Merton’s model, credit risk and volatility skews

Kendall rank correlation

Standard error

z-statistic

All data ImpVol Trad ImpVol – Trad

0.2836 0.2590 0.0247

0.0172 0.0173 0.0244

33.55 30.63 1.01

ImpVol

0.3967

0.0188

38.01

Trad

0.3101

0.0193

28.71

ImpVol – Trad

0.0866

0.0270

3.21

TABLE 3 Rank order correlation measures.

## Firm-by-firm (n = 86)

0.4230

0.0199

33.36

0.3929

0.0202

30.99

0.0301

0.0284

1.06

0.5409

0.0202

35.51

0.4386

0.0212

28.79

0.1023

0.0293

3.49

Spearman rank correlation

Standard error

z-statistic

ImpVol

0.2506

0.0239

20.37

0.3630

0.0280

20.35

Trad

0.2188

0.0241

17.78

0.3186

0.0285

17.87

ImpVol – Trad

0.0318

0.0340

0.94

0.0443

0.0400

1.11

## Day-by-day (n = 90)

Table gives Kendall rank order correlation measure and Spearman rank order correlation measure between the implied credit spread from Merton’s (1974) model and the five-year credit default swap spread. The ImpVol implementation is the implementation of Merton’s model we propose in Equations (8) and (9). The Trad implementation is the traditional implementation of Merton’s model in Equations (1) and (2). 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 A – B rows use Equation (A1) to test the difference between the A and B correlations.

There are a number of possible reasons why the ImpVol implementation ranks credit spreads better than the Trad implementation. Implied volatilities adjust to information more quickly than do historic volatilities. There is noise in the estimates of the historic volatility for the Trad implementation, and the Trad implementation requires the assumption that the historic volatility is the same as the instantaneous volatility. Also, we had only limited information on the company’s capital structure for the Trad implementation.

# 4.2 Impact of debt maturity

As the debt maturity date in the ImpVol implementation of Merton’s model changes the implied credit spread changes, but there is little effect on the ranking of implied credit spreads. This is illustrated in Table 4, which shows the rank order correlation between the implied credit spreads when T equals 1, 2, 5 and 10. We obtained similar results for the probability of default rankings and for the day-by-day and company-by-company analyses. For financial institutions that are interested only in ranking the creditworthiness of counterparties, this result may add to the attraction of the ImpVol implementation of Merton’s model.

## Research papers

www.journalofcreditrisk.com

17