Alexandros A. Zimbidis

Modeling Earthquake Risk via Extreme Value Theory and Pricing the Respective Catastrophe Bonds

The aim of the paper is twofold. Firstly, to analyze the historical data of the earthquakes in the boarder area of Greece and then to produce a reliable model for the risk dynamics of the magnitude of the earthquakes, using advanced techniques from the Extreme Value Theory. Secondly, to discuss briefly the relevant theory of incomplete markets and price earthquake catastrophe bonds, combining the model found for the earthquake risk and an appropriate model for the interest rate dynamics in an incomplete market framework. The paper ends by providing some numerical results using Monte Carlo simulation techniques and stochastic iterative equations.

Linear Generalized Stochastic Systems for Insurance Portfolios

We consider a typical portfolio of different insurance products and investigate the pricing process using the framework of a linear time invariant generalized stochastic discrete-time model. Moreover, we assume that, due to regulatory constraints, the resulting system is (regular) descriptor and calculate the solution using the tools of matrix pencil theory. Finally, we present a numerical application for two different portfolios.

Stochastic Control System for Mortality Benefits

Abstract This article investigates a stochastic control model for a pension fund which provides a variable death benefit to its members during the post-retirement period. The main framework model is described by two correlated fractional Brownian motions which correspond to investment and mortality risks, accordingly. Using the recent advanced results of stochastic control theory for fractional Brownian motion (fBm), we obtain the optimal Markovian control for the level of the death benefit. Finally, using a typical numerical example, we examine the effect of the Hurst exponent with respect to the different management decisions of the controlled variable.

Optimal Control for Non-Homogeneous Linear Systems Driven by Fractional Noises

The article considers the problem of optimal control of a non homogeneous linear system driven by fractional noises. Due to the high complexity of the specific problem, we use a scalar valued state variable and also a quadratic objective criterion. The entire analysis could be generalized within a multi dimensional framework. The investigation is restricted within the class of the Markovian controls. Actually, we calculate a sub optimal control for the original problem.

Optimal Premium Pricing for a Heterogeneous Portfolio of Insurance Risks

The paper revisits the classical problem of premium rating within a heterogeneous portfolio of insurance risks using a continuous stochastic control framework. The portfolio is divided into several classes where each class interacts with the others. The risks are modelled dynamically by the means of a Brownian motion. This dynamic approach is also transferred to the design of the premium process. The premium is not constant but equals the drift of the Brownian motion plus a controlled percentage of the respective volatility. The optimal controller for the premium is obtained using advanced optimization techniques, and it is finally shown that the respective pricing strategy follows a more balanced development compared with the traditional premium approaches.

Insurance pricing using H∞-control

Abstract The paper considers a typical insurance system “suffering” from the three standard “curses”: (a) the stochastic nature of claims, (b) the inherent delays in claims settlement and reserving process and (c) the uncertainty concept that endows many of its parameters and especially the investment process. We construct a general multidimensional model for pricing simultaneously one, two or more different insurance products. The responsible decision maker uses the incomplete information of claims and aims to balance the system by the means of a feedback mechanism. The robust stabilization controller of the system is obtained by the means of H∞-control using typical linear matrix inequalities. Finally, a numerical application is fully investigated providing further insight into the practical problem of pricing assuming the simplest case of a portfolio with a single product.

Optimal Management of a Variable Annuity Invested in a Black–Scholes Market Driven by a Multidimensional Fractional Brownian Motion

This article considers a defined-contribution pension scheme. It focuses in the post-retirement period and investigates the problem of controlling the level of payment of a variable annuity. The general version of the model is solved assuming a vector control variable differentiating the payments for each pensioner according to his age and an enhanced version for the market behavior, modeled via a multidimensional correlated fractional Brownian motion. Then, a reduced version of the basic model is also examined assuming an identical payment rate for all pensioners and a modified version of the typical Black-Scholes model driven by a standard fractional Brownian motion. Finally, a numerical application is developed for investigating the different investment strategies and also exploring the impact of the Hurst exponent in the final formula.

A Multi-Agent System for the Pay-As-You-GO (PAYGO) Social Security Scheme

Multi-Agent Systems (MAS) are suitable for dealing with applications where the environment is both dynamic and very complex, with several type of competitors to get involved. Thus, in this paper, we import MAS conceptualization into the well-known (quasi) Pay-As-You-GO social security scheme; practically useful for many Western Economies. First, we start our analysis with the individual agent and subsequently we move towards a system containing many (cognitive) social agents that considers relations, coordinations and interactions.

Premium and reinsurance control of an ordinary insurance system with liabilities driven by a fractional Brownian motion

This paper investigates the problem of premium and reinsurance control of an ordinary insurance system when liabilities are driven by a fractional Brownian motion. The reserve equation is considered using two alternative routes: the first with no reinsurance option, and the second with some controllable proportional reinsurance coverage. Recent results from the theory of fractional linear-quadratic control (fractional calculus) are discussed, partially extended and utilized to derive compact analytical formulae for the optimal functionals of the safety loading (consequently for the respective premium rate), and the volume of the retained risk (or equivalently, for the proportion of the reinsurance coverage).