Modeling Credit Risk in the Jump Threshold Framework
C.-Y. Chiu, Alec Kercheval
The jump threshold framework for credit risk modeling
developed by Garreau and Kercheval (2016) enjoys the advantages of
both structural and reduced form models. In their paper, the focus
is on multi-dimensional default dependence, under the assumptions
that stock prices follow an exponential Lévy process (i.i.d. log
returns) and that interest rates and stock volatility are
constant. Explicit formulas for default time distributions and
Basket CDS prices are obtained when the default threshold is
deterministic, but only in terms of expectations when the default
threshold is stochastic.
In this paper we restrict attention to the one-dimensional, single-name case in order to obtain explicit closed-form solutions for the default time distribution when the default threshold, interest rate, and volatility are all stochastic. When the interest rate and volatility processes are affine diffusions and the stochastic default threshold is properly chosen, we provide explicit formulas for the default time distribution, prices of defaultable bonds, and CDS premia. The main idea is to make use of the Duffie-Pan-Singleton method of evaluating expectations of exponential integrals of affine diffusions.