*Article* **Between** P **and** Q**: The** P Q **Measure for Pricing in Asset Liability Management**

**Marcel T. P. van Dijk 1,2,\*, Cornelis S. L. de Graaf 1 and Cornelis W. Oosterlee 2,3**


Received: 17 September 2018; Accepted: 21 October 2018; Published: 24 October 2018

**Abstract:** Insurance companies issue guarantees that need to be valued according to the market expectations. By calibrating option pricing models to the available implied volatility surfaces, one deals with the so-called risk-neutral measure Q, which can be used to generate market consistent values for these guarantees. For asset liability management, insurers also need future values of these guarantees. Next to that, new regulations require insurance companies to value their positions on a one-year horizon. As the option prices at *t* = 1 are unknown, it is common practice to assume that the parameters of these option pricing models are constant, i.e., the calibrated parameters from time *t* = 0 are also used to value the guarantees at *t* = 1. However, it is well-known that the parameters are not constant and may depend on the state of the market which evolves under the real-world measure P. In this paper, we propose improved regression models that, given a set of market variables such as the VIX index and risk-free interest rates, estimate the calibrated parameters. When the market variables are included in a real-world simulation, one is able to assess the calibrated parameters (and consequently the implied volatility surface) in line with the simulated state of the market. By performing a regression, we are able to *predict* out-of-sample implied volatility surfaces accurately. Moreover, the impact on the *Solvency Capital Requirement* has been evaluated for different points in time. The impact depends on the initial state of the market and may vary between −46% and +52%.

**Keywords:** insurance; Solvency II; risk-neutral models
