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Reconciling Discounting Adjustment Signs in XVA Formulas

Article Quant Q&A · Author: solid
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Summary

The note compares a discounting adjustment expressed as the expected discounted integral of the difference between a target rate and a base rate, multiplied by exposure, with a formula attributed to Piterbarg for a fully collateralized position. The apparent sign conflict comes from an error in the version of the Piterbarg formula quoted in the question: it contains a sign mistake, so the two expressions can be consistent once that formula is corrected.

The answer also emphasizes that the adjustment depends on clearly defining the base and target price representations, since those define the adjustment itself. The target discounting rate is general and may correspond to a funding rate in the example, but that identification is not universal. The note gives a conceptual resolution rather than a full derivation; readers should check the underlying formula and definitions before comparing signs across sources.

Key ideas

  • The apparent sign discrepancy is attributed to an incorrectly stated Piterbarg formula.
  • The adjustment must be defined relative to explicit base and target price representations.
  • The target discounting rate can equal a funding rate in some settings, but need not do so generally.
  • Matching notation and correcting the formula are necessary before comparing XVA expressions.

Tags

Fixed incomeDerivatives pricingRisk management
Full text
# Discounting price adjustment (XVA)


# Discounting price adjustment (XVA)












In the paper The Fundamental Representation of Pricing Adjustments (2025), the authors describe a fundamental representation of price adjustments and then compare this to well-known adjustments in the literature. In Chapter 3.2 ("Discounting: Piterbarg (2010)") of their paper, they analyze the discounting adjustment, where the P&L bleed is given by $Z = - (\hat{R} - R)V$.

They write the value adjustment as:

$$ U_0 = -E_0 \left( \int_0^T \exp \left( -\int_0^t \hat{R}_s ds \right) (\hat{R}_t-R_t)V_tdt \right) $$

and affirm that this expression is equivalent to the one in Piterbarg (2010), equation (3.2) in the fully collateralized case $C = V$, using the identifications $\hat{R} = r_F$ (funding rate), and $R = r_C$ (collateral rate)

$$ V_0 = \mathbb{E}_0 \left( \int_0^T e^{-\int_0^t r_F(s) ds} V_T \right) + \mathbb{E}_0 \left( \int_0^T e^{-\int_0^t r_F(s) ds} (r_F(t) - r_C(t)) C_t \, dt \right) $$

I feel like I might be missing something. The two formulas appear to differ in sign.

Can someone clarify whether the two formulas are indeed equivalent and help reconcile the notational differences?

## Answer by Ryan (score 1, accepted)

https://quant.stackexchange.com/a/83859

The fundamental representation stated in this question is consistent with that article, and indeed reduces to a corresponding formula from Piterbarg (2010). The problem is that the formula stated in this question for the "corresponding formula" from Piterbarg (2010) is not correct; it's suffered from a minus mishap. To improve the question, it would be worth stating what the base ($V$) and target ($\hat V$) price representations are (since these define the adjustment $U$ being discussed). Then the question should practically answer itself.

Regarding notation, $\hat R$ represents a very general target discounting rate. In certain situations, this will coincide with one's funding rate, like in this example. But it won't always. This should be clear from the other examples in the paper.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.