Basic Prerequisites: Black–Scholes–Merton (derivative valuation) and Credit Risk & the Merton Model (default probability, LGD). (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)
Scope note - two views of xVA. This is the risk/regulatory view: exposure profiles (EE/EPE/PFE), collateral & netting, SA-CCR, FVA/MVA and wrong-way risk as a risk-management problem. The complementary pricing/desk view - xVA as an adjustment to the derivative’s price, marked and hedged by the desk - lives at Pillar 3 · Counterparty Risk & xVA. Same mathematics, different job; use this folder to govern, that one to price.
1. Intuition & Practical Objective
A derivative is a bilateral contract. Between trade date and expiry the two parties each hold a claim on the other, and that claim changes sign as markets move. The credit risk in that claim - the risk that your counterparty is insolvent while owing you money - is counterparty credit risk (CCR). It is unlike lending: a bond exposes you to ≈ par; a swap exposes you to an uncertain, symmetric, path-dependent value that can be zero today and large tomorrow.
This folder is a hub: it (a) gives the fast formula lookup below (job #1 of this pillar), and (b) routes to six sub-pages that walk from raw intuition through exposure measurement, CVA/DVA, collateral/netting/SA-CCR, the failure modes, and the funding/capital extensions (FVA, MVA, KVA).
The central object is CVA - the credit valuation adjustment. It is the market price of counterparty default risk, the number that turns a risk-free derivative value into the value an actual, defaultable counterparty is worth:
The one-sentence essence. “CVA is the product of market risk and credit risk: it is the loss-given-default-weighted average of your positive exposure over the counterparty’s default-probability profile - an integral of an exposure path against a survival curve.”
2. Mathematical Ground Truth & Derivations
Quick-Reference Lookup (job #1). Notation: = portfolio value to the party computing the adjustment; = counterparty default time; = instantaneous default intensity; = recovery; ; = CDS/credit spread; a discount factor; the standard normal CDF.
| Quantity | Formula | Verified check (§3) |
|---|---|---|
| Positive / negative exposure | ; | Gregory (11.1)–(11.2) |
| EFV expected future value | - | |
| EPE expected positive exposure | profile () | |
| ENE expected negative exposure | ||
| PFE potential future exposure | -quantile of (= VaR) | - |
| Unilateral CVA (integral) | Gregory Eq 17.2 | |
| Unilateral CVA (discrete) | $m | |
| Spread approximation | Eq 17.4 | |
| Hazard from spread | Hull Eq 24.2 | |
| Bilateral CVA | $m | |
| CVA (bilateral) | Eq 17.7b | |
| DVA (bilateral) | $m | |
| FVA | (funding cost + benefit) | Eq 18.4 |
| MVA | Eq 20.1 | |
| SA-CCR EAD | BIS (13.16) | |
| SA-CCR replacement cost | BIS (13.19) | |
| SA-CCR PFE | BIS (13.21) |
Cost components by margin arrangement (Gregory Table 16.5): uncollateralised ⇒ CVA, FVA, KVA; collateralised ⇒ CVA (reduced), ColVA, KVA; collateralised with IM ⇒ CVA, ColVA, MVA; cleared ⇒ CVA, KVA, MVA.
The key structural fact. In the standard (no-wrong-way-risk) model, exposure, default probability and LGD are independent, so CVA factorises into a market-risk piece (EPE) and a credit-risk piece (PD × LGD). Wrong-way risk is precisely the failure of that factorisation - see 05 · Failure Modes.
3. Computational Implementation - the CVA formula engine
A minimal CVA/DVA engine: take a tabulated (already discounted) EPE/ENE profile of a 5-year interest-rate swap, a credit spread, and an LGD, and evaluate the discrete product form. Standard library only.
Read the signs literally: the swap is worth 0.72m less than its risk-free value because the counterparty (150bp spread) might default while we are in the money; own default risk adds +0.37m (DVA) back. The two do not cancel - the counterparty is the weaker credit.
4. Failure Modes & First-Principles Breakdowns
Hub signposts - the folder’s failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:
- Wrong-way risk (WWR). The independence assumption behind is false when exposure rises exactly as the counterparty deteriorates. In §3 of 05 a mild exposure/intensity correlation of on the same scale multiplies CVA by 2.4× () - and no amount of collateral reliably covers a jump.
- Exposure misestimation. EPE is a model output (vols, correlations, path dependence, collateral terms). Understating volatility understates CVA proportionally - a model on a true world reports CVA too low.
- Collateral & netting illusion. A zero-threshold CSA does not eliminate exposure: the margin period of risk (MPoR, ~10 days) leaves a residual, and cash-flow “collateral spikes” survive inside it. Netting only helps within a legally enforceable netting set.
- Default-probability provenance. Accounting (IFRS 13) and market practice use risk-neutral (CDS-implied) PDs; capital rules derecognise DVA. Mixing real-world and risk-neutral PDs, or double-counting DVA with FBA, produces a value nobody can hedge.
5. References
- Gregory, Jon: The xVA Challenge: A Valuation Adjustment Framework for Modern Derivatives Markets (5th ed., 2025)
- Brigo, Morini & Pallavicini: Counterparty Credit Risk, Collateral and Funding (2013, Wiley)
- Pykhtin & Zhu: A Guide to Modeling Counterparty Credit Risk (GARP Risk Review, 2007)
- Hull, John C.: Options, Futures, and Other Derivatives
- BCBS (2014): The Standardised Approach for Measuring Counterparty Credit Risk Exposures (BIS d317)
- BCBS (2017): Basel III CVA Risk Framework (BIS d325)
6. Connected Graph Bridges
- Foundational base: Black–Scholes–Merton · Credit Risk & the Merton Model
- Sub-pages (in-folder): 01 From Zero · 02 Exposure (EE/EPE/PFE) · 03 CVA & DVA · 04 Collateral, Netting & SA-CCR · 05 Failure Modes · 06 Advanced Extensions
- Sibling topics: Value at Risk & Expected Shortfall · Liquidity Risk & Margin Spirals · Stress Testing & Scenario Analysis