Merton’s model, credit risk and volatility skews

Implementation

Constant

Slope

R^{2 }

n

ImpVol

92.564 (1.017) 95.811 (1.137)

0.307 (0.008) 0.157 (0.007)

0.178

6,220

Trad

0.069

6,220

TABLE 1 Regression of CDS spreads against implied credit spreads.

The credit spreads are implied from the ImpVol implementation and the Trad implementation of Merton’s (1974) model. The ImpVol implementation is the implementation we propose in Equations (8) and (9). The Trad implementation is the traditional implementation of Merton’s model in Equations (1) and (2). Standard errors are shown in parentheses.

Mean

100.55

0.33

0.26

63

Median

76.17

0.15

0.23

53

Mean

96.31

0.22

0.29

63

Median

79.53

0.14

0.28

53

Trad

ImpVol

TABLE 2 Regression of CDS spreads against implied credit spreads firm by firm and day by day.

# Implementation

n

Constant

Slope R^{2 }Time series by ticker

Mean

90.95

0.88

0.24

37

Median

92.68

0.36

0.18

35

Mean

97.17

0.40

0.12

37

Median

99.37

0.21

0.06

35

## ImpVol

## Trad

Cross-section by day

Mean and median results for regressions of observed CDS spread against implied credit spread. The credit spreads are implied from the ImpVol implementation and the Trad implementation of Merton’s (1974) model. The ImpVol implementation is that proposed in Equations (8) and (9). The Trad implementation is the traditional implementation of Merton’s model in Equations (1) and (2). The upper panel gives results for time-series regressions done on a firm-by-firm basis (86 regressions), and the lower panel those for cross-sectional regressions done on a day-by-day basis (90 regressions). Only regressions with 30 or more observations are included.

to change through time. To explore these possibilities we carried out a separate regression for each firm and a separate regression for each day.

The results are shown in Table 2. For both the firm-by-firm regressions and the day-by-day regressions we report mean and median values for the constant, slope, R^{2}, and number of observations. Any firm for which there were less than 30 observations was not included in the firm-by-firm regressions. Any day for which there were less than 30 observations was not included in the day-by-by regression. This resulted in 86 firm-by-firm regressions and 90 day-by-day regressions. Table 2 shows that, on average, the implied spreads fit the CDS observed spreads much better when considered on a firm-by-firm or day-by-day basis than when fitting the entire sample. This is not surprising since the number of degrees of freedom is much larger in the firm-by-firm or day-by-day analysis.

# Research papers

www.journalofcreditrisk.com

13