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Merton’s model, credit risk and volatility skews - page 21 / 26





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Kendall rank correlation

Standard error


Spearman rank correlation

Standard error


Merton (1974) Merton (1976) Merton (1974) Merton (1976)

0.2836 0.2095 0.0741

0.0172 0.0175 0.0246

33.546 24.778 3.019

0.4230 0.3177 0.1053

0.0199 0.0208 0.0288

33.362 25.056 3.657

Data are pooled for all companies on all days. The table shows Kendalls rank order correlation measure and Spearmans rank order correlation measure between implied credit spread and the five-year credit default swap spread.

The standard error of the rank order correlation is an upper bound. The z-statistic tests whether the rank order correlation is significantly greater than zero. The last row uses Equation (A1) to test the difference between the Merton (1974) and Merton (1976) correlations.

Mertons model, credit risk and volatility skews

TABLE 6 Comparison of Mertons (1974) model with a model based on Mertons (1976) mixed jumpdiffusion process.

Just as Equations (8), (9) and (5) allowed us to imply credit spreads from option volatilities under the Merton (1974) model, Equations (11) and (12) allow us to do so under the Merton (1976) model. Option implied volatilities can be used to infer the default intensity and the stock volatility, and these parameters can be used to imply a credit spread. As in Section 4, this can be compared with the contemporaneous CDS spread. In doing the analysis it is necessary to assume a time to debt maturity, T, and a recovery rate, R. The magnitude of the resulting implied credit spread is sensitive to these assumptions but the relative ranking of outcomes is not.

These results allow us to provide an interesting test of the value of the struc- tural model underlying Merton (1974). Our null hypothesis is that Merton (1976) ranks the credit quality of companies as well as Merton (1974). Table 6 compares the performance of Merton’s (1976) model with Merton’s (1974) model. The 1976 model has statistically significant explanatory power, but in all cases the Merton (1974) model provides significantly better predictions of default proba- bilities and credit spreads at the 1% level.


The traditional approach to implementing Merton’s model involves estimating the instantaneous equity volatility and the debt outstanding by a particular future time. We have presented an alternative implementation where the inputs to the model are much simpler. All that is required to imply a credit spread is two implied volatilities. The alternative approach is particularly appropriate for firms that are known (or rumored) to have significant off-balance-sheet liabilities.

Our proposed implementation of Merton’s model outperforms a simple version of the traditional implementation of the model. It is reassuring that it also out- performs an alternative way of deriving credit spreads from implied volatilities that is based on a model with less structure.

Research papers



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