4.7.5 Failure Modes & Practice

Basic Prerequisites: PFE and 03 · CVA & DVA.

---

1. Intuition & Practical Objective

Every CVA formula in this folder rests on one hidden assumption: that exposure, default probability and LGD are independent. Real counterparties break that assumption. This page names the ways the standard model fails, so a practitioner knows which number to distrust and how much it moves.

The failures, in one line each:

1. Wrong-way risk (WWR) - exposure is high exactly when the counterparty is likely to default. The independence assumption is false, and CVA is understated.
2. Exposure misestimation - EPE is a model output; understate volatility, correlations, or path dependence and CVA falls proportionally.
3. Jump-to-default (JTD) - credit deteriorates not continuously but in jumps; a jump defeats collateral, defeats Greek hedging, and dominates the tail.
4. Collateral false comfort - margin works beautifully against continuous moves and is nearly useless against a correlated jump.

The headline number. Take the ATM forward (150bp spread, LGD 60%) whose independent CVA is −1.05. Add a mild positive dependence between exposure and default intensity, and CVA rises to −2.54 (2.4×) and then −3.69. The same trade, a different assumption about correlation - and correlation is precisely the parameter you cannot estimate reliably.

---

2. Mathematical Ground Truth & Derivations

2.1 General vs specific wrong-way risk

Gregory (Table 17.7) splits WWR into two kinds:

• General WWR - macro-driven: exposure and credit quality co-move with the economy (e.g. a sovereign paying local currency on an FX swap). Detectable historically, priceable.
• Specific WWR - structural, trade-specific: the derivative itself ties your exposure to the counterparty’s fate (e.g. buying protection on a name from a counterparty that is that name). Hard to detect, dangerous with naive correlations - to be avoided, not modelled.

2.2 Quantifying WWR: conditional EPE

The independent CVA formula uses $\text{EPE}(t,t_i)$, the unconditional expected exposure. WWR replaces it with the exposure conditional on default at that date:

$${\text{CVA}}^{\text{WWR}}=-\text{LGD}\sum _i\text{EPE}\left(t,t_i\mid t_i={\tau }_C\right)\text{PD}(t_{i-1},t_i).$$

A single dependence parameter drives a conditional-EPE formula (Gregory Appendix 17F): at a correlation of $+50\%$ the conditional EPE roughly doubles; at $-50\%$ it at least halves. The critical and counter-intuitive result: WWR increases as counterparty credit quality improves - the default of a strong name is a bigger shock, so a portfolio of AAA counterparties can carry the largest relative WWR.

2.3 Jump-to-default

Strong empirical support for jumps (Gregory §17.6.4): implied quanto jumps in CDS markets of 83% for AAA and 27% for BBB sovereigns (Levy–Levin 1999); euro-crisis EUR/USD jumps of 9–25% for Greece/Italy/Spain/Germany; Chung–Gregory (2019) financials 13.5%, non-financials 8%, sovereign 38.4%. Pure intensity models cannot reproduce this (Ehlers–Schönbucher 2006) - the exposure must be allowed to gap, not just diffuse, on the default event.

Jump-to-default P&L for a hedged book (Gregory Eq 21.2):

$$\text{JTD P\&L}=-\underset{\text{loss on default}}{\underbrace{\text{current exposure}\times \text{LGD}}}+\underset{\text{hedge payoff}}{\underbrace{\text{CDS notional}\times \text{CDS LGD}}}-\underbrace{\text{current CVA contribution}}.$$

A CDS hedge of the wrong maturity leaves JTD risk; buying short-dated protection is the (rarely available) fix.

2.4 Why collateral fails against WWR

Collateral is a timing hedge. If exposure approaches its peak gradually, margin can be posted along the way and protects you. If exposure jumps - as it does on a correlated default - the margin posted before the jump is based on the pre-jump value and is grossly insufficient. Pykhtin–Sokol (2013) show that with jumps and elevated post-default volatility, WWR erodes the collateral benefit, and it matters most for the systemic banks that post the most margin (Gregory §17.6.6).

---

3. Computational Implementation - wrong-way risk, jumps, misestimation

A path-wise Monte Carlo over the ATM forward where the default intensity is tied to the current exposure, ${\lambda }_i={\lambda }_0(1+\text{wwr}\cdot V_i/\text{scale})$ - chosen so the average hazard is unchanged (isolating the dependence, not just a higher PD). We also measure the jump case and the misestimation case. Standard library only.

Read the three blocks:

• WWR. The cumulative PD is held nearly constant across rows (~12%), so the change is pure dependence: at wwr = 0 the E[exposure | default] is ≈ 0 (independence, CVA −1.05); at wwr = +1 the conditional exposure is +35 and CVA more than doubles to −2.54. Negative dependence (right-way risk) shrinks CVA to −0.19.
• Jump. A 30% adverse jump on the default date raises CVA by 30% (−1.05 → −1.37); a 50% jump by 50%. Jumps are a first-order effect, not a correction.
• Misestimation. A model that uses $\sigma =15\%$ on a true $\sigma =25\%$ world reports CVA 40% too low (−0.64 vs −1.05) - and a conservative $\sigma =35\%$ overstates it by 38%. The error is roughly linear in volatility.

---

4. Failure Modes & First-Principles Breakdowns (numbered)

1. Wrong-way risk is unhedgeable in the tail. No single-name hedge removes a correlated default: the CDS protection pays precisely when the exposure jumps above its hedged level. Prefer to avoid specific WWR structurally (exclude legally connected trades from the netting set; SA-CCR sets EAD = 100% of value with LGD = 100% in these cases).
2. Correlation is not independence. Zero correlation is not the same as independence ($Y=X^2$ has zero correlation but total dependence). Reporting a zero WWR parameter as “no WWR” is a category error (Gregory §17.6.2).
3. Jump-to-default defeats the Greeks. A delta/CS01 hedge is calibrated to diffusive moves; the default event is a jump in a different risk factor. It must be managed with explicit JTD limits, not delta (Gregory §21.2.5).
4. Collateral is a timing hedge, not a default hedge. Against a gradual exposure, margin protects; against a correlated jump it does not. Assuming collateral eliminates WWR is the most expensive mistake on this page (Pykhtin–Sokol 2013).
5. Exposure model risk. Vols, correlations, grid resolution and path-dependence modelling all feed EPE, and errors are roughly proportional. A conservative exposure model is cheaper than an understated CVA.
6. Data is uninformative. Credit-quality/exposure co-movement is rare and regime-dependent; historical correlation estimates are noisy and backward-looking. This is why WWR is often addressed by structure (avoidance, limits, conservative add-ons) rather than by a correlation parameter.

---

5. References

• Gregory, Jon: The xVA Challenge (5th ed., 2025)
• Pykhtin & Sokol (2013): Modelling Wrong-Way Risk and Credit Valuation Adjustment
• Levy & Levin (1999) / Chung & Gregory (2019): empirical implied sovereign-jump magnitudes (the evidence base for
• Hull, Options, Futures, and Other Derivatives

---

6. Connected Graph Bridges

• Back: 04 · Collateral, Netting & SA-CCR
• Forward: MVA, WWR modelling) · Index Hub
• Sibling: Extreme Value Theory & Fat Tails (jump/tail modelling) · Liquidity Risk & Margin Spirals