基于α-稳定双随机复合过程的操作风险损失分布模型及其对资本配置的启示
Analytic Loss Distributional Approach Model for Operational Risk from the alpha-Stable Doubly Stochastic Compound Processes and Implications for Capital Allocation
摘要：
根据《巴塞尔协议二》标准，操作风险（OpRisk）高级计量方法对用于进行资本估计的统计模型类别并无规定。然而，使用损失分布法（LDA）范例来模拟与巴塞尔II业务线/事件类型对应的单个OpRisk损失过程已被广泛接受。在本文中，我们得到了一类新的双随机α-稳定的家庭LDA模型。这些模型提供了捕捉OpRisk典型的重尾损失过程的能力，同时还提供了复合过程年损失密度和分布以及聚合复合过程年损失模型的分析表达式。特别是，我们开发模型的年度损失过程中的两种情况。第一种情况考虑损失过程的随机强度参数，导致每年的非齐次复合泊松过程。在这种模型下，损失的到达过程将在一年内对增量进行独立计数。第二种情况考虑将年度损失过程离散化为每月增量，其中时间增量由二项式过程捕获，成功概率每年变化。这些模型中的每一个都将在LDA框架下与重尾严重性模型耦合，该模型由每个损失事件的损失金额的$α$稳定严重性组成。在本文中，我们将推导出这些模型下的年损失分布密度和分布的解析结果，并研究它们的性质。
英文摘要：
Under the Basel II standards, the Operational Risk (OpRisk) advanced measurement approach is not prescriptive regarding the class of statistical model utilised to undertake capital estimation. It has however become well accepted to utlise a Loss Distributional Approach (LDA) paradigm to model the individual OpRisk loss process corresponding to the Basel II Business line/event type. In this paper we derive a novel class of doubly stochastic alpha-stable family LDA models. These models provide the ability to capture the heavy tailed loss process typical of OpRisk whilst also providing analytic expressions for the compound process annual loss density and distributions as well as the aggregated compound process annual loss models. In particular we develop models of the annual loss process in two scenarios. The first scenario considers the loss process with a stochastic intensity parameter, resulting in an inhomogeneous compound Poisson processes annually. The resulting arrival process of losses under such a model will have independent counts over increments within the year. The second scenario considers discretization of the annual loss process into monthly increments with dependent time increments as captured by a Binomial process 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.
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