Theta

Formula

Let φ(d1) be the standard normal density, and
d1 = [ln(S/K) + (r − q + σ²/2) T] / (σ √T),   d2 = d1 − σ √T.

Θ_call = [ −S e^{−qT} φ(d1) σ / (2√T) − r K e^{−rT} N(d2) + q S e^{−qT} N(d1) ] / 365
Θ_put  = [ −S e^{−qT} φ(d1) σ / (2√T) + r K e^{−rT} N(−d2) − q S e^{−qT} N(−d1) ] / 365

Both are per calendar day. A trading-day theta divides the annual derivative by 252 instead;
multiply the value here by 365/252 to convert. q = 0 drops the last term in each bracket.
The Greeks calculator on this site uses these expressions.

What the sign means

• Negative theta means the model value falls as the clock runs, with every other input frozen. That is the usual case for a long option away from unusual rate and dividend combinations.
• The decay term −S e^{−qT} φ(d1) σ / (2√T) is shared. The rate and dividend terms differ for calls and puts, which is why a call theta and a put theta are not equal, unlike gamma and vega.
• See delta for the d1 that enters, and the Greeks calculator for the same per-day convention.

Pitfalls

• 365 versus 252. This function divides by 365, calendar days, matching the calculator. A desk that marks theta per trading day wants the other divisor. The two numbers differ by 365/252.
• One day is not infinitesimal. The derivative times 1/365 approximates the change from losing one calendar day of expiry. Very short-dated options have a theta that changes a lot over that same day, so the approximation is rougher.
• Frozen volatility. The derivative does not include a change in implied volatility as expiry approaches. Real premiums move for that reason too.

Python reference implementation

Requires scipy. Tested before publication; the generator re-executes it on every rebuild. MIT licensed. Download theta.py · All snippets

from math import exp, log, sqrt
from scipy.stats import norm

def bs_theta(S, K, T, r, sigma, call=True, q=0.0) -> float:
    """Black-Scholes-Merton theta per calendar day (annual derivative / 365).

    call=False returns the put. Multiply by 365/252 for a trading-day theta.
    """
    if S <= 0 or K <= 0 or T <= 0 or sigma <= 0:
        raise ValueError("S, K, T and sigma must be positive")
    sT = sigma * sqrt(T)
    d1 = (log(S / K) + (r - q + 0.5 * sigma * sigma) * T) / sT
    d2 = d1 - sT
    decay = -S * exp(-q * T) * norm.pdf(d1) * sigma / (2.0 * sqrt(T))
    if call:
        annual = decay - r * K * exp(-r * T) * norm.cdf(d2) + q * S * exp(-q * T) * norm.cdf(d1)
    else:
        annual = decay + r * K * exp(-r * T) * norm.cdf(-d2) - q * S * exp(-q * T) * norm.cdf(-d1)
    return float(annual / 365.0)

FAQ

Why is theta usually negative for a long option?

With the other inputs held fixed, less time means less chance of a large move and, for a call, less time for the prepaid forward to earn its rates. The model premium of a long option therefore usually falls as expiry gets closer. A short option has the opposite sign. This is a partial derivative, not a prediction of tomorrow's premium.

Does the function use trading days or calendar days?

Calendar days. The annual Black-Scholes derivative is divided by 365, which is what the Greeks calculator on this site displays. Multiply by 365/252 if you want a per-trading-day figure.