3.10.2 Exposure, Netting & Margin

Basic Prerequisites: 01 · From Zero.

1. Intuition & Practical Objective

Exposure is the raw material every xVA term is built on, and margin is the single most powerful tool for reshaping it. This page makes both precise. The practical objective: know, at every future time $t$ , what your counterparty could owe you (positive exposure), how to shrink it with netting and collateral, and how much residual exposure survives after margin - because that residual, over the margin period of risk (MPoR) , is exactly what the next pages price.

Two ideas drive everything:

Exposure metrics are integrals of the future value distribution. EPE, ENE, PFE are all just summaries of $V (t)$ ‘s distribution at each horizon.
Margin converts exposure into a gap risk. Variation margin (VM) tracks the running value; initial margin (IM) absorbs the MPoR gap - the moves between the last margin call and close-out. A zero-threshold, perfectly-collected two-way margin can drive exposure to near zero, but never exactly zero, because of the MPoR (Gregory Ch 7, Ch 9).

2. Mathematical Ground Truth & Derivations

2.1 Exposure metrics (Gregory Ch 11, App. 11A)

For a portfolio value $V (t) \sim N (μ, σ)$ at horizon $t$ , with $z = μ / σ$ :

EF V (t) = μ, PF E_{α} (t) = μ + σ Φ^{- 1} (α), EPE (t) = σ ϕ (z) + μ Φ (z), ENE (t) = σ ϕ (z) - μ Φ (- z) .

PFE is the $α$ -quantile of $V$ - exactly a Value-at-Risk number (Gregory 2.6; Hull 24.7). PFE(99%) is exceeded with probability $\leq 1%$ .
EPE averages all values with negatives set to zero; ENE averages the negatives (it is your counterparty’s positive exposure).
Both EPE and ENE rise with volatility $σ$ - an out-of-the-money option still has positive EPE because volatility can push it in-the-money. ENE $= σ ϕ (z) - μ Φ (- z) \leq 0$ , and EPE + ENE = EFV (partition of the value).

2.2 Netting (Gregory Ch 6, Hull 24.7)

Under one netting agreement the exposure is on the net value, so for a netting set with trades $j$ :

EP E^{NS} (t) = E [max (j \sum V^{j} (t), 0)] \leq j \sum EP E^{j} (t) .

Netting benefit grows with correlation: two offsetting trades (opposite sign) have a large netting reduction; directional trades have a small one. Netting is what makes CVA a netting-set-level quantity - you cannot sum standalone trade CVAs and get the right portfolio CVA (Gregory 17.4; the standalone sum $\geq$ netting-set CVA, Eq 17.10).

2.3 Margin / collateral (Gregory Ch 7)

Variation margin (VM) tracks the base valuation; initial margin (IM) is extra margin, independent of value, absorbing the MPoR risk. “Initial margin relates to the variability of the value rather than the value itself” (Gregory §7.3.4). IM is the mathematical opposite of a threshold: IM ≡ a negative threshold, so TH and IM are never used together.

Credit support amount (two-way VM, Gregory Eq 7.3):

CS A = max (V - K_{C}, 0) - max (- V - K_{P}, 0) - C,

where $K_{C}, K_{P}$ are the two parties’ thresholds and $C$ the margin already held. The MPoR model (Gregory Eq 15.3) - the residual exposure is the value minus the margin that was posted a full MPoR ago:

Exposure_{t} = max (V_{t} - C_{t - MP o R}, 0),

with $C$ positive if you receive margin, negative if you post it. Two reasons margin can’t eliminate exposure: (i) threshold undercollateralisation, and (ii) the MPoR/MTA discrete tracking error (Gregory §7.3.7). Regulatory MPoR: 5 days centrally cleared / repos, 10 days bilateral OTC (Gregory Ch 9; Hull 24.7).

2.4 Initial margin sizing (Gregory Ch 9, Hull 24)

The variance-covariance IM (Gregory §9.4.3) is a parametric VaR:

I M_{α, τ} = Φ^{- 1} (α) τ σ_{P},

with $Φ^{- 1} (0.99) = 2.33$ , $10/252 = 0.1992$ . Regulatory IM must cover a 99% one-tailed, 10-day horizon using data incorporating a significant stress period (BCBS–IOSCO 2015). The ISDA SIMM generalises this to a nested sequence of variance-covariance calculations over six risk classes (IR, credit-qualifying, credit-non-qualifying, equity, commodity, FX), with weighted sensitivities $W S = R W \times s \times CR$ (risk weight × net sensitivity × concentration factor) aggregated within a class by correlations - e.g. two-tenor delta margin $W S_{1}^{2} + W S_{2}^{2} + 2 ρ W S_{1} W S_{2}$ (Gregory Table 9.7). The standardised schedule is far more conservative (Gregory §9.4.1: >$8trn of schedule-based margin estimated for the in-scope population), which is why SIMM was built.

3. Computational Implementation - IM sizing and the SIMM aggregation

Stdlib only; reproduces Gregory’s re-verified variance-covariance cases and the SIMM two-tenor delta example.

The IM table shows the same $σ_{P} = 100$ at different horizons/levels, exactly the Gregory worked examples; the SIMM line shows how correlation (84% between the two tenors) shrinks margin below the gross sum.

4. Failure Modes & First-Principles Breakdowns

Margin call frequency ≠ MPoR. A daily margin call is consistent with a 10-day MPoR; the MPoR is a model parameter (delay to declare default + liquidation), not the call period. Confusing them understates the residual exposure (Gregory §7.3.3, §15.5.2).
MTA ⇒ path dependency. With a minimum transfer amount, today’s margin balance depends on the previous balance (Tables 7.6–7.7) - margin must be modelled on a continuous grid or with look-back points, not as an independent point estimate.
Threshold + MTA are additive. Margin is only called once exposure exceeds $T H + MT A$ ; adding rather than combining them is a classic implementation slip (Gregory §7.3.4).
IM is not VM. Subtracting initial margin from exposure to zero (as the old CEM allowed) drove EAD to zero - a key reason SA-CCR replaced it (Gregory 13.4.2). IM covers the MPoR gap, VM covers the running value; they are different risks.

5. References

Gregory, The xVA Challenge
Hull, Options, Futures, and Other Derivatives
BCBS–IOSCO (2015): Margin requirements for non-centrally cleared derivatives

6. Connected Graph Bridges

Back: 01 · From Zero
Forward: 03 · CVA & DVA · 04 · FVA & MVA · Index Hub
Sibling: VaR & Expected Shortfall (PFE = VaR)

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