On the conditional default probability in a regulated market: a structural approach

Click to increase image sizeClick to decrease image size

Acknowledgements

The authors would like to thank the Editor and two anonymous referees for their valuable comments and suggestions on an earlier version of this article. Supported by the LPMC at Nankai University and the Keygrant Project of the Chinese Ministry of Education (No. 309009). The research of Bo and Yang is also supported by the Basic Science Research Fund in Xidian University (No. JY10000970002).

Notes

†Note that CDP = 1 − CSP.

‡For the existence and uniqueness of a strong solution triple (Q _t , l _t , u _t ) _t≥0 for (2.1) with continuous sample path modification, see Lions and Sznitman (1984).

§Usually, the noisy source is assumed to be known. Then by lemma 3.2 in section 3, the distribution F _Y (t, dy) of the observed process Y _t = Q _t + ξ _t at time t can be given using the independence of Q and ξ.

†An explicit expression for the inverse Laplace transform η _v of ℒ(η _v ) does not seem to be possible to obtain. Instead, there are algorithms available for the numerical inversion of Laplace transforms (see Valkó and Vajda 2002, Valkó and Abate 2004 and section 4).

†As in Bo et al. (2006), if μ = 0 and σ = 1, the following power series l ^(λ) on [0, b] is a solution to equation (3.4):

where

with the convention . Expression (3.6) for l ^(λ) will be applied to our numerical experiment in section 4.

†Here we choose μ = 0 and σ = 1 so that we can use the explicit expression (3.6) in footnote 1 for the Laplace transform. On the other hand, we can transform the price band to [0, 1] by shift and scale transforms. For the spot interest rate, is a standard preference value in the literature. Selection of the other parameters was for convenience.

‡The sources of the ‘GWR’ algorithm can be found in Valkó and Abate (2004). In addition, Valkó and Abate provided a MATHEMATICA® package ‘Numerical Inversion of Laplace Transform with Multiple Precision’ at the web sites http://library.wolfram.com/infocenter/MathSource/4738/ and http://www.pe.tamu.edu/valko/Nil/.

†We apply proposition 2.1 of Alili et al. (2005) to calculate the CSPs via the inverse Laplace transform. Although the literature (Alili et al. 2005) presents three numerical methods for approaching the density of the first hitting time directly, the inverse Laplace transform seems to be more efficient than the others.

‡There is only one exception (ℓ(0.1, 0.2; 2.5)), which may be due to calculation error.