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

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%.

On the calibration of stochastic volatility models to estimate the real-world measure used in option pricing

“…The real-world measure is often neglected in favour of the risk-neutral measure due to the pricing of contingent claims. However, the real-world measure is extremely useful and important in risk-management and asset/liability applications (see, e.g., [18]); simulationbased analysis of trading and investment strategies (see, [16]); analysis of so-called xVA (counterparty credit, margin, and capital costs); investment based pricing and evaluation of asset price behaviour; and product development, to name a few.…”

Term Premia in Norwegian Interest Rate Swaps

“…This is due to the ACM assumption of serially uncorrelated excess return pricing errors, as opposed to serially uncorrelated yield pricing errors (see Section 2.4 of Adrian et al (2013)). 10 See Van Dijk et al (2018) for a thorough discussion of joint modeling of risk factors in the physical (P) and risk-neutral (Q) measures.…”

Editorial Statement for Mathematical Finance

“…Van Dijk et al (2018) propose improved regression models to estimate calibrated parameters (including the market variables in a real-world simulation), predict out-of-sample implied volatility surfaces, and evaluate the impact on the solvency capital requirement for different points in time. Korkmaz et al (2018) introduce and study a new three-parameter Pareto distribution.…”