The curvature charge

RiskAtoms Bank's equity desk owns 1,000 UK index puts. If the index fell 15%, their delta says they would gain £2.89m. Revalued in full they gain £7.11m: £4.22m more than delta sees. For the bank that is good news. Had it sold the puts instead, the same £4.22m would be a loss that the delta charge of L4.3.02: The sensitivities-based method: delta and vega never counted. The curvature charge counts it.

What delta misses

The delta charge treats every position as a straight line: risk weight times sensitivity. An option's value is a curve. L1.4.09: Convexity called the bend convexityConvexity: the bend in a value curve. A convex position gains more than its delta says when the market moves either way., and L3.2.06: Option Greeks across a book measured it across the desk as gammaGamma: the second derivative of value with respect to the underlying. Long options have positive gamma, short options negative.. For a big move, the gap between the curve and the line can be large. The SBM therefore adds a curvature chargeCurvature charge: the SBM's charge for the part of an option's move that delta does not capture, measured by full revaluation under an up and a down shock., one per risk class, alongside delta and vega. This is the separate charge for options promised in L1.7.06: Options: calls and puts and L2.5.04: Sensitivities.

Shock, revalue, subtract delta

For each risk factor, shock it up and down and revalue every position in full. For equity and FX the shock is a relative move equal to the delta risk weight; for rates, credit spreads and commodities it is a parallel shift by the highest delta risk weight in the bucket (MAR21.98–21.99). Then take away the part delta already charged, RW × s. What is left, with the sign flipped so that a loss is positive, is the curvature risk positionCurvature risk position (CVR): minus the loss beyond delta when a risk factor is shocked by its risk weight. Positive means the position loses more than delta predicts., CVR+ for the up shock and CVR− for the down shock (MAR21.5). For the bank's puts, bucket 12, RW = 15%:

Both CVRs are negative: in each direction the puts do better than delta says. Going up, they can lose no more than their £0.95m of value, while delta predicts a loss of £2.89m; going down, they gain more and more as they move into the money. That is long gamma.

Max and floors

The curvature formulas are built so that convexity can only cost capital, never earn it. Three devices do it (M-A11: Min, max, floors and caps):

• In the squares, each CVR is floored at zero first: $\max(CVR_k, 0)^2$. A negative CVR adds nothing.
• In the cross terms, the switch ψψ (psi): a switch in the curvature formulas that is 0 when both CVRs in a cross term are negative and 1 otherwise, so two gains cannot offset anything. is 0 when both CVRs are negative, so two gains cannot combine into a negative number that cuts the charge. The correlations are the delta correlations squared (MAR21.100): 64% between two indices in bucket 12.
• The bucket takes the worse side, $K_b = \max(K_b^{+}, K_b^{-})$, and the whole thing sits under $\sqrt{\max(0, \cdot)}$.

With one factor in one bucket, as on this desk, it all collapses to $\max\big(\max(CVR^{+},0),\ \max(CVR^{-},0)\big)$. For the bank's puts: max(0, 0) = 0. That is the zero in the SBM table of L4.3.02: The sensitivities-based method: delta and vega in every scenario, and in L4.1.04: Risk-weighted assets.

For risk factor k and instruments i with delta sensitivity $s_{ik}$ (MAR21.5, 21.98–21.101; the formulas are images in MAR21 and were read as text in the PRA's Market Risk: Advanced Standardised Approach (CRR) Part, Article 325g):

• $x_k^{(RW\pm)}$: risk factor k shocked up or down by the curvature risk weight; $\psi(x, y) = 0$ if $x < 0$ and $y < 0$, else 1.
• $\rho_{kl}$ and $\gamma_{bc}$: the delta correlations, squared (MAR21.100–21.101), then moved for each correlation scenario as in L4.3.02: The sensitivities-based method: delta and vega. If $K_b^{+} = K_b^{-}$, the side with the larger sum of CVRs is used.
• The link to gamma is a second-order Taylor expansion (M-B03: Taylor expansion): $V(x+h) - V(x) - h V'(x) \approx \tfrac12 h^2 V''(x)$, so $CVR^{\pm} \approx -\tfrac12\, \Gamma\, (RW\, x)^2$, the same on both sides. The exact CVRs differ on each side because gamma itself changes over a 15% move.

Start with the bank's puts: two green gaps, no charge. Switch to sold and the gaps turn red. Then shrink the shock to 5% and watch the Taylor estimate close in on the exact CVRs; raise the volatility and see the curve flatten.

£ millions. UK index at 8,000, 1,000 puts at £10 a point, strike 7,600, three months, implied volatility 16%.

Curvature is MAR21.5 and MAR21.96–21.101. Statements on jurisdictions are as at 3 November 2026 as at 3 Nov 2026.

• Delta is a straight line; options bend. The curvature charge catches the loss beyond delta when each risk factor is shocked by its risk weight.
• CVR± = −[V(x±) − V(x) ∓ RW × s], from full revaluation; by Taylor, about −½ Γ (RW·x)².
• Floors at zero, ψ and K_b = max(K+, K−) mean convexity can only add capital: long gamma earns no credit.
• The bank's long puts: CVR+ −1.959, CVR− −4.218, charge 0. The same puts sold: £4.218m, set by the down shock.
• Correlations in the curvature formulas are the delta correlations squared, moved by the same three scenarios.

Curvature charge

Curvature charge: the SBM's charge for the part of an option's move that delta does not capture, measured by full revaluation under an up and a down shock.

Curvature risk position (CVR)

Curvature risk position (CVR): minus the loss beyond delta when a risk factor is shocked by its risk weight. Positive means the position loses more than delta predicts.

Convexity

Convexity: the bend in a value curve. A convex position gains more than its delta says when the market moves either way.

Gamma

Gamma: the second derivative of value with respect to the underlying. Long options have positive gamma, short options negative.

ψ (psi)

ψ (psi): a switch in the curvature formulas that is 0 when both CVRs in a cross term are negative and 1 otherwise, so two gains cannot offset anything.