Default prediction with the Merton-type structural model based on the NIG Lévy process. (English) Zbl 1354.91159

Summary: C. R. Merton’s model [“On the pricing of corporate debt: the risk structure of interest rates”, J. Finance 29, No. 2, 449–470 (1974; doi:10.1111/j.1540-6261.1974.tb03058.x)] has long been a standard for estimating company’s probability of default (PD) for listed companies. The major advantage of Merton’s model is the use of current market prices to determine the probability of default. The logic behind the model is simple; the market prices best reflect all the relevant information (being forward looking estimates of company’s prospect) and should be (and are) superior to the balance sheet disclosures, which at best are ex post realisations of company’s performance. It is thus a pity that the benefits (strengths) of Merton’s model are hindered by a significant shortcoming of the model namely the assumption of normally distributed returns.
As numerous authors point out [O. E. Barndorff-Nielsen, Scand. J. Stat. 24, No. 1, 1–13 (1997; Zbl 0934.62109); Finance Stoch. 2, No. 1, 41–68 (1998; Zbl 0894.90011); K. Prause, The generalized hyperbolic model: estimation, financial derivatives, and risk measures. Freiburg i. Br.: Univ. Freiburg i. Br., Mathematische Fakultät (1999; Zbl 0944.91026); E. Eberlein, in: Lévy processes. Theory and applications. Boston: Birkhäuser. 319–336 (2001; Zbl 0982.60045); C. Brambilla et al., Far East J. Math. Sci. (FJMS) 97, No. 1, 101–119 (2015; Zbl 1409.91259)], stock returns are not normally distributed which significantly limits the use of model in practice. Moreover the estimates of PDs can be biased downwards exposing the banks to the possibility of undercapitalisation and systematic shocks.
It is the purpose of this paper to remedy this situation. Firstly we extend the Merton model by allowing for normal inverse Gaussian (NIG) distributed returns. As several authors point out using the examples of options [W. Schoutens and J. Cariboni, Lévy processes in credit risk. Chichester: John Wiley & Sons (2009; Zbl 1192.91008)], NIG in most cases provides a robust statistical platform for estimating stock returns. We further extend our approach by constructing a robust EM algorithm for estimating PDs within the Merton NIG framework.
We also test the reliability of the NIG improved Merton model against classical Merton’s model for estimating PDs. Applying our results to Ljubljana stock exchange we find that the PD estimates using classical Merton’s model are biased, whereas the estimates from NIG Merton’s model are robust.


91G40 Credit risk
60G51 Processes with independent increments; Lévy processes
60H30 Applications of stochastic analysis (to PDEs, etc.)
62P05 Applications of statistics to actuarial sciences and financial mathematics


Full Text: DOI


[1] Merton, C. R., On the pricing of corporate debt: the risk structure of interest rates, J. Finance, 29, 2, 449-470, (1974)
[2] Altman, E. I., Default recovery rates and LGD in credit risk modelling and practice: an updated review of the literature and empirical evidence, (Jones, S.; Hensher, D. A., Advances in Credit Risk Modelling and Corporate Bankruptcy Prediction, (2008), Cambridge University Press), 175-206
[3] Crosbie, P.; Bohn, J., Modeling default risk, (2003), Moody’s KMV Company
[4] Liu, B.; Kocagil, A. E.; Gupton, G. M., Fitch equity implied rating and probability of default model, (2007), FitchSolutions
[5] Barndorff-Nielsen, O. E., Normal inverse Gaussian distributions and stochastic volatility modelling, Scand. J. Statist., 24, 1, 1-13, (1997) · Zbl 0934.62109
[6] Barndorff-Nielsen, O. E., Processes of normal inverse Gaussian type, Finance Stoch., 2, 1, 41-68, (1997) · Zbl 0894.90011
[7] Prause, K., The generalized hyperbolic model: estimation, financial derivatives, and risk measures, (1999), Albert-Ludwigs-Universität Freiburg, (Doctoral dissertation) · Zbl 0944.91026
[8] Eberlein, E., Application of generalized hyperbolic Lévy motions to finance, (Barndorff-Nielsen, O.; Resnick, S.; Mikosch, T., Lévy Processes, (2001), Birkhäuser Boston), 319-336 · Zbl 0982.60045
[9] Brambilla, C.; Gurny, M.; Ortobelli Loza, S., Structural credit risk models with Lévy processes: the VG and NIG cases, Far East J. Math. Sci., 97, 1, 101-119, (2015) · Zbl 1409.91259
[10] Özkan, F., Lévy processes in credit risk and market models, (2002), Albert-Ludwigs-Universität Freiburg, (Doctoral dissertation) · Zbl 1001.91055
[11] Kalemanova, A.; Schmid, B.; Werner, R., The normal inverse Gaussian distribution for synthetic CDO pricing, J. Deriv., 4, 3, 80-94, (2007)
[12] Schoutens, W.; Carboni, J., Lévy processes in credit risk, (2009), Wiley · Zbl 1192.91008
[13] Duan, J.-C., Maximum likelihood estimation using price data of the derivative contract, Math. Finance, 4, 2, 155-167, (1994) · Zbl 0884.90026
[14] Duan, J.-C., Correction: maximum likelihood estimation using price data of the derivative contract, Math. Finance, 10, 4, 461-462, (2000)
[15] J.-C. Duan, G. Gauthier, J.-G. Simonato, On the Equivalence of the KMV and Maximum Likelihood Methods for Structural Credit Risk Models, Working paper, 2005.
[16] Rice, J. A., Mathematical statistics and data analysis, (2007), Duxbury Press Belmont, CA
[17] Black, F.; Scholes, M., The pricing of option and corporate liabilities, J. Polit. Econ., 81, 3, 637-654, (1973) · Zbl 1092.91524
[18] Regulation (EU) No 575/2013 of the European Parliament and of the Council of 26 June 2013 on prudential requirements for credit institutions and investment firms, 2013.
[19] Abramowitz, M.; Stegun, I. A., Handbook of mathematical functions with formulas, graphs, and mathematical tables, (1964), Dover New York · Zbl 0171.38503
[20] Rydberg, T. H., The normal inverse Gaussian Lévy process: simulation and approximation, Comm. Statist. Stochastic Models, 13, 4, 887-910, (1997) · Zbl 0899.60036
[21] Raible, S., Lévy processes in finance: theory, numerics, and empirical facts, (2000), Albert-Ludwigs-Universität Freiburg, (Doctoral dissertation) · Zbl 0966.60044
[22] Karlis, D., An EM type algorithm for maximum likelihood estimation of the normal-inverse Gaussian distribution, Statist. Probab. Lett., 57, 1, 43-52, (2002) · Zbl 0996.62015
[23] K. Aas, I.H. Haff, NIG and Skew Student’s t: Two special cases of the Generalised Hyperbolic distribution, 2005.
[24] Paolella, M. S., Intermediate probability: A computational approach, (2007), John Wiley & Sons, Ltd. Chichester, England · Zbl 1149.60002
[25] Gerber, H. U.; Shiu, E. S.W., Option pricing by esscher transforms, Trans. Soc. Actuar., 46, 99-191, (1994)
[26] Eberlein, E.; Keller, U., Hyperbolic distributions in finance, Bernoulli, 1, 3, 281-299, (1995) · Zbl 0836.62107
[27] Eberlein, E.; Keller, U.; Prause, K., New insights into smile, mispricing, and value at risk: the hyperbolic model, J. Bus., 71, 3, 371-405, (1998)
[28] Albrecher, H.; Predota, M., On Asian option pricing for NIG Lévy processes, J. Comput. Appl. Math., 172, 1, 153-168, (2004) · Zbl 1107.91042
[29] Rasmus, S.; Asmussen, S.; Wiktorsson, M., Pricing of some exotic options with NIG-Lévy input, (Bubak, M.; van Albada, G.; Sloot, P.; Dongarra, J., Computational Science — ICCS 2004, Lecture Notes in Computer Science, vol. 3039, (2004), Springer Berlin, Heidelberg), 795-802 · Zbl 1102.91330
[30] Hubalek, F.; Sgarra, C., Esscher transforms and the minimal entropy martingale measure for exponential Lévy models, Quant. Finance, 6, 2, 125-145, (2006) · Zbl 1099.60033
[31] Jovan, M., The Merton structural model and IRB compliance, Metodološki zvezki, 7, 1, 39-57, (2010)
[32] R Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2016. URL https://www.R-project.org/.
[33] Ahčan, A., Testing the sustainability of growth of the LJSEX in the January 2000 to may 2010 period, Organizacija (Kranj), 44, 2, 47-58, (2011)
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