3.10.6 Advanced Extensions

Basic Prerequisites: 05 · Failure Modes.

1. Intuition & Practical Objective

CVA/DVA/FVA/MVA price the economic costs of a derivative. But a derivative also consumes regulatory capital (a charge the bank must hold, costing shareholders a return), and the entire xVA family has to be managed - priced, hedged, and allocated by a dedicated desk. This page is the launchpad to the practitioner layer: CVA capital under Basel/FRTB, wrong-way risk modelling, and the operation of an xVA desk. The objective is to know what a real bank’s xVA function actually computes and why.

Three ideas structure the page:

Capital is a different animal. The CVA capital charge capitalises the volatility of CVA (a market-risk charge), separate from and generally additive to the default-risk charge, and it derecognises DVA (Gregory §13.3). It is portfolio-level - a single number across counterparties - which creates a pricing problem.
WWR is the modelling frontier. Intensity models cannot reproduce observed wrong-way behaviour (quanto effects, sovereign jumps); the strongest evidence is jump-based, not diffusion (Gregory §17.6.4).
The xVA desk is a utility, not a profit centre. It prices, hedges, and explains xVA P&L, intermediates with Treasury, and optimises across the xVA terms - a cross-asset credit-hybrid business (Gregory Ch 21).

2. Mathematical Ground Truth & Derivations

2.1 CVA capital (Gregory Ch 13; Hull 24)

The default-risk (IRB) capital charge is the familiar $RC = E A D \times L G D \times (P D_{99.9%} - P D) \times M A$ (Gregory Eq 13.1, the Vasicek large-homogeneous-pool / Gordy form). The CVA capital charge is separate and capitalises CVA volatility. The BA-CVA (basic) reduced form (Eq 13.3):

K_{re d u ce d} = ρ (c \sum SC V A_{c})^{2} + (1 - ρ^{2}) c \sum SC V A_{c}^{2}, SC V A_{c} = \frac{1}{α} R W_{c} NS \sum M_{NS} E A D_{NS} D F_{NS},

with supervisory correlation $ρ = 50%$ , $α = 1.4$ , risk weights $R W_{c}$ by sector/rating bucket, and $D F_{NS}$ a maturity discount. Crucial diversification property: for $n$ equal counterparties (Gregory Eq 13.23–13.24),

K_{per cpty} = SC V A \cdot ρ^{2} + \frac{1 - ρ ^{2}}{n} n \to \infty SC V A \cdot ρ,

so diversification is only half-rewarded (capital floors at 50% of the gross sum, never zero) - Basel does not let you diversify away CVA risk. SA-CVA (the FRTB-style standardised version) is sensitivity-based: per-bucket $K_{b} = \sum_{k} W S_{k}^{2} + \sum_{k \neq = l} ρ_{k l} W S_{k} W S_{l} + R \sum_{k} [W S_{k}^{H d g}]^{2}$ then $\sum_{b} K_{b}^{2} + \sum_{b \neq = c} γ_{b c} K_{b} K_{c}$ scaled by $m_{C V A}$ (Eqs 13.10–13.12); the hedging disallowance $R = 0.01$ (√R = 10% of gross hedge sensitivities) stops perfect hedging.

2.2 Wrong-way risk modelling (Gregory 17.6)

Three approaches:

Intensity approach: a stochastic credit-spread process correlated with the exposure drivers; default generated from the intensity; conditional EPE from defaulted paths. Tractable but weak - even ±100% correlation understates the true effect.
Structural approach: map exposure and default onto a bivariate distribution (e.g. Gaussian copula); no revaluation needed; stronger effect, opaque calibration.
Jump approaches: the strongest empirical support - quanto effects in CDS markets and sovereign FX (Italian CDS 2011; Levy–Levin implied sovereign jumps: 83% for AAA down to 27% for BBB). Intensity (diffusion) models cannot reproduce this (Ehlers–Schönbucher 2006). This is the WWR-theoretical cliff-edge.

2.3 The xVA desk (Gregory Ch 21)

The desk prices incremental xVA, hedges it (manage xVA like an option book), and explains its P&L. The P&L explain identity (Gregory Eq 21.1 / Table 21.8):

δ x V A = \frac{\partial x V A}{\partial S} Δ S + \frac{\partial x V A}{\partial E} Δ E + \frac{\partial x V A}{\partial t} Δ t + \frac{\partial ^{2} x V A}{\partial S \partial E} Δ S Δ E,

where the cross-gamma term (joint moves in exposure and credit - e.g. Brexit: rates down, spreads wider) is typically unhedgeable. The desk’s P&L reconciles theta (time decay) + deltas + gamma + defaults + funding/capital costs to the traded P&L. KVA vs MVA trade-off (Gregory §20.4): posting IM raises MVA but lowers KVA; the optimum is below full regulatory IM because KVA relief under SA-CCR has diminishing returns.

3. Computational Implementation - capital, P&L explain, and the IM/KVA optimum

Stdlib only; reproduces Gregory’s re-verified BA-CVA diversification and the Table 21.8 P&L arithmetic.

Three practitioner results, all visible: Basel’s BA-CVA half-rewards diversification (capital floors at 50% of the gross sum); a real xVA desk P&L reconciles exactly (theta + deltas + UX = traded P&L, where UX absorbs gamma/cross-gamma/vega); and the optimal posted IM is ~75% of full regulatory - posting some IM buys cheap capital relief, posting all of it costs more MVA than the remaining KVA is worth (Gregory §20.4).

4. Failure Modes & First-Principles Breakdowns

DVA in capital. CVA capital must derecognise DVA (BCBS 2011d), while accounting (IFRS 13) requires it - treating them consistently inside capital is an error that overstates or double-counts own-credit (Gregory §13.3.1).
CVA hedges can increase capital. Under BA-CVA/SA-CVA, proxy and misaligned hedges consume more capital than they relieve (the “split hedge” / overhedging problem, §13.3.6, §13.5.3). Hedging accounting CVA and minimising CVA capital are different objectives.
WWR via diffusion. Intensity (diffusion) WWR models understate the true jump behaviour; sovereign/quanto evidence demands jump models (Gregory §17.6.4). A diffusion-only WWR is systematically under-pessimistic.
The 5% floor. SA-CCR capital can never reach zero even with huge IM or very negative value ( $Fl oor = 5%$ , Eq 13.21) - pricing as if capital vanishes with perfect collateralisation misprices (Gregory §19.2.3, §13.4.3).
xVA desk as profit centre. A desk targeted at positive P&L will warehouse credit risk (profitable in expectation via the real-world/risk-neutral PD gap) and inflate short-term volatility - Gregory’s consensus is a utility with a zero/slightly-negative P&L target (§21.1.4).

5. References

Gregory, The xVA Challenge
Hull, Options, Futures, and Other Derivatives
Brigo & Mercurio, Interest Rate Models
Basel Committee (BCBS 2015, 2017): The standardised approach for measuring counterparty credit risk exposures (SA-CCR) and the FRTB-CVA framework; BCBS–IOSCO (2015) bilateral margin rules.

6. Connected Graph Bridges

Back: 05 · Failure Modes · Index Hub
Credit theory: Credit Risk & the Merton Structural Model · Stress Testing & Scenario Analysis
Risk measures: VaR & Expected Shortfall · Extreme Value Theory & Fat Tails
Base: Black–Scholes–Merton · Interest Rate Models

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