Cross-gamma: Definition & Examples

Cross-gamma is the mixed second derivative of a position’s value with respect to two different underlying asset prices. It measures how the delta on one asset changes when the other asset’s price moves. A hedge can therefore need adjusting even when the price of the asset being hedged has stayed still.

For a value function $V(S_1,S_2)$ , hold time, volatilities, correlation assumptions and all other pricing inputs fixed. Then

\Gamma_{12}=\frac{\partial^2 V}{\partial S_1\partial S_2} =\frac{\partial\Delta_1}{\partial S_2}, \qquad \Delta_1=\frac{\partial V}{\partial S_1}.

If the second partial derivatives are continuous in a neighbourhood, the order of differentiation agrees:

\Gamma_{12}=\Gamma_{21}

. Ordinary single-asset gamma,

\Gamma_{11}

, instead measures how asset 1’s delta changes with asset 1’s own price. These are different entries of the same second-derivative matrix. Duffie and Pan describe cross-market delta–gamma exposures; NIST §1.5(i) gives the mixed-partial definitions and symmetry condition.

A hedge changes because the other price moves

Use the deliberately simple synthetic function

V(S_1,S_2)=0.01S_1S_2.

Here $S_1$ and $S_2$ are numerical price quotes in USD per unit of their respective assets, and $V$ is USD for one synthetic position. The coefficient carries the units needed to turn the product of those quotes into USD value. This function teaches sensitivities; it is not a quoted option price or a claim of an arbitrage-free pricing model.

At $S_1=100$ and $S_2=50$ :

Quantity	Value	Meaning
$V$	USD 50	Initial synthetic value
$\Delta_1=0.01S_2$	0.50	USD value per USD move in price 1
$\Delta_2=0.01S_1$	1.00	USD value per USD move in price 2
$\Gamma_{12}$	0.01	USD value per product of USD price moves
$\Gamma_{11},\Gamma_{22}$	Both zero	No curvature along either fixed-other-price slice

Cross-gamma’s units are $\text{USD value}/(\text{USD price}_1\times\text{USD price}_2)$ , abbreviated inverse USD when the asset-unit conventions are fixed. Report position size and quote conventions with the number; a price-level derivative is not automatically a sensitivity to percentage returns.

Now let only asset 2 fall from 50 to 47. Asset 1 remains at 100, but

\Delta_1

falls from 0.50 to 0.47. The change is

0.01(-3)=-0.03

. A hedge sized to the original asset-1 delta has become stale because the other underlying moved.

Fictional value V = 0.01 S1 S2. Left: delta1 rises from 0.45 to 0.55 as S2 rises from USD 45 to USD 55, with delta1 0.50 at S2 50. Right: moves of magnitude USD 2 in S1 and USD 3 in S2 produce a cross residual of plus USD 0.06 for matching signs and minus USD 0.06 for opposite signs.

Original synthetic illustration, not market prices. The residual subtracts both initial delta effects from the exact value change. Pure gammas are zero; the interaction survives.

Why the mixed P&L term has no extra half

For a sufficiently smooth value function, the second-order Taylor approximation at the initial prices is

\begin{aligned} \Delta V\approx{}&\Delta_1\Delta S_1+\Delta_2\Delta S_2\\ &+\tfrac12\Gamma_{11}(\Delta S_1)^2 +\Gamma_{12}\Delta S_1\Delta S_2 +\tfrac12\Gamma_{22}(\Delta S_2)^2. \end{aligned}

The Hessian form is

\tfrac12\Delta\mathbf S^{\mathsf T}H\Delta\mathbf S

. Its two off-diagonal entries contribute

\Gamma_{12}\Delta S_1\Delta S_2

and

\Gamma_{21}\Delta S_2\Delta S_1

. Under symmetry these are equal, so the half multiplies their sum. Writing a further half in front of the single combined mixed term would undercount it. See Pinkham §12.4 for the quadratic expansion and its remainder.

Return to the synthetic function. Move asset 1 by $+2$ and asset 2 by $-3$ in USD price units:

\begin{aligned} \text{Initial-delta contribution}&=0.50(2)+1.00(-3)=-2.00,\\ \text{Cross contribution}&=0.01(2)(-3)=-0.06,\\ \text{Exact value change}&=0.01(102)(47)-50=-2.06. \end{aligned}

All three results are USD. The second-order expression is exact here because this bilinear polynomial has no higher-order terms. For a general model, the omitted remainder matters as moves grow or sensitivities vary sharply. Kinks and discontinuities can also invalidate a smooth local expansion.

The sign is revealing: positive cross-gamma contributes positively when the price moves have the same sign and negatively when they have opposite signs. It does not determine total P&L. Even a position initially hedged against both deltas can retain this interaction, alongside pure-gamma, time, volatility, funding and other effects absent from this instantaneous toy calculation.

Estimate it with four valuations

A central mixed finite difference bumps both prices. Define $V_{ab}=V(S_1+ah,S_2+bk)$ for signs $a,b\in\{-1,+1\}$ and positive price bumps $h,k$ . Then

\widehat\Gamma_{12} =\frac{V_{++}-V_{+-}-V_{-+}+V_{--}}{4hk}.

For $h=k=1$ , the synthetic values in that order are 51.51, 49.49, 50.49 and 48.51 USD. Their signed combination is USD 0.04; dividing by $4hk$ returns 0.01 in cross-gamma units.

This stencil is exact for this polynomial. For a general smooth deterministic function, it converges to the mixed derivative as both bumps shrink; with sufficiently smooth fourth derivatives, its truncation error is of order

h^2+k^2

. Tiny bumps can amplify valuation noise and floating-point cancellation. Keep other inputs and valuation conventions consistent across all four calls, and check stability across sensible bump sizes. The Greeks and Numerical Sensitivities lesson develops the related bias/noise problem. Its preview is public; the full lesson requires membership.

Cross-gamma is not correlation risk

Correlation describes the joint behaviour of asset returns. A pricing model’s sensitivity to its correlation input is

\partial V/\partial\rho_{12}

, whereas cross-gamma differentiates with respect to two prices. The quantities can be related within a specified model, but they are not interchangeable. Fengler and Schwendner §2.1 distinguish the off-diagonal gamma matrix from correlation sensitivity.

For example, a separable portfolio $V=f(S_1)+g(S_2)$ has zero cross-gamma even if the assets’ returns are strongly correlated. Conversely, cross-gamma is a property of the value function that can be calculated without specifying a statistical distribution for today’s price shocks.

Nor is it vanna: vanna mixes a price input with a volatility input. When reading a risk report, identify the two differentiated coordinates, their units, the inputs held fixed and whether the number is a raw derivative or a pre-scaled bump P&L.

For the calculus underneath these distinctions, Partial Derivatives covers controlled sensitivities and mixed partials. The preview is public; the full lesson requires membership.

Sources

Duffie and Pan, An Overview of Value at Risk, “Delta-Gamma Exposures of Cross-Market Derivatives”: cross-gamma as a change in one underlying’s delta with another underlying’s price.
NIST Digital Library of Mathematical Functions, §1.5, equations 1.5.5–1.5.6 and 1.5.18: mixed partials, symmetry and the two-variable Taylor expansion.
Henry C. Pinkham, Analysis, Convexity, and Optimization, §12.4: the second-order Hessian expansion and a remainder smaller than squared displacement locally.
Matthias R. Fengler and Peter Schwendner, Correlation Risk Premia for Multi-Asset Equity Options, §2.1: separate spot, volatility and correlation sensitivities. The paper’s historical market observations are not treated here as current conditions.

A hedge changes because the other price moves

Why the mixed P&L term has no extra half

Estimate it with four valuations

Cross-gamma is not correlation risk

Sources

Related terms

Correlation

Delta

Gamma