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Merton’s model, credit risk and volatility skews - page 22 / 26

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

Two predictions made by Merton’s model are:

there should be a positive relationship with positive convexity between credit

spreads and at-the-money volatilities; and there should be a positive relationship between credit spreads and volatility

skews when the at-the-money volatility is high.

The first prediction is strongly supported by the data. The second is also supported by the data but somewhat less strongly than the first.

APPENDIX

Kendall and Gibbons (1990) provide a great deal of information on the statistical properties of the Kendall and Spearman rank correlation measures. For n > 10, the probability distribution of Kendall’s rank order correlation, rk, conditional on no rank order correlation between the variables is approximately normal with a mean of zero and a variance of [2(2n + 5)][9n(n – 1)]. The z-statistic for testing the null hypothesis that rk is zero is therefore

3rk n(n 1) 2(2n + 5)

F o r n > 3 0 t h e p r o b a b i l i t y d i s t r i b u t i o n o f S p e a r m a n s r a n k o r d e r c o r r e l a t i o n , r s conditional on no rank order correlation is approximately normal with a mean of zero and a variance of 1 ⁄ (n – 1). The z-statistic for testing the null hypothesis that , r s i s z e r o i s t h e r e f o r e r s n 1 .

When the rank order correlation is non-zero, Kendall and Gibbons show that the standard deviation of the estimate of rk depends on the true value of rk and other unknown quantities concerned with the arrangement of the ranks in the p a r e n t p o p u l a t i o n . T h e s a m e i s t r u e o f r s . T h e e s t i m a t e d v a l u e o f r k c a n b assumed to be drawn from a normal distribution with a mean of ρk and a vari- e a n c e o f a t m o s t 2 ( 1 ρ k 2 ) n , w h e r e ρ k i s t h e t r u e K e n d a l l r a n k o r d e r c o r r e l a t i o n . T h e e s t i m a t e d v a l u e o f r s c a n b e a s s u m e d t o b e d r a w n f r o m a n o r m a l d i s t r i b u t i o n w i t h a m e a n o f ρ s a n d a v a r i a n c e o f a t m o s t 3 ( 1 ρ s 2 ) n , w h e r e ρ s i s t Spearman’s rank order correlation. In practice, ρk and ρs are set equal to the esti- h e t r u e m a t e s , r k a n d r s , i n t h e s e f o r m u l a s .

These results enable us to construct a conservative test of whether there is a significant difference between two rank order correlations. For example, suppose we observe a Spearman rank order correlation of rs,1 from a sample of n1 and a Spearman rank order correlation of rs,2 from a sample of n2. The z-statistic for testing whether they are significantly different is

rs,1 rs, 2

(A1)

r n r n s s , , 1 2 1 2 2 2 3 1 3 1 ( ) + ( )

Journal of Credit Risk

Volume 1/Number 1, Winter 2004/05

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