### Delong, Ł., Dhaene, J., Barigou, K., 2019, Fair valuation of insurance liability cash-flow streams in continuous time: Theory

Published Paper,
05-03-2019

We investigate fair (market-consistent and actuarial) valuation of insurance liability cash-flow streams in continuous time. We first consider one-period hedge-based valuations, where in the first step, an optimal dynamic hedge for the liability is set up, based on the assets traded in the market and a quadratic hedging objective, while in the second step, the remaining part of the claim is valuated
via an actuarial valuation. Then, we extend this approach to a multi-period setting by backward iterations for a given discrete-time step $h$, and consider the continuous-time limit for $h\to 0$. We formally derive a partial differential equation for the valuation operator which satisfies the continuous-time limit of the multi-period, discrete-time iterations and prove that this valuation operator is actuarial and market-consistent. We show that our continuous-time fair valuation operator has a natural decomposition into the best estimate of the liability and a risk margin. The dynamic hedging strategy associated with the continuous-time fair valuation operator is also established. Finally, the valuation operator and the hedging strategy allow us to study the dynamics of the net asset value of the insurer.

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### Delong, Ł., 2019, Asymptotic optimality of a first-order approximate strategy for an exponential utility maximization problem with a small coefficient of wealth-dependent risk aversion

Working Paper,
02-01-2019

In Delong (2019) we investigate an exponential utility maximization problem for an insurer who faces a stream of non-hedgeable claims. We assume that the insurer's risk aversion coefficient consists of a constant risk aversion and a small amount of wealth-dependent risk aversion. We apply perturbation theory and expand the equilibrium value function of the optimization problem on the parameter $\epsilon$ controlling the degree of the insurer's risk aversion depending on wealth. We derive a candidate for the first-order approximation to the equilibrium investment strategy. In this paper we formally show that the zeroth-order investment strategy $\pi_0^*$ postulated by Delong (2019) performs better than any strategy $\pi_0$ when we compare the asymptotic expansions of the objective functions up to order $\mathcal{O}(1)$ as $\epsilon\rightarrow 0$, and the first-order investment strategy $\pi_0^*+\pi_1^*\epsilon$ postulated by Delong (2019) is the equilibrium strategy in the class of strategies $\pi^*_0+\pi_1\epsilon$ when we compare the asymptotic expansions of the objective functions up to order $\mathcal{O}(\epsilon^2)$ as $\epsilon\to 0$, where $\epsilon$ denotes the parameter controlling the degree of the insurer's risk aversion depending on wealth.

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### Delong, Ł., 2013, Backward Stochastic Differential Equations with Jumps and their Actuarial and Financial Applications

Book,
10-04-2013

Backward Stochastic Differential Equations with jumps can be used to solve problems in both finance and insurance.

This book will make BSDEs more accessible to those who are interested in applying these equations to actuarial and financial problems. It will be beneficial to students and researchers in mathematical finance, risk measures, portfolio optimization as well as actuarial practitioners.

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