Theta Decay: Why Options Lose Value Every Day

If you've heard one piece of options folklore, it's probably "options are a wasting asset" — buy one and, all else equal, it's worth a little less tomorrow than today, just from time passing. Theta is the Greek that measures exactly how much less.

What theta measures

Theta is the rate of change of an option's price with respect to time — ∂C/∂T — but with a sign flip, since T (time to expiration) is counted down as time passes, not up. It's usually quoted as the dollar amount an option's price is expected to drop by, per day, holding everything else (stock price, volatility, rates) fixed. A theta of −0.05 means: if nothing else changes, tomorrow this option is worth about 5 cents less than today.

For a call:

Θ = −(S·n(d1)·σ) / (2√T) − r·K·e^(−rT)·N(d2)

where n(d1) (lowercase n) is the normal probability density function at d1 — not the cumulative one — and the rest of the terms are the familiar Black-Scholes inputs. The formula looks dense, but the two terms have a clean reading: the first is decay from the option's time value eroding, and the second is a smaller adjustment from the cost of carrying the discounted strike.

Why time value erodes, intuitively

An option's price is made of two pieces: intrinsic value (what it'd be worth if exercised right now — max(S − K, 0) for a call) and time value (everything above that, reflecting the chance the stock still moves further in your favor before expiration). Intrinsic value doesn't decay from time passing alone. Time value does — because every day that passes is one fewer day for the stock to make a favorable move, and "fewer days remaining" mechanically shrinks the probability-weighted range of outcomes the option's price reflects.

Why decay accelerates near expiration

Theta isn't linear — an option loses value slowly with months to go and fast in its final days. The 1/√T term in the formula is the reason: as T shrinks toward zero, 1/√T grows without bound, so the same fixed amount of time value gets compressed into an ever-shrinking window. A rough way to feel this: an at-the-money option with 90 days left might lose a few cents a day; the same option with 3 days left can lose that same few cents in an hour. This is why "theta gang" option sellers specifically target short-dated, near-the-money options — that's where decay is fastest and most collectible.

Who's on which side

Every option has a buyer and a seller, and theta is symmetric but opposite for each: the buyer's position loses value from decay (they're long theta-negative exposure), and the seller's position gains from it (they're short the option, so decay works in their favor, collecting what the buyer loses). This is why "selling premium" strategies — covered calls, cash-secured puts, credit spreads — are often framed as "getting paid for time passing." They're not free money; they're compensation for taking on the risk the buyer is paying to offload, priced through the option's premium.

Computing theta in Python

import math

def norm_cdf(x):
    return 0.5 * math.erfc(-x / math.sqrt(2))

def norm_pdf(x):
    return math.exp(-0.5 * x**2) / math.sqrt(2 * math.pi)

def call_theta(S, K, T, r, sigma):
    d1 = (math.log(S / K) + (r + 0.5 * sigma**2) * T) / (sigma * math.sqrt(T))
    d2 = d1 - sigma * math.sqrt(T)
    term1 = -(S * norm_pdf(d1) * sigma) / (2 * math.sqrt(T))
    term2 = -r * K * math.exp(-r * T) * norm_cdf(d2)
    annual_theta = term1 + term2
    return annual_theta / 365  # convert to a per-day figure

S, K, T, r, sigma = 100, 100, 1, 0.05, 0.20
print(round(call_theta(S, K, T, r, sigma), 4))  # ≈ -0.0176 per day

Dividing by 365 at the end matters — the raw formula gives theta in "per year of T," and nobody thinks in per-year decay for an instrument that expires in weeks. Converting to per-day is what actually matches how theta gets quoted on a broker's options chain.