4.7.1 Counterparty Risk & xVA from Zero

Basic Prerequisites: Black–Scholes–Merton (a derivative has a value that moves) - nothing else.

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1. Intuition & Practical Objective

This page builds the why of counterparty credit risk and xVA with no prior risk-management knowledge. The objective is one idea: when you trade a derivative, your profit depends on a promise - and promises can be broken at the worst possible moment. Pricing the derivative risk-free is therefore incomplete; you must subtract the expected cost of that broken promise.

Start with the dumbest question: a swap is signed at zero value for both sides - where does the risk come from? Tomorrow the market moves. If the move is in your favour, you are in the money (ITM) - the counterparty owes you, and if they go bankrupt right now you have a claim in the bankruptcy queue. If the move is against you, you owe them, and their bankruptcy frees you from paying. That is the whole asymmetry: exposure is positive or negative depending on the market, and only the positive states hurt you.

Three steps:

1. Exposure is conditional and one-sided. Your loss on default is $\max (V,0)$ - the positive part of the portfolio value. A negative value is not a loss; it is a liability you simply stop paying. So the risk lives entirely in the in-the-money states.

2. Exposure is uncertain and symmetric. A loan exposes you to nearly its full face value from day one. A swap exposes you to a value that starts at zero, wanders, and at default is whatever the market says. The number you are exposed to is itself random - which is why quantifying it needs simulation, not a formula plug-in.

3. CVA is a price, not a provision. The credit valuation adjustment is what the market charges today for that future possibility of loss: the discounted, LGD-weighted, default-probability-weighted average of your positive exposure. It is the market price of counterparty default risk, quoted like a spread (in basis points) and hedged with CDS.

The whole xVA family is the same trick applied to other costs: CVA/DVA = credit, FVA = funding, MVA = initial-margin funding, KVA = capital, ColVA = collateral.

One number to internalise. A 5-year ATM forward, volatility 25%, counterparty spread 150bp, LGD 60%: the risk-free value is $0, but CVA is about −1.05 on a $100 unit notional. That “1%” is the price of the promise, and it is charged before the trade is done.

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2. Mathematical Ground Truth & Derivations

Exposure, defined. Let $V(t)$ be the value (to you) of the portfolio. Gregory’s definitions (Eqs 11.1–11.2):

$$\text{positive exposure}=\max (V(t),0),\text{negative exposure}=\min (V(t),0)\le 0.$$

From these come the four exposure statistics that run through the whole subject:

$$\text{EFV}=\mathbb{E}[V(t)],\text{EPE}=\mathbb{E}\left[V(t{)}^{+}\right],\text{ENE}=\mathbb{E}\left[V(t{)}^{-}\right]\le 0,\text{PFE}=\text{quantile of }V(t{)}^{+}.$$

Note the identity $\text{EPE}+\text{ENE}=\text{EFV}$: the two one-sided means split the expected value. For an at-the-money forward $\text{EFV}\approx 0$, so $\text{EPE}\approx -\text{ENE}$ - the expected gain and expected loss are mirror images.

PFE is a VaR. The potential future exposure at confidence $\alpha$ is exactly a value-at-risk on the portfolio value - the loss exceeded with probability $\le 1-\alpha$ (Gregory §2.6, §11.1.5). This is the bridge to Value at Risk & Expected Shortfall: same machinery, different application.

A worked five-scenario sample (Gregory Spreadsheet 11.1). Five equally likely future values:

$$V=(70,50,30,-10,-30).$$

Then

$$\text{EFV}=\frac{70+50+30-10-30}{5}=22,\text{EPE}=\frac{70+50+30}{5}=30,\text{ENE}=\frac{-10-30}{5}=-8,\text{PFE}=70.$$

None of these four numbers is the others: they answer four different questions about the same distribution. Confusing them is the standard beginner error.

When the value is normal (Gregory Appendix 11A), $V\sim \mathcal{N}(\mu ,{\sigma }^2)$ with $z=\mu /\sigma$:

$$\text{EFV}=\mu ,\text{PFE}(\alpha )=\mu +\sigma {\Phi }^{-1}(\alpha ),\text{EPE}=\sigma \phi (z)+\mu \Phi (z),\text{ENE}=\mu -\text{EPE}.$$

Why CVA exists. Start from “the value is whatever the risk-free model says”. That assumes the counterparty always pays. Relax it: with probability the counterparty defaults (time $\tau$, probability from its survival curve) and you then lose the LGD fraction of whatever you are owed. CVA is that expected, discounted loss:

$$\text{CVA}=-\text{LGD}\mathbb{E}\left[1_{\tau \le T}D(0,\tau )V(\tau {)}^{+}\right].$$

The minus sign is the convention: CVA is a subtraction from the risk-free value. Everything in this folder is an elaboration of that one expectation.

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3. Computational Implementation - metrics and the normal exposure formulas

Part (A) evaluates the four statistics on the five-scenario sample of §2 (reproducing Gregory’s worked numbers exactly). Part (B) evaluates the closed-form normal-distribution exposure formulas. Standard library only.

Read the (B) block: at $\sigma =2$ the expected exposure is $2.17$ on a mean of $2$; double the volatility and EPE rises to $2.79$ and PFE jumps from $6.65$ to $11.31$. Exposure is created by uncertainty - a contract that never moves has zero EPE. (The ENE figures $-0.17,-0.79$ are the magnitudes Gregory quotes as positive.)

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4. Failure Modes & First-Principles Breakdowns

1. The “risk is today’s value” trap. Current value is one point on a distribution. Exposure is about the future: a trade at zero today can have large EPE (the ATM forward above). Confusing current exposure with PFE/EPE is the root error.
2. Sign confusion between PFE, EPE and VaR. PFE is a quantile (a threshold), EPE is a mean (an average). They move in the same direction under volatility but are not interchangeable, and neither is an expected loss.
3. Expected value ≠ risk when the counterparty is weak. A positive EFV is not a windfall if the counterparty may not pay; it is exactly the thing CVA discounts away. Positive exposure and counterparty default are the two ingredients that must be multiplied, not added.
4. One-sided loss, two-sided contract. Because only $\max (V,0)$ is at risk, a naive “notional × PD × LGD” (a loan-style calculation) wildly overstates CCR: derivative exposure is far below notional, and sometimes zero.

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5. References

• Gregory, Jon: The xVA Challenge (5th ed., 2025)
• Pykhtin & Zhu: A Guide to Modeling Counterparty Credit Risk (2007)
• Hull: Options, Futures, and Other Derivatives

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6. Connected Graph Bridges

• Base: Black–Scholes–Merton · Credit Risk & the Merton Model
• Continue: PFE · Index Hub
• Sibling: Value at Risk & Expected Shortfall (PFE is a VaR)