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Merton’s model, credit risk and volatility skews - page 10 / 26





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John C. Hull, Izzy Nelken and Alan D. White


In this section we test whether five-year credit spreads implied from our imple- mentation of Merton’s model and the traditional implementation are consistent with the observed five-year CDS spreads. For the traditional implementation we estimated the equity value, historical equity volatility and the outstanding debt in the way described in Section 3.3. Equations (1) and (2) were used to compute the asset value and asset volatility and Equation (5) was then used to compute the credit spread. For our implementation we used the 50- and 25-delta implied put volatilities in Equations (8) and (9) to imply the leverage ratio and the asset volatility.12 As in the case of the traditional implementation, Equation (5) is then used to compute the credit spread. In the balance of the discussion we shall refer to our implementation of Merton’s model as the “ImpVol” implementation and the traditional implementation as the “Trad” implementation.

There are a number of reasons why we should expect differences between the credit spreads implied from Merton’s model and observed CDS spreads. Merton’s model is not a perfect representation of reality because companies do not usually issue only zero-coupon debt and because a number of factors besides the value of their assets are liable to influence a company’s decision to default on its obligations. Also, CDS spreads are likely to be slightly different from bond yield spreads for the reasons listed in Hull, Predescu and White (2004). Finally, the credit spread backed out from Merton’s model is the spread between the yields on zero-coupon bonds, while a CDS credit spread is (at least, approxi- mately) the spread between the yields on par yield bonds.

Table 1 shows the results of regressing CDS spreads against the spreads implied from Merton’s model using the two implementation approaches. Given the nature of the model generating the implied spreads, it is unlikely that errors in the implied spreads are normally distributed. However, the regression does provide a first attempt at describing the relationship (if any) between the implied and observed spreads.

The results in Table 1 reveal a positive relationship between the observed CDS spreads and the implied spreads that is roughly similar for both implementations. The mean CDS observed spread is about 95 basis points higher than the mean implied spread for both models. The R2 of the regressions indicate that the ImpVol implementation provides a better fit to the observed data than the Trad implementation.

It is possible that there are factors other than those suggested by Merton’s model that affect CDS spreads. Figure 1 shows a scatter diagram of the CDS spread versus the implied spread from the ImpVol model for Merrill Lynch, Dow Chemicals and Bowater. Figure 2 shows the same for the Trad model. These figures suggest that the relation between the CDS spread and the implied spread may be different for different firms. It is also possible that macroeconomic variables cause the relationship between the CDS spread and the implied spread


This involves determining the value of κ that produces options with deltas of 0.50 and 0.25.

Journal of Credit Risk

Volume 1/Number 1, Winter 2004/05

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