Merton’s model, credit risk and volatility skews

On the basis of the regression R^{2}, the results also show that the implied spreads match CDS spreads better on a firm-by-firm basis than when looking across firms on a particular day. What this means is that the models seem to work better at explaining how the observed credit spreads for a firm change over time than they do at discriminating between different firms at a single time. The two implemen- tations are comparable when applied on a firm-by-firm basis, but the ImpVol implementation appears to provide a better fit than the Trad implementation when applied cross-sectionally.

# 4.1 Rank correlations

The results presented so far indicate that both implementations of Merton’s model are consistent with the data in the sense that there is a positive relationship between the model predictions and the observed data. There is also some evi- dence that the relation between the implied credit spread and observed credit spread is non-linear.

To address the apparent non-linearity we could test alternative non-linear models. However, the nature of the non-linearity is not known and it may differ from firm to firm. A general approach to fitting data that are subject to an unknown non- linear relationship is to linearize the data by translating observations to ranks. Formally, if

y = f (x)

for some monotonic increasing function, f, and we have a set of observations {(x_{1}, y_{1}), (x_{2}, y_{2}), …, (x_{n}, y_{n})}, then

r(y_{i }{y_{1}, y_{2},…, y_{n}}) = r(x_{i }{x_{1}, x_{2},…, x_{n}})

where r(a{b}) is the rank of a within the set b. This linearization works per- fectly in the deterministic case described here, but difficulties arise when variables are observed with error. However, there is a well-developed literature on rank correlation which will allow a more formal and robust distribution-free test to determine which version of the model is more consistent with the data.^{13 }

There are two measures of rank order correlation in the literature: Kendall’s and Spearman’s. To explain how they are calculated, suppose that we have n observations on two variables. (In our application the variables are five-year implied credit spread and five-year observed CDS spread). As a first step we calculate the rank of each observation on each variable.

13 In the context of our tests, it is interesting to note that proponents of the commercial use of Merton’s model claim that, although estimated default probabilities and credit spreads are not accurate, the models rank the credit quality of companies well. See, for example, Kealhofer (2003a, 2003b).

# Research papers

www.journalofcreditrisk.com

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