Optimal Exercise Strategies for Operational Risk Insurance via Multiple Stopping Times

Abstract

In this paper we demonstrate how to develop analytic closed form solutions to optimal multiple stopping time problems arising in the setting in which the value function acts on a compound process that is modified by the actions taken at the stopping times. This class of problem is particularly relevant in insurance and risk management settings and we demonstrate this on an important application domain based on insurance strategies in Operational Risk management for financial institutions. In this area of risk management the most prevalent class of loss process models is the Loss Distribution Approach (LDA) framework which involves modelling annual losses via a compound process. Given an LDA model framework, we consider Operational Risk insurance products that mitigate the risk for such loss processes and may reduce capital requirements. In particular, we consider insurance products that grant the policy holder the right to insure k of its annual Operational losses in a horizon of T years. We consider two insurance product structures and two general model settings, the first are families of relevant LDA loss models that we can obtain closed form optimal stopping rules for under each generic insurance mitigation structure and then secondly classes of LDA models for which we can develop closed form approximations of the optimal stopping rules. In particular, for losses following a compound Poisson process with jump size given by an Inverse-Gaussian distribution and two generic types of insurance mitigation, we are able to derive analytic expressions for the loss process modified by the insurance application, as well as closed form solutions for the optimal multiple stopping rules in discrete time (annually). When the combination of insurance mitigation and jump size distribution does not lead to tractable stopping rules we develop a principled class of closed form approximations to the optimal decision rule. These approximations are

On Asymptotics of Optimal Stopping Times

“…Related analyses may be found in [8,9]. Optimal stopping also features in financial and economic applications, such as in the pricing of American options [10], games [11], operational risk insurance [12], dynamic pricing [13] and more complicated buying-selling problems [14][15][16].…”

An Optimal Decision Rule for a Multiple Selling Problem with a Variable Rate of Offers

“…Another example is a buying-selling problem with two stops allowed (see [19,21]), when Z m 1 ,m 2 = y m 2 − y m 1 . For other examples of the gain function, see [13,20]. In this paper, the gain function takes the Form (1).…”

“…The multiple selling problem described above, which belongs to a class of sequential decision problems, can be considered within the framework of the optimal stopping theory [2] and its extension, the theory of optimal multiple stopping rules [3][4][5]. Various applications of optimal stopping problems can be found in many fiellds including job search [6,7], engineering [8,9], games [10], resource allocation [11], and insurance [12,13]. Different multiple selling problems have been considered in the literature.…”

“…In [16], the author considers a problem of selling several identical commodities in a finite time interval with potential customers arrive according to a Poisson process. In [13], the authors apply optimal multiple stopping rules to analyse a class of insurance products where the policy holder has the right to insure several of its annual operational losses in a finite time horizon. The general formulation of a discrete time optimal stopping problem under the criterion of a fractional reward is given in [17].…”