Abstract
Default probability distributions are often defined in terms of their conditional default probability distribution, or their hazard rate. By their definition, they imply a unique probability density function. The applications of default probability distributions are varied, including the risk premium model used to price default bonds, reliability measurement models, insurance, etc. Fractional probability density functions (FPD), however, are not in general conventional probability density functions (Tapiero and Vallois, Physica A,. Stat. Mech. Appl. 462:1161–1177, 2016). As a result, a fractional FPD does not define a fractional hazard rate. However, a fractional hazard rate implies a unique and conventional FPD. For example, an exponential distribution fractional hazard rate implies a Weibull probability density function while, a fractional exponential probability distribution is not a conventional distribution and therefore does not define a fractional hazard rate. The purpose of this paper consists of defining fractional hazard rates implied fractional distributions and to highlight their usefulness to granular default risk distributions. Applications of such an approach are varied. For example, pricing default bonds, pricing complex insurance contracts, as well as complex network risks of various granularity, that have well defined and quantitative definitions of their hazard rates.
Highlights
Default probability distributions are often defined in terms of their conditional default probability distribution, or their hazard rate
Some fractal properties have been demonstrated by a study of high-density foreign exchange (FX) in data that the mean size of the absolute values of price changes followed a “fractal” scaling law - a power of the observation time-interval size (Muller et al, 1990, 1993)
In an autocorrelation study with high-density data (Dacorogna et al 1993), the absolute values of price changes behave like a “fractional noise” (Mandelbrot and Van Ness 1968) rather than the absolute price changes expected from a GARCH process: the memory of the volatility declines hyperbolically with time
Summary
Default probability distributions are often defined in terms of their conditional default probability distribution, or their hazard rate. A hazard rate implies a unique conventional probability density and its cumulative distribution. A fractional hazard rate, defined by the application of a fractional operator, implies as well a conventional probability density and its cumulative distribution. The purposes of this paper are to define fractional hazard rates and their properties as well as their implied distributions which we apply to several examples associated with insurance and risk models. In this paper we suggest a fractional hazard rate to define conventional fractional distributions and suggest that the fractional hazard rate is a reasonable and consistent approach to define fractional default distributions and useful in the definition of risk models that define the increased (or decreased) risks that occur due to model granularity Applications of this approach are considered as well, including default bonds reliability as well as insurance. We shall consider a number of examples and their applications below
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