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We introduce the class of actuarial-consistent valuation methods for insurance liabilities which depend on both financial and actuarial risks, which imposes that all actuarial risks are priced via standard actuarial principles. We propose to extend standard actuarial principles by a new actuarialconsistent procedure, which we call "two-step actuarial valuations". In the coherent setting, we show that actuarial-consistent valuations are equivalent to two-step actuarial valuations. We also discuss the connection with "two-step market-consistent valuations" from Pelsser and Stadje (2014). In particular, we discuss how the dependence structure between actuarial and financial risks impacts both actuarial-consistent and market-consistent valuations.

International audience; This book covers recent developments in the interdisciplinary fields of actuarial science, quantitative finance, risk- and asset management. The authors are leading experts from academia and practice who participated in Innovations in Insurance, Risk- and Asset Management, an international conference held at the Technical University of Munich in 2017. The topics covered include the mathematics of extreme risks, systemic risk, model uncertainty, interest rate and hybrid models, alternative investments, dynamic investment strategies, quantitative risk management, asset liability management, liability driven investments, and behavioral finance. This timely selection of topics is highly relevant for the financial industry and addresses current issues both from an academic as well as from a practitioner's point of view.

In this paper, we introduce two classes of indices which can be used to measure the market perception concerning the degree of dependency that exists between a set of random variables, representing different stock prices at a fi xed future date. The construction of these measures is based on the theory of comonotonicity. Both types of herd behavior indices are model-free and risk-neutral, derived from available option data. Depending on its particular de finition, each index represents a particular aspect of the market sentiment concerning future co-movement of the underlying stock prices. ispartof: FEB Research Report AFI_1597 nrpages: 32 status: published

In this paper we employ the L´evy copula model to determine basket option prices. More precisely, basket option prices are determined by replacing the real basket with an appropriate approximation. For the approximate basket we determine the underlying characteristic function and hence we can derive the related basket option prices by using the Carr-Madan formula. Two approaches are considered. In the first approach, we replace the arithmetic sum by an appropriate geometric sum, whereas the second approach can be considered as a three-moments-matching method. Numerical examples illustrate the accuracy of our approximations; several L´evy models are calibrated to market data and basket option prices are determined. In a last part we show how our newly designed basket option pricing formula can be used to define implied L´evy correlation by matching model and market prices for basket options. Our main finding is that the implied L´evy correlation smile is flatter than its Gaussian counterpart. Furthermore, if (near) at-the-money option prices are used, the corresponding implied Gaussian correlation estimate is a good proxy for the implied L´evy correlation. ispartof: FEB Research Report AFI_1494 pages:1-36 status: published

ispartof: FEB Research Report AFI_1496 pages:1-30 status: published

This paper proposes different diffusion processes to model herd behavior indices such as the Herd Behavior Index (HIX) or the comonotonicity index (CIX). These models arise by combining popular mean-reverting processes with simple algebraic functions mapping the definition domain of the underlying mean-reverting process to the unit interval. The such obtained Itˆo processes preserve, to some extent, the mean-reverting trend of the underlying process while satisfying the fundamental properties of the so-called herd behavior indices. In the numerical study, we calibrate the different model settings to time series data for a period spanning from January 2000 until October 2009 and investigate their ability to predict the future behavior of herd behavior indices. ispartof: FEB Research Report AFI_1490 pages:1-24 status: published

ispartof: FEB Research Report AFI_1381 pages:1-14 status: published

ispartof: FEB Research Report AFI_1383 pages:1-23 status: published

ispartof: FEB Research Report: AFI_1386 pages:1-28 status: published

ispartof: Theory of Probability and Mathematical Statistics vol:88 pages:1-14 status: published

In this paper we show that under appropriate moment conditions, the supermodular ordered random vectors X = (X1;X2; : : : ;Xn) and Y = (Y1; Y2; : : : ; Yn) with equal expected utilities (or distorted expectations) of the sums X1 +X2 +: : :+Xn and Y1 +Y2 +: : :+Yn for an appropriate utility (or distortion) function, must necessarily be equal in distribution, that is X d=Y . The results in this paper can be considered as generalizations of the results of Cheung (2010), who presents necessary conditions related to the distribution of X1 + X2 + : : : + Xn for the random vector X = (X1;X2; : : : ;Xn) to be comonotonic. ispartof: FEB Research Report AFI_1377 pages:1-23 status: published

Distorted expectations can be expressed as mixtures of quantiles. In this note, we show that this statement is essentially true, but that one has to be careful with the correct formulation of it. Furthermore, the proofs of the additivity property for distorted expectations of a comonotonic sum that appear in the literature often do not cover the case of a general distortion function. We present a straightforward proof for the general case, making use of the appropriate expressions for distorted expectations in terms of mixtures of quantiles. ispartof: FEB Research Report AFI_1273 pages:1-11 status: published

ispartof: FBE Research Report AFI_1265 pages:1-56 status: published

In this paper, we propose a new fear index based on (equity) option surfaces of an index and its components. The quanti¯cation of the fear level will be solely based on option price data. The index takes into account market risk via the VIX volatility barometer, liquidity risk via the concept of implied liquidity, and systemic risk and herd-behavior via the concept of comonotonicity. It thus allows us to measure an overall level of fear (excluding credit risk) in the market as well as to identify precisely the individual importance of the distinct risk components (market, liquidity or systemic risk). As a side result we also derive an upperbound for the VIX. ispartof: FBE Research Report AFI_1155 pages:1-17 status: published

We introduce a new and easy to calculate measure for systemic risk in financial markets. This measure is baptized the Herd Behavior Index (HIX). It is model-independent and forward looking, based on observed option data. In order to determine the degree of systemic risk or herd behavior in a financial market one should compare the observed market situation with the extreme (theoretical) situation under which the whole system is driven by a single factor. The Herd Behavior Index (HIX) is defined as the ratio of an option-based estimate of the risk-neutral variance of the market index and an option-based estimate of the corresponding variance of this extreme market situation. Using the theory of comonotonicity, the extreme situation can easily be backed out of the observed option quotes. The HIX can be determined for any market index provided an appropriate series of vanilla options is traded on this index as well as on its components. As an illustration, we determine historical values of the 30-days implied Herd Behavior Index for the Dow Jones Industrial Average, covering the period January 2003 to October 2009. ispartof: FBE Research Report AFI_1157 pages:1-31 status: published