3.10.5 Failure Modes & Real-World Practice

Basic Prerequisites: 04 · FVA & MVA.

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1. Intuition & Practical Objective

The xVA formulas in this folder rest on assumptions that are quietly strong. This page names them precisely, so a practitioner knows which assumption to distrust and how the failure shows up in money terms. The objective is not cynicism - it is knowing exactly where the model is an approximation so the residual risk can be measured and managed (the same discipline as Black–Scholes’ failure modes).

The failures, in one line each:

1. Wrong-way risk (WWR) - exposure and counterparty credit are assumed independent; in reality default is more likely when exposure is high, so CVA is underpriced.
2. DVA / FBA double-counting - two frameworks price the same negative-exposure benefit twice.
3. Jump-to-default (JTD) - CVA is a smooth function of spread, but real defaults are jumps; the delta hedge cannot protect the jump loss.

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2. Mathematical Ground Truth & Derivations

Where the assumptions live. Every CVA/DVA in this folder assumes, explicitly or implicitly (Gregory 17.2, 17.6):

• (A1) Independence of exposure, default probability, and LGD (no WWR).
• (A2) Exposure is computed unconditionally, not conditional on default.
• (A3) Default is a diffusion - spread gamma is continuous; the close-out is the base value.

(A1)/(A2) fail ⇒ WWR. The correct exposure is $EPE(t,t_i|{\tau }_C=t_i)$ - exposure given the counterparty defaults at $t_i$. With a single correlation $\rho$ between the exposure and credit drivers, corr +50% roughly doubles EPE and −50% at least halves it (Gregory §17.6.2). Crucially, WWR increases with the counterparty’s credit quality - the default of a strong name is a bigger shock - and matters most for the systemic parties that post margin (Pykhtin–Sokol 2013: jump-based WWR erodes the collateral benefit precisely when the margin is posted).

(A3) fails ⇒ JTD. Gregory Eq 21.2:

$$JTDP\&L=-(\text{Current exposure})\times LGD+(\text{CDS hedge notional})\times LG{D}_{CDS}-\text{current xVA contribution}.$$

For an ITM portfolio without a single-name hedge, JTD is always negative - a jump through the exposure is a first-order loss no smooth hedge captures (Gregory §21.2.5).

The double-counting structure (Gregory §18.2.5). DVA (own default avoids paying the derivative) and FBA (negative exposure gives a funding benefit) describe the same cash flow. Burgard–Kjaer reconcile: $FBA\equiv DVA$ when the funding spread equals $LG{D}_P{\lambda }_P$ and all terms are additive at netting-set level - so the frameworks are consistent only if you do not take both. The two acceptable packages:

$$CVA+\underset{\text{symmetric funding}}{\underbrace{FCA+FBA}}\text{or}\underset{\text{bilateral}}{\underbrace{CVA+DVA}}+FCA.$$

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3. Computational Implementation - the failures in numbers

Three experiments, stdlib only.

Experiment 1 - WWR in Monte Carlo. A common stress factor $X$ drives both exposure (loading $\rho$) and default ($X>x$). Conditional-on-default EPE is compared with the unconditional (no-WWR) number.

Experiment 2 - DVA / FBA double-counting.

Experiment 3 - Jump-to-default P&L (Eq 21.2).

WWR is real and asymmetric: at +50% correlation the conditional EPE is 1.85× the unconditional (the independence model undercharges by ~half); at −50% it collapses to 0.34× (right-way risk, overcharge). The JTD experiment shows the hedge is doing real work - it turns a guaranteed −26 loss into +10 - but a hedge needs the right notional and maturity, which is exactly the management problem of 06 · Advanced Extensions.

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

1. The independence delusion (A1 fails). Almost every basic CVA treats exposure, PD and LGD as independent. Under real WWR this underprices CVA - by roughly 2× at +50% correlation. The safe fix is conditional exposure $EPE(\cdot |\tau )$ (Gregory §17.6).
2. Unconditional exposure (A2 fails). Computing EPE from market paths alone is the implicit no-WWR assumption. It is not a neutral choice - it is a bet that default and exposure are uncorrelated.
3. Double-counting DVA with FBA. Including both discounts the negative-exposure benefit twice. Two acceptable packages, never a mix (Gregory §18.2.5).
4. Jumps break the smooth-CVA view (A3 fails). CVA’s spread-gamma is continuous; a real default is a jump. Delta-hedging CVA with CDS protects the spread move, not the jump - the JTD loss above is the unhedged piece (Gregory §21.2.5).
5. Credit limits vs CVA. Credit limits are a portfolio concentration control; CVA is a price. CVA rewards fewer counterparties (netting), limits reward more (diversification) - conflating the two misprices and misallocates (Gregory §3.1.5).

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5. References

• Gregory, The xVA Challenge
• Pykhtin, Michael & Andrew Sokol (2013): Exposure under systematic credit impact
• Rosen, Dan & David Saunders (2012): CVA the wrong way
• Hull & White (2012, 2014) / Kenyon & Green (2014) / Morini & Prampolini (2010): the FVA/DVA double-counting debate.

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6. Connected Graph Bridges

• Back: 04 · FVA & MVA · Index Hub
• Forward: 06 · Advanced Extensions
• Sibling: Credit Risk & the Merton Structural Model · VaR & Expected Shortfall