gamma

Black-76

Computes Gamma of the option on a future using the Black-76 model (sensitivity to delta change).

Avg. gas

1,704

Max abs. error

3.2e-15

when γ < 1

Max rel. error

5e-12

when γ ≥ 1

Signature

solidity

function gamma(
    uint128 future,
    uint128 strike,
    uint32  timeToExp,
    uint64  volatility,
    uint64  rate
) internal pure returns (uint256 gammaOut)

Parameters

Name	Type	Description
future	uint128	Current future price in 18-decimal fixed-point format.
strike	uint128	Strike price, 18-decimal fixed-point. Precision-tuned for the no-arbitrage band against the future — see Bounds.
timeToExp	uint32	Time to expiration in seconds. timeToExp == 0 is allowed (handled as expired).
volatility	uint64	Annualized implied volatility, 18-decimal fixed-point (e.g. 60% → 6e17).
rate	uint64	Annualized risk-free (discount) rate, 18-decimal fixed-point.

Returns

Name	Type	Description
gammaOut	uint256	Gamma in 18-decimal fixed-point. Γ ≥ 0; identical for call and put under put-call parity.

Bounds

Bound	Value
MIN_FUTURE	1e-6 smallest allowed future price (`1e12`)
MAX_FUTURE	1e15 largest allowed future price (`1e33`)
MAX_STSP_RATIO	5× (strike must lie within [future/5, future·5])
MAX_EXPIRATION	32 years (1,009,152,000 seconds)
MAX_RATE	400% annual (`4e18`)

Behavior

Validates all five inputs against module-wide constants and reverts with a typed error on any violation.
Returns a single value — call and put gamma are identical under put-call parity, so there is no tuple.
Fast-path on expiration: when timeToExp == 0, returns 0.
Composes three DeFiMath primitives — ln, sqrtTime (specialized sqrt for years), and exp (for the density φ(d₁)), plus expPositive for the discount.
Pure internal function; no external calls, no storage. Inlined into the caller's bytecode at compile time.

How it works

gamma is the second derivative of option value with respect to the future — the rate at which delta changes. Under Black-76 it is the discounted standard normal density at d₁ scaled by 1 / (F·σ·√T):

\Gamma = \frac{e^{-rT}\,\varphi(d_1)}{F \, \sigma \sqrt{T}}, \qquad \varphi(d_1) = \frac{e^{-d_1^2 / 2}}{\sqrt{2\pi}}

d_1 = \frac{\ln(F/K) + \tfrac{\sigma^2}{2} T}{\sigma \sqrt{T}}

The density φ(d₁) is evaluated as Math.exp(−d₁²/2) / √(2π), divided by future · σ√T, and discounted by e^(−rT) (from Math.expPositive(rT)). Gamma is symmetric across the call/put boundary, so only one value is returned.

Precision follows the dual-metric rule: a relative bound of 5e-12 where γ ≥ 1 (low vol / short time) and an absolute bound of 3.2e-15 where γ < 1, at future = $1,000 — head-to-head measurements live in defimath-compare.

Errors

Error	Trigger
FutureLowerBoundError	`future ≤ MIN_FUTURE`
FutureUpperBoundError	`future ≥ MAX_FUTURE`
StrikeLowerBoundError	`strike · 5 < future`
StrikeUpperBoundError	`future · 5 < strike`
TimeToExpiryUpperBoundError	`timeToExp ≥ MAX_EXPIRATION`
RateUpperBoundError	`rate ≥ MAX_RATE`

Example

solidity

import "defimath-lib/contracts/derivatives/Black76.sol";

uint256 g = Black76.gamma(
    1000e18,         // future = $1,000
    1050e18,         // strike = $1,050
    90 days,         // 90 days to expiry
    0.60e18,         // 60% annualized vol
    0.05e18          // 5% discount rate
);
// g ≈ 0.0013e18 (per $1 move in the future)

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