theta

Binary options

Computes Theta for binary cash-or-nothing call and put options using the Black-Scholes model (per day).

Avg. gas

3,161

Max abs. error

1e-14

Signature

solidity

function theta(
    uint128 spot,
    uint128 strike,
    uint32  timeToExp,
    uint64  volatility,
    uint64  rate
) internal pure returns (int128 thetaCall, int128 thetaPut)

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
thetaCall	int128	Binary call theta per day for unit payout in 18-decimal fixed-point. Signed.
thetaPut	int128	Binary put theta per day for unit payout in 18-decimal fixed-point. Signed.

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 theta per day (the annual figure divided by 365) for both call and put. Binary theta is signed and can be either sign depending on moneyness.
Volatility has no explicit revert — it's bounded only by its uint64 type (max ≈ 1.84e19, i.e. ~1840% annualized).
Fast-path on expiration: when timeToExp == 0, returns (0, 0).
Composes five DeFiMath primitives — ln, sqrtTime (specialized sqrt for years), expPositive (the discount factor), exp (the density φ(d₂)), and stdNormCDF (the carry term). Its higher gas reflects that fuller composition.
Pure internal function; no external calls, no storage. Inlined into the caller's bytecode at compile time.

How it works

Binary theta is the derivative of the cash-or-nothing price with respect to the passage of time. DeFiMath returns it per day — the annualized figure divided by 365:

\Theta_{call} = \frac{1}{365}\left[ r \, e^{-rT} \Phi(d_2) + e^{-rT} \varphi(d_2)\left(\frac{d_1}{2T} - \frac{r}{\sigma\sqrt{T}}\right) \right]

\Theta_{put} = \frac{1}{365}\left[ r \, e^{-rT} \Phi(-d_2) - e^{-rT} \varphi(d_2)\left(\frac{d_1}{2T} - \frac{r}{\sigma\sqrt{T}}\right) \right]

The shared decay term e^(−rT)·φ(d₂)·(d₁/2T − r/σ√T) is computed once and reused for both; only the sign and the carry term r·e^(−rT)·Φ(±d₂) differ. The density uses Math.exp, the discount factor Math.expPositive, the carry CDF Math.stdNormCDF, and √T / ln(spot/strike) use Math.sqrtTime / Math.ln.

On the supported domain the magnitude stays well below 1, so the suite enforces an absolute error only — 1e-14 per day — with no relative bound. Head-to-head measurements 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/BinaryOptions.sol";

(int128 thetaCall, int128 thetaPut) = BinaryOptions.theta(
    1000e18,         // spot = $1,000
    1050e18,         // strike = $1,050
    30 days,         // 30 days to expiry
    0.60e18,         // 60% annualized vol
    0.05e18          // 5% risk-free rate
);
// thetaCall, thetaPut per day (signed by moneyness)

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