A Subordinated CIR Intensity Model with Application to Wrong-Way Risk CVA

Abstract

Credit Valuation Adjustment (CVA) pricing models need to be both flexible and tractable. The survival probability has to be known in closed form (for calibration purposes), the model should be able to fit any valid Credit Default Swap (CDS) curve, should lead to large volatilities (in line with CDS options) and finally should be able to feature significant Wrong-Way Risk (WWR) impact. The Cox-Ingersoll-Ross model (CIR) combined with independent positive jumps and deterministic shift (JCIR++) is a very good candidate : the variance (and thus covariance with exposure, i.e. WWR) can be increased with the jumps, whereas the calibration constraint is achieved via the shift. In practice however, there is a strong limit on the model parameters that can be chosen, and thus on the resulting WWR impact. This is because only non-negative shifts are allowed for consistency reasons, whereas the upwards jumps of the JCIR++ need to be compensated by a downward shift. To limit this problem, we consider the two-side jump model recently introduced by Mendoza-Arriaga & Linetsky, built by time-changing CIR intensities. In a multivariate setup like CVA, time-changing the * Voie du Roman Pays 34, intensity partly kills the potential correlation with the exposure process and destroys WWR impact. Moreover, it can introduce a forward looking effect that can lead to arbitrage opportunities. In this paper, we use the time-changed CIR process in a way that the above issues are avoided. We show that the resulting process allows to introduce a large WWR effect compared to the JCIR++ model. The computation cost of the resulting Monte Carlo framework is reduced by using an adaptive control variate procedure.

Affine Term-Structure Models: A Time-Changed Approach with Perfect Fit to Market Curves

“…Usually, we consider the general case of wrong-way risk (WWR) effect, obtained by introducing a correlation between the Brownian drivers. For the CIR++ we assume dW t dW V t = ρdt, whereas for the TC-CIR, we apply the synchronisation procedure devised in Mbaye and Vrins (2018) in order to preserve the correlation after timechanging the intensity process. In the special case where the default time of the counterparty is independent from the discounted exposure (i.e., ρ = 0, that is no wrong-way risk) one can easily deduce from (23) the independent CVA formula…”

“…The analytical tractability of the resulting model is preserved to some extend. This model has been recently applied to counterparty credit risk Mbaye and Vrins (2018). In this work, we exploit the time change idea in yet another way, to solve a completely different problem.…”

“…seeMbaye and Vrins (2018) for the implementation of the adaptive control variate applied on CVA computation.…”

Affine term structure models: A time‐change approach with perfect fit to market curves

“…Usually, we consider the general case of the WWR effect, obtained by introducing a correlation between Brownian drivers. For CIR++, we assume that $$dW_tdW^V_t=\rho dt$$, whereas for TC‐CIR, we apply the synchronization procedure suggested in Mbaye & Vrins (2018) to preserve the correlation after time‐changing the intensity process. In the special case where the default time of the counterparty is independent of the discounted exposure (i.e., $$\rho =0$$, that is, no WWR), one can easily deduce from Equation (53) the independent CVA formula $$\begin{equation} \mathrm{CVA}^\perp = -(1-R)\int _0^T\mathbb {E}{\left[V^+_{u}\right]}d\,\mathbb {E}{\left[e^{-\Lambda _u}\right]}=(1-R)\int _0^T f^\lambda (u) {\mathbb {E}}{\left[V^+_u\right]}P^\lambda (u)du.$$…”

“…The analytical tractability of the resulting model is preserved to some extent. This model has recently been applied to counterparty credit risk (Mbaye & Vrins, 2018). In this work, we exploit the time‐change idea in yet another way to solve a completely different problem.…”