T=1

1.000

0.978

0.954

0.988

T=2

0.878

1.000

0.946

0.974

T=5

0.826

0.800

1.000

0.947

T = 10

0.908

0.861

0.809

1.000

18

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

TABLE 4 Rank order correlations for different values

of the debt maturity, T.

T=1

T=2

T=5

T = 10

Table gives Spearman and Kendall rank order correlations betwee Merton’s model for different values of the debt maturity, , using the triangular portion of the matrix shows the Spearman correlation, the Kendall correlation.

n the credit spreads implied from ImpVol implementation. The upper lower triangular portion shows the

# 5 PROPERTIES OF MERTON’S MODEL

In this section we show that Merton’s model makes certain predictions about the nature of the relationship between implied volatilities and credit spreads and test whether the predictions are supported by the data. We define ATMVOL for matu- rity τ as the implied volatility for an option with a delta of 0.50. We define SKEW for maturity τ as the implied volatility for an option with maturity τ and a delta of 0.25 minus the implied volatility for an option with maturity τ and a delta of 0.50.

Our implementation of Merton’s model can be used to relate T-year credit spreads to ATMVOL and SKEW for particular values of τ and T. Figure 3 shows the theoretical relationship between credit spread and ATMVOL for different values of SKEW when T = 5 and τ = 0.1667. The figure shows that there is a pro- nounced positive relationship with positive convexity. Similar results are obtained for other values of T and τ. Figure 4 shows the relationship between credit spread and SKEW for different values of ATMVOL when T = 5 and τ = 0.1667. There is very little relationship for low values of ATMVOL (volatility less than 50%) but a strongly positive relationship for higher levels of ATMVOL.

To test whether these properties of the model are supported by the data we performed a linear regression of observed CDS spread (CREDSPR) against ATMVOL and SKEW for our data:

# CREDSPR = a + b × ATMVOL + c × SKEW

ATMVOL and SKEW are quite highly correlated. To address this co-linearity issue the regressions were done twice, once orthogonalizing SKEW with respect to ATMVOL and once orthogonalizing ATMVOL with respect to SKEW. To explore both the convexity of the relation between CREDSPR and ATM and the increasing slope of the relation between CREDSPR and SKEW for higher ATM volatilities, we partitioned the sample into subsets based on the ATM volatility.^{15 }The results are given in Table 5.

15 The partitioning was based on the average ATM volatility for each firm. As a result, each firm appears in only one subset.

## Journal of Credit Risk

Volume 1/Number 1, Winter 2004/05