Implied fractional hazard rates and default risk distributions

Charles S. Tapiero; Pierre Vallois

doi:10.1186/s41546-017-0015-6

Article Contents

2017, Volume 2: 2. Doi: 10.1186/s41546-017-0015-6

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Implied fractional hazard rates and default risk distributions

Charles S. Tapiero^1, , and
Pierre Vallois²

1 Topfer Chair Distinguished Professor of Financial Engineering and Technology Management, Department of Finance and Risk Engineering, New York University Tandon School of Engineering, 6 Metro Tech, 22201 Brooklyn, New York, USA;
2 Universite de Lorraine, Institut de Mathematiques Elie Cartan de Lorraine, INRIA. BIGS, CNRS UMR 7502 BP 239, F-54506, Vandoeuvre-Les-Nancy, France

Charles S. Tapiero, cst262@nyu.edu

Received: September 26, 2016

Revised: January 16, 2017

Published: September 2020

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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.

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