Hedging Valuation Adjustment - Fact and Friction

Abstract

We develop a simple and generic expression for the impact of transaction costs on the value of a derivative portfolio, expressed as a ‘Hedging Valuation Adjustment’ (HVA). We provide expressions for the HVA in two cases: when it is included as part of the value to be hedged, and when it is left as an unhedged reserve. The hedged case shows an interesting feature we term ‘imaginary volatility’. We show numerical results, and extend the formalism to the pricing of individual trades and to the multi-asset case.

Summary

This foundational paper introduces the concept of a Hedging Valuation Adjustment (HVA) — a valuation adjustment that captures the expected cost of transaction friction (bid-ask spreads) incurred by a delta-hedging strategy. The key insight is that friction costs can be treated as an adjustment to the frictionless price, analogous to how CVA, FVA and other XVAs adjust the risk-free value. The HVA is driven by the book’s gamma squared: $U\propto -\int \mathbb{E}[{\mathcal{D}}_r\cdot (\sigma \Gamma /D{)}^2\cdot \psi ]\mathrm{d}u$, where $\Gamma$ is the portfolio gamma, $D$ the delta threshold for rehedging, $\sigma$ the volatility, and $\psi$ the friction cost function.

The paper develops two variants: a hedged HVA (where the HVA itself is included in the value being hedged, evaluated under the risk-neutral measure $\mathbb{Q}$) and an unhedged HVA (treated as a reserve, evaluated under the real-world measure $\mathbb{P}$). The hedged case introduces a remarkable feature: the total value $\hat{V}$ satisfies a Black-Scholes-type PDE but with an imaginary effective volatility ${\hat{\sigma }}^2={\sigma }^2(1-2{\hat{V}}_{xx}\psi /D^2)$, which can become negative for options with large positive gamma and high friction costs. This imaginary vol implies the option value can become negative — a dramatic admission that the no-arbitrage hedging argument breaks down when friction overwhelms the hedging benefit.

The formalism extends naturally to multi-asset portfolios via a trace formulation $U=-\int \mathbb{E}[{\mathcal{D}}_r\mathrm{Tr}(\mathbf{F}\mathbf{\Gamma }\mathbf{\Sigma }\mathbf{\Gamma })]\mathrm{d}u$, where $\mathbf{F}$ is a relative cost matrix and $\mathbf{\Sigma }$ the covariance matrix. The extension to new trade pricing yields a “new trade HVA” that is approximately twice the naive single-trade estimate, due to the quadratic dependence on book gamma.

Key Contributions

• Introduction of the Hedging Valuation Adjustment (HVA) as a new XVA-type quantity measuring expected friction costs
• Derivation of closed-form Feynman-Kac representations for both hedged ($\mathbb{Q}$-measure) and unhedged ($\mathbb{P}$-measure) HVA
• Discovery of “imaginary volatility” — the effective volatility becomes imaginary when friction costs are sufficiently large, implying option values can become negative
• Extension to multi-asset books via the trace formulation with relative cost matrix
• Discrete-time correction using Siegmund’s overshoot: $D^{\prime }=D+\beta |\Gamma |\sigma \sqrt{\Delta t}$ with $\beta \approx 0.5826$
• New trade HVA allocation showing the marginal impact is approximately $2\times$ the naive estimate due to quadratic gamma dependence

Methodology

The derivation starts from a standard hedging portfolio $\Pi =\hat{V}+aX+b$ with a delta-threshold rehedging strategy: buy/sell $D$ units of underlying whenever the net delta exceeds $D$. The expected P&L is set to zero over each rehedging interval $\delta t$, yielding a modified Black-Scholes PDE with a source term $(\sigma \Gamma /D{)}^2\psi (D,X_t)$. The friction rate $f(X_t,t)\approx (\sigma \Gamma /D{)}^2\psi$ arises from the expected rehedging frequency ${\mathbb{E}}_t[\delta t]\approx (D/\sigma \Gamma {)}^2$ and expected cost $\psi (D,X_t)$ per event. Splitting $\hat{V}=V+U$ (risk-free value plus HVA) and applying the nonlinear Feynman-Kac formula yields the HVA as a discounted expectation of the friction rate. The unhedged case differs in hedging only $V$ (not $\hat{V}$), leading to a convection-diffusion PDE under $\mathbb{P}$ rather than $\mathbb{Q}$.

Key Findings

• The HVA has the form of a discounted integral of friction rate, proportional to ${\sigma }^2{\Gamma }^2\psi /D^2$
• For the hedged case, the HVA is evaluated under $\mathbb{Q}$; for the unhedged case, under $\mathbb{P}$
• Imaginary volatility arises when ${\hat{\sigma }}_{\hat{V}}^2={\sigma }^2(1-2{\hat{V}}_{xx}\psi /D^2)<0$, implying positive theta and potentially negative option values
• Discrete-time monitoring introduces a correction via the Siegmund overshoot factor, which is numerically very accurate
• Numerical tests on 20Y ATM options show the HVA with discrete adjustment keeps average P&L flat at zero across varying thresholds, rates, and drifts
• The multi-asset HVA mixes risks through the covariance matrix, projecting portfolio-level cross-gammas onto individual asset hedging costs

Important References

1. Leland 1985 - Option pricing with transactions costs — seminal paper on option pricing with transaction costs that introduced the modified volatility approach
2. Burgard and Kjaer 2013 - Funding Strategies Funding Costs — framework for FVA that the HVA formalism is designed to be consistent with
3. Deep xVA Solver - A Neural Network Based Counterparty Credit Risk Management Framework — deep BSDE approach noted as potentially applicable to the nonlinear HVA PDE

Atomic Notes

• Hedging Valuation Adjustment
• friction rate
• imaginary volatility
• delta-threshold hedging strategy
• new trade HVA
• multi-asset HVA trace formulation

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