The Recalibration Conundrum - Hedging Valuation Adjustment for Callable Claims

Abstract

The dynamic hedging theory only makes sense in the setup of one given model, whereas the practice of dynamic hedging is just the opposite, with models fleeing after the data through daily recalibration. This is quite of a quantitative finance paradox. In this paper we revisit Burnett (2021) & Burnett and Williams (2021)‘s notion of hedging valuation adjustment (HVA), originally intended to deal with dynamic hedging frictions, in the direction of recalibration and model risks. Specifically, we extend to callable assets the HVA model risk approach of Bénézet and Crépey (2024). The classical way to deal with model risk is to reserve the differences between the valuations in reference models and in the local models used by traders. However, while traders’ prices are thus corrected, their hedging strategies and their exercise decisions are still wrong, which necessitates a risk-adjusted reserve. We illustrate our approach on a stylized callable range accrual representative of huge amounts of structured products on the market. We show that a model risk reserve adjusted for the risk of wrong exercise decisions may largely exceed a basic reserve only accounting for valuation differences.

Summary

This paper extends the HVA (hedging valuation adjustment) concept from European to callable claims in the context of model risk. HVA is defined as $\text{HVA}=-\text{va}(\text{pnl})$: the fair-value correction needed to restore the martingale property of the trader’s P&L process. For callable claims, HVA includes not only the valuation gap between local and reference models but also the expected cost of suboptimal exercise decisions made under the wrong model.

The paper treats HVA as a full, non-perturbative, O(1) quantity — numerically, HVA = 181 on notional 100 for a “bad trader.” This is a fundamentally different concept from the friction-based HVA in the XVA HJB paper.

Key Contributions

• Extension of HVA model risk framework from European to callable claims
• Decomposition of HVA into misvaluation term + expected cost of wrong exercise + supermartingale correction (Proposition 2.1, eq. 10)
• “Darwinian model risk” framework: adverse selection of models by traders
• Numerical demonstration that model risk reserve including exercise error can vastly exceed simple valuation differences

Key Findings

• HVA${}_0^{\text{bad}}$ = 181, HVA${}_0^{\text{nsb}}$ = 69 on notional 100 — HVA exceeds the notional for a “bad” trader
• KVA (capital for model risk tail) is dominated by HVA by factor > 4 in the callable case
• The “not-so-bad” trader (who switches to the reference model when the local model fails) has roughly half the HVA
• Model risk reserve including exercise error “may largely exceed” a basic valuation-difference reserve

Critical Notes

Different definition of HVA from the XVA HJB paper

The XVA HJB paper defines HVA as the cost of hedging frictions (market impact), entering at $O(\epsilon )$ in their perturbative expansion. Bénézet-Crépey-Essaket define HVA as a model risk adjustment — the cost of using the wrong model for hedging and exercise. These are conceptually different quantities that happen to share the same acronym. The Crépey HVA is O(1) and can exceed the notional; the XVA HJB paper’s HVA is a friction penalty that vanishes as frictions vanish. Both definitions trace back to Burnett (2021), who introduced HVA for hedging frictions — the Crépey group has since reinterpreted it for model risk. The XVA HJB paper’s claim that “HVA is $O(\epsilon )$, not $O(1)$” is valid for friction-HVA under their scaling, but does not apply to model-risk-HVA.

Atomic Notes

• XVA perturbative expansion

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