Wrong-way risk (WWR) refers to the adverse dependence between a counterparty’s exposure and its default probability: the exposure to a counterparty tends to increase precisely when that counterparty is more likely to default. This correlation is notoriously difficult to model because it requires capturing the joint distribution of market exposures and credit events. Regulators such as Basel III historically simplified CVA risk charges by ignoring explicit WWR due to this complexity.
In structural credit models, wrong-way risk arises naturally through the correlation structure between the Brownian drivers of the non-defaultable risk factors (which drive portfolio exposure) and the defaultable risk factors (bank and counterparty asset processes). Specifically, the full (d+2)-dimensional correlation matrix rho includes off-diagonal blocks rho_{i,j} for i in the non-defaultable set A and j in the defaultable set E, which directly capture the dependence between exposure movements and default-triggering dynamics. This is a key advantage of the structural approach: since default is endogenously derived from the firm’s asset dynamics, the correlation between exposure and default likelihood is specified at the level of the driving Brownian motions.
An alternative approach to WWR, proposed by Brigo and Vrins (2018), uses a change-of-measure technique that incorporates the adverse dependence into the drift of the exposure process, adjusting the dynamics so that higher exposure states coincide with higher default risk. This involves an infinite sequence of measure changes but yields a practical compromise between mathematical rigour and tractability. In the context of deep BSDE methods, the Girsanov-based drift tilting serves a dual purpose: it both increases default frequency for training and naturally accommodates the WWR correlation structure through the full (d+2)-dimensional forward SDE.
Key Details
- In a structural model, WWR is parameterised by rho_{i,j} for i in A (non-defaultable) and j in E (defaultable)
- Higher correlations between market factors and default processes lead to larger WWR effects
- Traditional Monte Carlo approaches to WWR require nested simulations, which are computationally prohibitive
- The deep BSDE approach with structural credit modelling handles WWR implicitly through the correlated forward SDE, without requiring separate WWR adjustments
- Numerical experiments use a full 7x7 correlation matrix with non-trivial off-diagonal entries between market and default factors
Textbook References
The xVA Challenge (Gregory, 2020)
- Section 17.6.1 (pp. 514—516): Overview of WWR. WWR is the unfavourable dependence between exposure (EPE) and counterparty credit quality. Classic examples across asset classes: put options on correlated stocks, FX forwards with sovereigns, interest rate swaps in recessionary environments, commodity swaps (right-way), and CDS contracts (specific WWR). Table 17.7 distinguishes general WWR (macro-economic, potentially modellable) from specific WWR (structural/causal, should be avoided or stress-tested).
- Section 17.6.2 (pp. 516—518): Quantification. The simplest approach replaces unconditional EPE with EPE conditional on counterparty default: EPE(t_i | t_i = tau_C). A single-correlation model shows that 50% correlation approximately doubles EPE, while -50% (right-way risk) halves it. WWR increases with counterparty credit quality because default of a stronger party represents a larger shock.
- Section 17.6.3 (pp. 518—522): WWR models. Three approaches: (i) intensity approach — stochastic credit spread correlated with exposure drivers, tractable but produces only weak dependence; (ii) structural approach — maps exposure and default distributions onto a bivariate copula, stronger dependence but opaque correlation parameter; (iii) parametric approach (Hull-White 2011) — links default probability to exposure via a functional relationship, calibrated to historical data or what-if scenarios.
- Section 17.6.4 (pp. 522—524): Jump approaches for specific WWR. FX rate jumps at sovereign default time. The CDS quanto basis (USD vs local currency CDS) provides market-observable calibration data. Implied jumps: 9%—25% for European sovereigns, average 38.4% for Japanese sovereign, 13.5% for Japanese financials.
- Section 17.6.5 (pp. 524—525): Credit derivatives and WWR. Buying CDS protection creates unavoidable WWR via correlation between reference entity and counterparty default. At 60% correlation, CVA is ~50 bps (one-fifth of the 250 bps CDS premium); at 100% correlation, the fair premium drops to recovery value (~100 bps).
- Section 17.6.6 (pp. 525—526): Collateralisation and WWR. Jump approaches show margin is nearly useless against WWR; continuous approaches suggest margin is effective. The truth is between and depends on the counterparty type. Counterparties that actively margin (banks) tend to be highly systemic with extreme WWR, while non-margining counterparties (corporates) are less systemic.
- Section 18.3.7 (p. 563): Wrong-way funding risk. Positive correlation between funding spread and market variables (e.g., interest rates and funding spreads) creates FVA WWR. In symmetric FVA, the effect is amplified because FCA and FBA move in the same direction.