite some relevant research on predicting the implied volatility surfaces (see, e.g., Cont et al. (2002), Mixon (2002) and Audrino and Colangelo (2010)), it is common practice to use option pricing models that are only calibrated at time *t* = 0, thereby assuming that the risk-neutral measure is independent of the state of the market (see, for example, Bauer et al. (2010) and Devineau and Loisel (2009)). This is however not in line with historical observations, where we see that the implied volatility surface does depend on the state of the market. Another drawback of this approach is that the resulting SCR is *pro-cyclical*, i.e., the SCR is relatively high when the market is in crisis and relatively low when the market is stable. The undesired effect of pro-cyclicality is that it can aggravate a downturn Bikker and Hu (2015).

In this paper, we investigate the impact of relaxing the assumption that the risk-neutral measure is considered to be independent of the state of the market and develop the so-called *VIX Heston model*, which depends on the current and also on simulated implied volatilities. This approach, which we have named here the PQ *approach*, takes into account the Q measure information at time *t* = 0 and simulates risk-neutral model parameters (thus, future implied volatility surfaces are obtained by means of simulation) based on historically observed relations with some relevant market variables such as the VIX index.

As is well-known, the VIX index is a volatility measure for the S&P-500 index, which is calculated by the Chicago Board Options Exchange (CBOE) (see CBOE (2015)), and it is therefore directly linked to the implied volatility surface. Consequently, extracting information from the VIX index is frequently studied (see, e.g., Duan and Yeh (2012)) and our approach in this paper is based on the methodology presented in Singor et al. (2017), where the development of the Heston model parameters for the S&P-500 index options and the VIX index have been analysed.

The contribution of our present research is two-fold. First, we discuss the justification of using a risk-neutral model with time-dependent parameters. By means of a hedge test, we show that hedging strategies that take into account the changes in the implied volatility surface significantly outperform those strategies that do not, both in simulation and with empirical tests. This leads to the conclusion that the time-dependent risk-neutral measure can be used for the evaluation of future embedded option prices. Secondly, we show the impact of our new approach. For that, we use real data from 2007 to 2016 and compute the SCR on a monthly basis with a constant Q measure and also with the VIX Heston model where this assumption is relaxed. We conclude that the VIX Heston model predicts out-of-sample implied volatility surfaces accurately and computes more conservative and stable SCRs. The impact of using the new approach on the SCR depends on the initial state of the market and may vary from −46% to +52% in our experiments. Moreover, we see that the SCR that is computed with the VIX Heston model is significantly less pro-cyclical, for example, it is lower in the wake of the 2008 credit crisis, as it incorporates the likely normalisation.

The outline of this paper is as follows. In Section 2, we give the definition of the SCR. In Section 3, we explain the dynamic VIX Heston model. In Section 4, we present the hedge tests with the corresponding results, followed by Section 5 where we present the numerical VIX Heston results and the impact of the using PQ dynamics on the SCR. Section 6 concludes.
