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Under the Basel II standards, the Operational Risk (OpRisk) advancedmeasurement approach is not prescriptive regarding the class of statisticalmodel utilised to undertake capital estimation. It has however become wellaccepted to utlise a Loss Distributional Approach (LDA) paradigm to model theindividual OpRisk loss process corresponding to the Basel II Businessline/event type. In this paper we derive a novel class of doubly stochasticalpha-stable family LDA models. These models provide the ability to capture theheavy tailed loss process typical of OpRisk whilst also providing analyticexpressions for the compound process annual loss density and distributions aswell as the aggregated compound process annual loss models. In particular wedevelop models of the annual loss process in two scenarios. The first scenarioconsiders the loss process with a stochastic intensity parameter, resulting inan inhomogeneous compound Poisson processes annually. The resulting arrivalprocess of losses under such a model will have independent counts overincrements within the year. The second scenario considers discretization of theannual loss process into monthly increments with dependent time increments ascaptured by a Binomial process with a stochastic probability of successchanging annually. Each of these models will be coupled under an LDA frameworkwith heavy-tailed severity models comprised of $\alpha$-stable severities forthe loss amounts per loss event. In this paper we will derive analytic resultsfor the annual loss distribution density and distribution under each of thesemodels and study their properties.

Analytic Loss Distributional Approach Model for Operational Risk from  the alpha-Stable Doubly Stochastic Compound Processes and Implications for  Capital Allocation