20

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

# TABLE 5 Regression of observed CDS spread against the at-the-money volatility

and the volatility skew.

## Orthogonalized SKEW

## Orthogonalized ATMVOL

## Restriction

a

b

c

a

b

c

R^{2 }n

None

– 27.61* (3.68)

336.47* (8.84)

57.83 (36.86)

75.63* (1.81)

328.26* (10.27)

657.50* (31.72)

0.1889 6,220

ATMVOL < 40%

7.62† (3.72)

222.96* (10.28)

14.13 (33.75)

72.31* (1.58)

220.68* (11.63)

313.74* (29.82)

0.1318 3,088

40% < ATMVOL

– 38.99* (7.62)

371.91* (16.45)

166.69† (66.95)

86.42* (3.48)

346.91* (19.28)

794.51* (57.15)

0.1413 3,132

Table presents results of a regression of observed CDS spread (CREDSPR) against the at-the-money volatility (ATMVOL) and the volatility skew (SKEW): CREDSPR = a + b × ATMVOL + c × SKEW. SKEW is measured as the 25-delta volatility less the at-the-money volatility. Volatilities are measured as decimals and the CDS spreads are in basis points. Standard errors are shown in parentheses. To address co-linearity issues, the regressions were done first with SKEW orthogonalized with respect to ATMVOL and then with ATMVOL orthogonalized with respect to SKEW. *Denotes significance at the 1% level. †Denotes significance at the 5% level.

# 5.1 Relation between credit spread and ATM volatility

When SKEW is orthogonalized with respect to ATMVOL, Table 5 shows that, as predicted, there is a significantly positive slope for the ATMVOL for the full sam- ple. When the sample is split into two roughly equal subsets based on ATMVOL, we find that the coefficient of the ATMVOL is consistently significantly positive. Further, the coefficient of the ATMVOL is higher in the high-volatility subset than it is in the low-volatility subset. This is consistent with the convex relation- ship predicted by the model. The increase in slope is statistically significant.

When ATMVOL is orthogonalized with respect to SKEW, some of the impact of the positive relation between CREDSPR and ATMVOL is liable to be captured by the SKEW coefficient. However, it is reassuring that the coefficient ATMVOL is still always significantly positive at the 1% level. Furthermore, it is signifi- cantly higher for the high-volatility subset than for the low-volatility subset. Table 5 therefore provides strong evidence for a convex positive relation between CREDSPR and ATMVOL.

# 5.2 Relation between credit spread and skew

When ATMVOL is orthogonalized with respect to SKEW the relation between CREDSPR and SKEW is always significantly positive at the 1% level. When the sample is split into two groups, the coefficient of SKEW is significantly higher in the high-volatility group than in the low-volatility group, and the difference is significant.

When SKEW is orthogonalized with respect to ATMVOL, the coefficient of SKEW is not significant overall. It is not significant for the lower-volatility subset

## Journal of Credit Risk

Volume 1/Number 1, Winter 2004/05