friction rate

The friction rate $f(X_t,t)$ is the expected rate of loss due to transaction costs from day-to-day delta hedging. It is the fundamental building block of the Hedging Valuation Adjustment — the HVA is simply the discounted integral of the friction rate over the portfolio’s lifetime.

Under a delta-threshold hedging strategy with threshold $D$, volatility $\sigma$, book gamma $\Gamma$, and friction cost function $\psi (N,x)$ per transaction of $N$ units at price $x$, the friction rate is:

$f(X_t,t)\approx {\left(\frac{\sigma (X_t,t)\Gamma (X_t,t)}{D}\right)}^2\psi (D,X_t).$

This follows from two observations: (1) the expected time between rehedging events is $\mathbb{E}[\delta t]\approx (D/\sigma \Gamma {)}^2$, since the delta drifts at rate $\Gamma \cdot \sigma \mathrm{d}W$ and hits the threshold $D$ after time $\sim (D/\sigma \Gamma {)}^2$; and (2) each rehedging event costs approximately $\psi (D,X_t)$. The ratio gives a friction rate that is linear in time: $\mathbb{E}[\mathrm{d}\varphi ]\sim \mathcal{O}(\mathrm{d}t)$.

Key Details

• The friction rate is quadratic in gamma — this is the source of the factor-of-2 multiplier in the new trade HVA and the super-contingency multiplier
• In the multi-counterparty setting, the friction bleed decomposes as $\mathrm{d}\varphi =\mathrm{d}{\varphi }_{\mathrm{drag}}+{\sum }_c{\Psi }_c\mathrm{d}{J}_c$, separating drag friction from default-triggered costs
• The friction cost function $\psi (D,x)=F_0+F_{1}D+F_{2}Dx$ captures fixed costs, per-unit spreads, and proportional costs
• The friction rate defines the PDE source term that distinguishes the HVA equation from the standard Black-Scholes PDE

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concept