delta

Black-Scholes

Computes Delta for both call and put options using the Black-Scholes model (sensitivity to spot price change).

Avg. gas

1,661

Max abs. error

1.2e-13

Signature

solidity

function delta(
    uint128 spot,
    uint128 strike,
    uint32  timeToExp,
    uint64  volatility,
    uint64  rate
) internal pure returns (int128 deltaCall, int128 deltaPut)

Parameters

Name	Type	Description
spot	uint128	Current spot price, 18-decimal fixed-point.
strike	uint128	Strike price, 18-decimal fixed-point. Precision-tuned for the no-arbitrage band against spot — 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 rate, 18-decimal fixed-point.

Returns

Name	Type	Description
deltaCall	int128	Call delta in 18-decimal fixed-point. δcall ∈ [0, 1].
deltaPut	int128	Put delta in 18-decimal fixed-point. δput = δcall − 1 ∈ [−1, 0].

Bounds

Bound	Value
MIN_SPOT	1e-6 smallest allowed spot price (`1e12`)
MAX_SPOT	1e15 largest allowed spot price (`1e33`)
MAX_STSP_RATIO	5× (strike must lie within [spot/5, spot·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 both call and put delta from a single Φ(d₁) evaluation — δput = δcall − 1 by put-call parity, so the second value is free.
Volatility has no explicit revert — it's bounded only by its uint64 type (max ≈ 1.84e19, i.e. ~1840% annualized). The MIN_VOL_IV / MAX_VOL_IV constants apply only to the impliedVolatility solver, not the greeks.
Fast-path on expiration: when timeToExp == 0, delta collapses to its degenerate expiry value (0 or ±1 by moneyness) without running the pricer.
Composes three DeFiMath primitives — ln, sqrtTime (specialized sqrt for years), and stdNormCDF. Delta needs no discount factor, so there is no exp call.
Pure internal function; no external calls, no storage. Inlined into the caller's bytecode at compile time.

How it works

delta is the first derivative of option value with respect to spot. Under Black-Scholes it is simply the standard normal CDF of d₁:

\delta_{call} = \Phi(d_1), \qquad \delta_{put} = \Phi(d_1) - 1

d_1 = \frac{\ln(S/K) + \left(r + \tfrac{\sigma^2}{2}\right) T}{\sigma \sqrt{T}}

The function annualizes timeToExp (seconds) by dividing by SECONDS_IN_YEAR, scales volatility by √T with Math.sqrtTime, forms ln(spot/strike) with Math.ln, and evaluates Φ(d₁) with Math.stdNormCDF. Because put delta differs from call delta by exactly 1, both are returned from one CDF evaluation — roughly halving gas versus computing them separately.

Delta is bounded to [−1, 1], so the suite enforces an absolute error only — no relative bound. The 1.2e-13 max absolute error is the bound enforced at spot = $1,000 across a full sweep of strike, time, vol, and rate — head-to-head measurements against other libraries live in defimath-compare.

Errors

Error	Trigger
SpotLowerBoundError	`spot ≤ MIN_SPOT`
SpotUpperBoundError	`spot ≥ MAX_SPOT`
StrikeLowerBoundError	`strike · 5 < spot`
StrikeUpperBoundError	`spot · 5 < strike`
TimeToExpiryUpperBoundError	`timeToExp ≥ MAX_EXPIRATION`
RateUpperBoundError	`rate ≥ MAX_RATE`

Example

solidity

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

(int128 deltaCall, int128 deltaPut) = BlackScholes.delta(
    1000e18,         // spot = $1,000
    980e18,          // strike = $980
    60 days,         // 60 days to expiry
    0.60e18,         // 60% annualized vol
    0.05e18          // 5% risk-free rate
);
// deltaCall ≈ 0.58e18, deltaPut ≈ -0.42e18

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