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January  2017, 2: 2 doi: 10.1186/s41546-017-0015-6

## Implied fractional hazard rates and default risk distributions

 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

Received  September 27, 2016 Revised  January 17, 2017 Published  June 2017

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.
Citation: Charles S. Tapiero, Pierre Vallois. Implied fractional hazard rates and default risk distributions. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 2-. doi: 10.1186/s41546-017-0015-6
##### References:
 [1] Baillie, RT:Long memory processes and fractional integration in econometrics. J. Econometrics 73, 5-59(1996) [2] Baleanu, D, Diethhlem, K, Scallas, E, Trujillo, J:Fractional Calculus:Models and Numerical Methods, CNC Series on Complexity, Nonlinearity and Chaos, vol. 3. World Scientific, Singapore (2010) [3] Barlow, R, Proschan, F:Mathematical Theory of Reliability. Wiley, New York (1965) [4] Beran, J:Statistical methods for data with long-range dependence, Statistical. Science 7(1992), 404-427(1992) [5] Berry, AC:The Accuracy of the Gaussian Approximation to the Sum of Independent Variates. Trans. Am.Math. Soc 49(1), 122-136 (1941) [6] Caputo, M:Linear model of dissipation whose Q is almost frequency dependent II. Geophys. Res 13, 529-539 (1967) [7] Cox, JJ, Tait, N:Reliability, Safety and Risk Management. Butterworth-Heinemann (1991) [8] Dacorogna, M, Muller, UA, Nagler, RJ, Olsen, RB, Pictet, OV:A geographical model for the daily and weekly seasonal volatility in the FX market. J. Int. Money Finance 12(4), 413-438 (1993) [9] Grunwald, AK:Fiber "begrenzte" Derivationen und deren Anwendung. Z. Math. Phys. 12(8), 441-480(1867) [10] Hilfer, R:Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) [11] Jumarie, G:Fractional differential calculus for non-differentiable functions. Lambert Academic Publishing, Saarbrucken (2013) [12] Korolev, VY, Shevtsova, IG:On the upper bound for the absolute constant in the Berry-Esseen inequality.Theory Probab. Appl 54(4), 638-658 (2010a) [13] Korolev, V, Shevtsova, I:An improvement of the Berry-Esseen inequality with applications to Poisson and mixed Poisson random sums. Scand. Actuarial J 1, 25 (2010b) [14] Laskin, N:Fractional Poisson Process. Commun. Nonlinear Sci. Numerical Simul 8(3-4), 201-213 (2003) [15] Liouville, J:Sur le calcul des differentielles à indices quelconques. J. Ecole Polytechnique 13, 71 (1832) [16] Letnikov, AV:Theory of differentiation of fractional order. Math Sb 3, 1-7 (1868) [17] Mandelbrot, BB:The variation of certain speculative prices. J. Bus 36, 394-419 (1963) [18] Mandelbrot, BB, Van Ness, JW:Fractional Brownian motions, fractional noises and applications. SIAM Rev 10, 422-437 (1968) [19] Meltzer, R, Klafter, Y:The Restaurant at the end of the random walk:recent developments in the description of anomalous transport by fractional dynamic. J. Phys. A. Math Gen 37, R161-R208(2004) [20] Miller, KS, Ross, B:An Introduction to the Fractional Calculus and Fractional Differential Equations.Wiley, New York (1993) [21] Muller, UA, Dacorogna, MM, Olsen, RB, Pictet, OV, Schwarz, M, Morgenegg, C:Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis.J. Banking Finance 14, 1189-1208 (1990) [22] Muller, UA, Dacorogna, MM, Dave, RD, Pictet, OV, Olsen, RB, Ward, JR:Fractals and Intinsic Time-A Challenge to Econometricians. Presented in an opening lecture of the XXXIXth International Conference of the Applied Econometrics Association (AEA) (1993) [23] Pillai, PRN:On Mittag-Leffler functions and related distributions. Ann. Inst. Statist. Math 42, 157-161(1990) [24] Podlubny, I:Fractional Differential Equations:An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego (1999) [25] Shevtsova, IG:Sharpening of the upper bound of the absolute constant in the Berry-Esseen inequality.Theory Probab. Appl 51(3), 549-553 (2007) [26] Shevtsova, IG:On the absolute constant in the Berry-Esseen inequality. Collection Papers Young Scientists Fac. Comput. Math. Cybernet. 5, 101-110 (2008) [27] Tapiero, CS:Reliability Design and RVaR. International Journal of Reliability, Quality and Safety Engineering (IJRQSE) 12(4), 347-353 (2005) [28] Tapiero, CS, Vallois, P:Fractional Randomness, Physica A, Stat. Mech. Appl 462, 1161-1177 (2016) [29] Tapiero, CS, Tapiero, O, Jumarie, G:The price of granularity And fractional finance. Risk Decis. Anal 6(1), 7-21 (2016)

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##### References:
 [1] Baillie, RT:Long memory processes and fractional integration in econometrics. J. Econometrics 73, 5-59(1996) [2] Baleanu, D, Diethhlem, K, Scallas, E, Trujillo, J:Fractional Calculus:Models and Numerical Methods, CNC Series on Complexity, Nonlinearity and Chaos, vol. 3. World Scientific, Singapore (2010) [3] Barlow, R, Proschan, F:Mathematical Theory of Reliability. Wiley, New York (1965) [4] Beran, J:Statistical methods for data with long-range dependence, Statistical. Science 7(1992), 404-427(1992) [5] Berry, AC:The Accuracy of the Gaussian Approximation to the Sum of Independent Variates. Trans. Am.Math. Soc 49(1), 122-136 (1941) [6] Caputo, M:Linear model of dissipation whose Q is almost frequency dependent II. Geophys. Res 13, 529-539 (1967) [7] Cox, JJ, Tait, N:Reliability, Safety and Risk Management. Butterworth-Heinemann (1991) [8] Dacorogna, M, Muller, UA, Nagler, RJ, Olsen, RB, Pictet, OV:A geographical model for the daily and weekly seasonal volatility in the FX market. J. Int. Money Finance 12(4), 413-438 (1993) [9] Grunwald, AK:Fiber "begrenzte" Derivationen und deren Anwendung. Z. Math. Phys. 12(8), 441-480(1867) [10] Hilfer, R:Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) [11] Jumarie, G:Fractional differential calculus for non-differentiable functions. Lambert Academic Publishing, Saarbrucken (2013) [12] Korolev, VY, Shevtsova, IG:On the upper bound for the absolute constant in the Berry-Esseen inequality.Theory Probab. Appl 54(4), 638-658 (2010a) [13] Korolev, V, Shevtsova, I:An improvement of the Berry-Esseen inequality with applications to Poisson and mixed Poisson random sums. Scand. Actuarial J 1, 25 (2010b) [14] Laskin, N:Fractional Poisson Process. Commun. Nonlinear Sci. Numerical Simul 8(3-4), 201-213 (2003) [15] Liouville, J:Sur le calcul des differentielles à indices quelconques. J. Ecole Polytechnique 13, 71 (1832) [16] Letnikov, AV:Theory of differentiation of fractional order. Math Sb 3, 1-7 (1868) [17] Mandelbrot, BB:The variation of certain speculative prices. J. Bus 36, 394-419 (1963) [18] Mandelbrot, BB, Van Ness, JW:Fractional Brownian motions, fractional noises and applications. SIAM Rev 10, 422-437 (1968) [19] Meltzer, R, Klafter, Y:The Restaurant at the end of the random walk:recent developments in the description of anomalous transport by fractional dynamic. J. Phys. A. Math Gen 37, R161-R208(2004) [20] Miller, KS, Ross, B:An Introduction to the Fractional Calculus and Fractional Differential Equations.Wiley, New York (1993) [21] Muller, UA, Dacorogna, MM, Olsen, RB, Pictet, OV, Schwarz, M, Morgenegg, C:Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis.J. Banking Finance 14, 1189-1208 (1990) [22] Muller, UA, Dacorogna, MM, Dave, RD, Pictet, OV, Olsen, RB, Ward, JR:Fractals and Intinsic Time-A Challenge to Econometricians. Presented in an opening lecture of the XXXIXth International Conference of the Applied Econometrics Association (AEA) (1993) [23] Pillai, PRN:On Mittag-Leffler functions and related distributions. Ann. Inst. Statist. Math 42, 157-161(1990) [24] Podlubny, I:Fractional Differential Equations:An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego (1999) [25] Shevtsova, IG:Sharpening of the upper bound of the absolute constant in the Berry-Esseen inequality.Theory Probab. Appl 51(3), 549-553 (2007) [26] Shevtsova, IG:On the absolute constant in the Berry-Esseen inequality. Collection Papers Young Scientists Fac. Comput. Math. Cybernet. 5, 101-110 (2008) [27] Tapiero, CS:Reliability Design and RVaR. International Journal of Reliability, Quality and Safety Engineering (IJRQSE) 12(4), 347-353 (2005) [28] Tapiero, CS, Vallois, P:Fractional Randomness, Physica A, Stat. Mech. Appl 462, 1161-1177 (2016) [29] Tapiero, CS, Tapiero, O, Jumarie, G:The price of granularity And fractional finance. Risk Decis. Anal 6(1), 7-21 (2016)
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