## Abstract

We estimate default measures for US banks using a model capable of handling volatility clustering like those observed during the Global Financial Crisis (GFC). In order to account for the time variation in volatility, we adapted a GARCH option pricing model which extends the seminal structural approach of default by Merton (J Finance 29(2):449, 1974) and calculated “distance to default” indicators that respond to heightened market developments. With its richer volatility dynamics, our results better reflect higher expected default probabilities precipitated by the GFC. The diagnostics show that the model generally outperforms standard models of default and offers relatively good indicators in assessing bank failures.

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## Notes

- 1.
There is overwhelming empirical evidence that GARCH models dominate the benchmark constant volatility Black–Scholes model achieving significant overall improvements in pricing performance. See Lehar et al. (2002), Hsieh and Ritchken (2005), Christoffersen et al. (2013) and Christoffersen et al. (2013).

- 2.
As pointed out by Vassalou and Xing (2004) the theoretical distribution implied by the Merton model is the normal distribution. On the contrary, the KMV approach utilizes their own default database to derive an empirical distribution relating the distance-to-default to a default probability. In this regard, unlike the default probability calculated by KMV the probability measure in Eq. (4) may not correspond to the true probability of default in large samples.

- 3.
\(N(-DD)\) is then the corresponding implied probability of default and sometimes called the expected default frequency (or EDF).

- 4.
In the literature, there are several approaches [See Duan and Wang (2012) for their pros and cons].

- 5.
We are very grateful to the anonymous referee for pointing out several points like this one.

- 6.
The details of this classification of short term debt are elaborate and can be found in Harada et al. (2010).

- 7.
Estimating implied default probabilities form CDS spreads is not a straight forward task as the process requires certain assumptions such as a value for the recovery rate. More importantly, this only yields risk-neutral probabilities unless we convert them into real world probabilities, which further requires an assumption on the value of risk aversion. Despite these differences we assume that the monotonic relationship between our default probabilities and CDS spreads is strong and therefore exploit the relationship.

- 8.
We cannot rule out the possibility that the right-hand-side variables are endogenous as all variables could potentially capture some common economic conditions and hence the coefficients may reflect correlations.

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## Acknowledgements

We thank the late Peter Christoffersen, Lynne Evans, Jens Hagendorff, Thomas Mazzoni and Aydin Ozkan for useful comments on an earlier version of the paper. In addition, we gratefully acknowledge very useful comments from an anonymous referee. Any remaining errors are our responsibility.

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## Appendices

### Appendix A: List of the sampled U.S. banks

See below table.

Distressed: | Distressed: | Distressed: | |||
---|---|---|---|---|---|

Banks | 1=Yes | Banks | 1=Yes | Banks | 1=Yes |

JPMORGAN CHASE BK NA | FULTON BK | COBIZ BK | |||

BANK OF AMER NA | 1 | SUSQUEHANNA BK PA | FIRST BK | ||

CITIBANK NA | 1 | PROSPERITY BK | WILSHIRE ST BK | 1 | |

WACHOVIA BK NA | 1 | CITIZENS BUS BK | FIRST RGNL BK | 1 | |

WELLS FARGO BK NA | SILICON VALLEY BK | MAINSOURCE BK | |||

U S BK NA | COLUMBUS B&TC | 1 | MACATAWA BK | 1 | |

SUNTRUST BK | NATIONAL PENN BK | FIRST CMNTY BK NA | |||

NATIONAL CITY BK | 1 | CENTRAL PACIFIC BK | 1 | MERCANTILE BK MI | 1 |

REGIONS BK | 1 | NBT BK NA | AMERIS BK | 1 | |

STATE STREET B&TC | AMCORE BK NA | 1 | COUNTY BK | 1 | |

BRANCH BKG&TC | COMMUNITY BK NA | SOUTHSIDE BK | |||

PNC BK NA | WESTAMERICA BK | NEWBRIDGE BK | 1 | ||

CAPITAL ONE NA | BANNER BK | STERLING NB | |||

KEYBANK NA | UNITED BK | LAKE CITY BK | |||

MANUFACTURERS & TRADERS TC | WESBANCO BK | TRI CTY BK | |||

COMERICA BK | HANMI BK | 1 | UNIVEST NB&TC | ||

FIFTH THIRD BK | FRONTIER BK | 1 | FIRST BK | ||

NORTHERN TC | CHEMICAL BK | FIRST MRCH BK NA | |||

HUNTINGTON NB | 1 | BANCFIRST | INTERVEST NB | 1 | |

M&I MARSHALL & ILSLEY BK | 1 | BUSEY BK | PEOPLES BK NA | ||

FIRST TENNESSEE BK NA MMPHS | MIDWEST B&TC | 1 | CAMDEN NB | ||

COLONIAL BK NA | 1 | RENASANT BK | FIDELITY BK | 1 | |

ASSOCIATED BK NA | 1 | IBERIABANK | CARDINAL BK | ||

ZIONS FIRST NB | IMPERIAL CAP BK | 1 | CENTURY B&TC | ||

WEBSTER BK NA | 1 | FIRST CMNTY BK | 1 | STOCK YARDS B&TC | |

TCF NB | HANCOCK BK | FIRST BK OF BEVERLY HILLS | 1 | ||

CITY NB | PRIVATEBANK & TC | FIRST UNITED B&TC | |||

COMMERCE BK NA | S&T BK | SUMMIT CMNTY BK | |||

BANK OF OK NA | INTEGRA BK NA | 1 | PEAPACK GLADSTONE BK | ||

FROST NB | FIRST FNCL BK NA | WEST BK | 1 | ||

FIRST-CITIZENS B&TC | REPUBLIC B&TC | SIMMONS FIRST NB |

BANCO POPULAR | SANDY SPRING BK | BANK OF THE SIERRA | |||
---|---|---|---|---|---|

BANCORPSOUTH BK | COLUMBIA ST BK | BANK OF KY | |||

Distressed: | Distressed: | Distressed: | |||

Banks | 1=Yes | Banks | 1=Yes | Banks | 1=Yes |

VALLEY NB | COMMUNITY TR BK INC | CITIZENS & NORTHERN BK | |||

EAST WEST BK | 1 | BOSTON PRIVATE B&TC | ROYAL BK AMERICA | 1 | |

BANK OF HAWAII | BANK OF THE OZARKS | HAWTHORN BK | |||

FIRSTMERIT BK | OLD SECOND NB | 1 | HEARTLAND B&TC | ||

CATHAY BK | 1 | CAPITAL CITY BK | GERMAN AMERICAN BC | ||

INTERNATIONAL BK OF CMRC | WASHINGTON TC | BAYLAKE BK | 1 | ||

TRUSTMARK NB | LAKELAND BK | FIRST NB OF LONG ISLAND | |||

CORUS BK NA | 1 | CITY NB OF WV | ALLIANCE BK | 1 | |

UMPQUA BK | VINEYARD BK NA | 1 | LORAIN NB | ||

UMB BK NA | GREAT SOUTHERN BK | COLUMBIA RIVER BK | 1 | ||

FIRST MIDWEST BK | STILLWATER NB&TC | REPUBLIC FIRST BK | 1 | ||

MB FNCL BK NA | SEACOAST NB | 1 | NORTHRIM BK | ||

OLD NB | BANK OF THE CASCADES | 1 | FIRST MID-IL B&T NA |

### Appendix B: Risk-neutralization of the HN-GARCH

This risk-neutral pricing framework relies on the following risk-neutralization of the processes in Eqs. (6) and (7):

where \(\gamma ^*=\gamma +\lambda \) and \(z_t^*=z_t+\lambda \sqrt{h_t}\). Above, we simply transformed the GARCH process under the physical measure in equations (6) and (7) to a risk-neutral one \(\mathbf {Q}\) in which the underlying asset earns the risk-free rate: \(\mathbf {E}^{\mathbf {Q}}[e^{\ln (A_t/A_{t-1})}]=\exp {(r_t)}\). To this end, following Christoffersen et al. (2013), we use the stochastic discount factor:

The no-arbitrage condition, \(\mathbf {E}^{\mathbf {Q}}[e^{\ln (A_t/A_{t-1})}]=e^{r_t}\), yields

with the implication that \(v_{t-1}=-\lambda \) and hence \(z_t^*=z_t+\lambda \sqrt{h_t}\) under the risk-neutral-\(\mathbf {Q}\) measure.

### Appendix C: The characteristic function

The characteristic function of the HN-GARCH model is represented by a set of difference equations:

with coefficients

Note that \(M_t\) and \(N_t\) are implicitly functions of *T* and \(\phi \). This system of difference equations can be solved backwards using the terminal condition \(M_T = N_T =0\).

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### Cite this article

Kenc, T., Cevik, E.I. & Dibooglu, S. Bank default indicators with volatility clustering.
*Ann Finance* (2020). https://doi.org/10.1007/s10436-020-00369-x

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### Keywords

- Default risk
- Structural credit risk models
- Contingent claims
- GARCH option pricing
- Bank defaults

### JEL Classifications

- G01
- G21
- G28