4.7.2 Exposure Profiles

Basic Prerequisites: 01 · From Zero and Black–Scholes–Merton (lognormal simulation).

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1. Intuition & Practical Objective

CVA needs one input more than any other: the exposure profile - the expected positive value of the portfolio at every future date, conditional on the counterparty having survived so far. This page builds that profile from scratch and separates the four statistics a desk actually quotes:

$$\underset{\text{expected exposure}}{\underbrace{\text{EE}(t)=\mathbb{E}[V(t{)}^{+}]}},\underset{\text{expected positive exposure}}{\underbrace{\text{EPE}=\frac{1}{T}{\int }_0^T\text{EE}(t)dt}},\underset{\text{expected negative exposure}}{\underbrace{\text{ENE}(t)=\mathbb{E}[V(t{)}^{-}]}},\underset{\text{quantile}}{\underbrace{{\text{PFE}}_{\alpha }(t)}}.$$

The practical objective is the shape, because the shape is the answer. A forward’s exposure grows monotonically (the further out you look, the more the spot can move away from the strike). A swap’s exposure is humped - large in the middle, collapsing to zero at maturity, because near maturity there are few remaining payments to differ. Quantifying that shape is the job of a Monte Carlo over risk-factor paths.

Why simulation, not a formula? The exposure at date $t$ is $\max (V(t),0)$ where $V$ depends on the whole future path of rates/FX/vol and on collateral terms. EPE is an option-like quantity - and for a swap it is exactly a strip of co-terminal (“diagonal”) swaptions (Sorensen–Bollier 1994). That is the deep point: exposure quantification is at least as hard as pricing the instrument.

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2. Mathematical Ground Truth & Derivations

2.1 Defining the profile and averaging it

For a grid $0=t_0<t_1<\cdots <t_m=T$:

$$\text{EE}(t_i)=\mathbb{E}\left[\max (V(t_i),0)\right],\text{EPE}=\frac{1}{m}\sum _{i=1}^m\text{EE}(t_i)(\text{time-weighted if uneven}),$$ $$\text{ENE}(t_i)=\mathbb{E}\left[\min (V(t_i),0)\right],{\text{PFE}}_{\alpha }(t_i)=\inf \{x:\mathbb{P}(V(t_i{)}^{+}\le x)\ge \alpha \}.$$

For CVA the profile is taken discounted to today: replace $\text{EE}(t_i)$ by $D(0,t_i)\text{EE}(t_i)$ (Gregory §17.2.2, citing Jamshidian 1989 to avoid a convexity bias).

2.2 A tractable example: the at-the-money forward

Take a forward to buy the asset at $K$ at time $T$, on a spot that follows geometric Brownian motion under $\mathbb{Q}$:

$$d{S}_t=r{S}_{t}dt+\sigma S_{t}d{W}_t,S_t=S_0\exp \left[(r-\frac{1}{2}{\sigma }^2)t+\sigma \sqrt{t}Z\right].$$

Set $K=S_0{e}^{rT}$ (the no-arbitrage forward price, so the forward is struck at-the-money and worth zero today). The value at $t$ is the discounted forward gap

$$V(t)=S_t-K{e}^{-r(T-t)}=S_t-S_0{e}^{rt},$$

so $\mathbb{E}[V(t)]=0$ and the forward has symmetric exposure. Because $S_t$ is lognormal, the expected positive exposure has a clean closed form:

$$\boxed{\text{EE}(t)=S_0{e}^{rt}\left[2\Phi \left(\frac{\sigma \sqrt{t}}{2}\right)-1\right]}$$

(derivation: for lognormal $X$ with mean $S_0{e}^{rt}$, $\mathbb{E}[(X-K{)}^{+}]=\mathbb{E}[X]\Phi (d_1)-K\Phi (d_2)$ with $d_1=\sigma \sqrt{t}/2,d_2=-\sigma \sqrt{t}/2$).

Two immediate consequences, both visible in the numbers below:

• EE(t) increases with $t$ - the profile of a forward is monotone, unlike a swap’s hump.
• EE(t) increases with $\sigma$ - exposure is manufactured by uncertainty. At $\sigma =40\%$ the 5-year EE is roughly four times the $\sigma =10\%$ value.

2.3 Why a swap looks different

An interest-rate swap’s exposure is humped because it is a portfolio of co-terminal swaptions: each future floating payment fixes against a forward that can move, so the value-to-be-replaced peaks in mid-life and goes to zero as the last payments wash out (Gregory §15.1.3). The forward above is the atom; a swap is a sum of such atoms with a decaying annuity weight.

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3. Computational Implementation - an exposure profile by Monte Carlo

Simulate the ATM forward of §2.2, tabulate EE/ENE, compute the average EPE and the 95% PFE, and confirm the Monte Carlo against the closed form. Standard library only.

Three things to verify by eye: (i) the Monte Carlo EE tracks the closed form within sampling error (~0.5%); (ii) $\text{EPE}+\text{ENE}=0$ to within noise (symmetric ATM forward); (iii) the 95% PFE (133.7) is far above the average EPE (16.4) - a reminder that PFE answers a different question from EPE, and that most of the tail sits in a small fraction of paths.

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4. Failure Modes & First-Principles Breakdowns

1. Roll-off / grid misses. Exposure jumps exist at cash-flow dates, exercise dates, break clauses and resets. A coarse time grid that skips them understates EPE (Gregory §15.3.1 “roll-off risk”). Put critical dates into the grid or use look-back points.
2. Parameter misestimation propagates linearly. EPE scales roughly with volatility; a volatility estimate that is 40% too low produces a CVA that is ~40% too low. Better to be conservative here than precise.
3. Wrong measure. CVA/FVA are risk-neutral computations (market-implied vols and drifts); PFE/IMM are often physical. Mixing them in one Monte Carlo silently biases both (Gregory §15.3.3).
4. The ATM-forward intuition does not generalise. A swap’s profile is humped and can be much larger than a forward’s; assuming monotone growth, or treating EPE as a constant, mis-states the mid-life peak that dominates CVA.
5. Correlation = exposure too. For multi-currency or multi-asset portfolios the EPE depends on correlations, which are far harder to estimate than volatilities. Under-modelling correlation is the quiet source of exposure surprise (see 05 · Failure Modes).

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5. References

• Gregory, Jon: The xVA Challenge (5th ed., 2025)
• Sorensen & Bollier (1994): the co-terminal-swaption representation of swap exposure (EPE = swaption payoff × risky duration).
• Pykhtin & Zhu (2007): A Guide to Modeling Counterparty Credit Risk
• Hull, Options, Futures, and Other Derivatives

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6. Connected Graph Bridges

• Back: 01 · From Zero
• Forward: 03 · CVA & DVA · Index Hub
• Base: Black–Scholes–Merton · Stochastic Calculus & Itô