Analytic loss distributional approach models for operational risk from the α-stable doubly stochastic compound processes and implications for capital allocation

Under the Basel II standards, the Operational Risk (OpRisk) advanced measurement approach is not prescriptive regarding the class of statistical model utilized to undertake capital estimation. It has however become well accepted to utilize a Loss Distributional Approach (LDA) paradigm to model the individual OpRisk loss processes corresponding to the Basel II Business line/event type. In this paper we derive a novel class of doubly stochastic α-stable family LDA models. These models provide the ability to capture the heavy tailed loss processes typical of OpRisk, whilst also providing analytic expressions for the compound processes annual loss density and distributions, as well as the aggregated compound processes’ annual loss models. In particular we develop models of the annual loss processes in two scenarios. The first scenario considers the loss processes with a stochastic intensity parameter, resulting in inhomogeneous compound Poisson processes annually. The resulting arrival processes of losses under such a model will have independent counts over increments within the year. The second scenario considers discretization of the annual loss processes into monthly increments with dependent time increments as captured by a Binomial processes with a stochastic probability of success changing annually. Each of these models will be coupled under an LDA framework with heavy-tailed severity models comprised of α-stable severities for the loss amounts per loss event. In this paper we will derive analytic results for the annual loss distribution density and distribution under each of these models and study their properties.