Credit Derivative Evaluation and CVA Under the Benchmark Approach

Abstract

In this paper, we discuss how to model credit risk under the benchmark approach. Firstly we introduce an affine credit risk model. We then show how to price credit default swaps (CDSs) and introduce credit valuation adjustment (CVA) as an extension of CDSs. In particular, our model can capture right-way—and wrong-way exposure. This means, we capture the dependence structure of the default event and the value of the transaction under consideration. For simple contracts, we provide closed-form solutions. However, due to the fact that we allow for a dependence between the default event and the value of the transaction, closed-form solutions are difficult to obtain in general. Hence we conclude this paper with a reduced form model, which is more tractable.

Appendix: Laplace Transforms for Square-Root Processes

The following results are based on Lennox (2011) and are contained in Baldeaux and Platen (2013) as Propositions 7.3.8 and 7.3.9. Consider a square-root process $X = \left\{ X_t, \, t \ge 0 \right\}$, where

$$\begin{aligned} d X_t = (a - b X_t ) dt + \sqrt{2 \sigma X_t} d W_t \end{aligned}$$

(34)

with $X_0=x >0$.

Proposition 7.1

Assume that $X = \left\{ X_t, \, t \ge 0 \right\}$ is given by (34) and that $\frac{2 a }{\sigma } \ge 2$. Let $\beta = 1 + m - \alpha + \frac{\nu }{2}$, $m=\frac{1}{2} \left( \frac{a}{\sigma } - 1 \right)$, and $\nu = \frac{1}{\sigma } \sqrt{(a - \sigma )^2 + 4 \mu \sigma }$. Then if $m > \alpha - \frac{\nu }{2} -1$,

$$\begin{aligned}&E \left( \exp \left\{ - \mu \int ^t_0 \frac{ds}{X_s} \right\} X^{- \alpha }_t \right) \\&\quad = \frac{1}{2^{\nu } x^m} \exp \left\{ - \frac{b x}{\sigma \left( e^{bt} -1 \right) } + b m t \right\} \left( \frac{b \exp \{ b t \} }{( e^{b t} -1 )\sigma } \right) ^{-m + \alpha - \frac{\nu }{2}}\\&\qquad \times \, \left( \frac{b^2 x}{\sigma ^2 \sinh ^2 (\frac{bt}{2})} \right) ^{\nu /2} \frac{\Gamma (\beta )}{\Gamma (1+ \nu )} _1 F_1 \left( \beta , 1 + \nu , \frac{bx}{\sigma \left( e^{b t} -1 \right) } \right) . \end{aligned}$$

where ${}_1{}F_1$ denotes the confluent hypergeometric function. Finally, we present Proposition 2.0.42 from Lennox (2011).

Proposition 7.2

Assume that $X = \left\{ X_t, \, t \ge 0 \right\}$ is given by Eq. (34) and that $\frac{2 a }{\sigma } \ge 2$. Define $A=b^2+4 \mu \sigma$, $m=\frac{1}{\sigma } \sqrt{(a- \sigma )^2 + 4 \sigma \nu }$, $\beta = \frac{\sqrt{A x}}{\sigma \sinh ( \frac{\sqrt{A} t}{2} )}$, and $k = \frac{\sqrt{A} + b \tanh ( \frac{\sqrt{A} t}{2} )}{2 \sigma \tanh (\frac{\sqrt{A} t}{2})}$. Then if $a > ( 2 \alpha -3 ) \sigma$, for $\mu >0$, $\nu \ge 0$,

$$\begin{aligned}&E \left( X^{- \alpha }_t \exp \left\{ - \nu \int ^t_0 \frac{ds}{X_s} - \mu \int ^t_0 X_s ds \right\} \right) \\&\quad = \frac{ \sqrt{A} x^{\frac{1}{2} - \frac{a}{2 \sigma }} }{2 \sigma \sinh ( \frac{\sqrt{A} t}{2} ) } \left( \frac{\beta }{2} \right) ^m \exp \left\{ \frac{ b (x + at) - \sqrt{A} x \coth ( \frac{\sqrt{A} t}{2} ) }{2 \sigma } \right\} k^{- (1 + \frac{a}{2 \sigma } + \frac{1}{2} + \frac{m}{2} - \alpha )}\\&\qquad \times \, \frac{\Gamma ( 1 + \frac{a}{2 \sigma } + \frac{1}{2} + \frac{m}{2} - \alpha )}{\Gamma (1+m)} _1 F_1 \left( 1 - \alpha + \frac{a}{2 \sigma } + \frac{1}{2} + \frac{m}{2}, 1+ m, \frac{\beta ^2}{4k} \right) . \end{aligned}$$