Theta: Time Decay

Options Are Wasting Assets

Every option carries an expiration date, and that changes its whole character compared to just owning the stock. Every day that passes without the underlying moving in your favor, the option loses a little value — not because anything bad happened, but simply because there's now less time left for things to go your way. That erosion has a name, time decay, and a Greek letter attached to it: theta (Θ).

Theta is the option price's partial derivative with respect to time:

Θ = ∂V / ∂t

By convention it's quoted as the dollar change per calendar day, and for a long option it's almost always negative — time only ever works against you when you're the one holding the option.

Own a call with Θ = −0.05 and you're bleeding roughly $0.05 a day from decay alone, everything else held equal. Sit through a three-day weekend and that's about $0.15 gone with the stock not having moved a cent.

The Formula

For a European call, Black-Scholes gives:

Θ_call = [−S · n(d₁) · σ / (2√T) − r · K · e^−rT · N(d₂)] / 365

and for a put:

Θ_put = [−S · n(d₁) · σ / (2√T) + r · K · e^−rT · N(−d₂)] / 365

n(·) here is the standard normal PDF, not the CDF you saw earlier — a different function, easy to mix up. Dividing by 365 just converts the annualized number into something per-day. The first term, built around n(d₁), is always negative and captures pure optionality bleeding away. The second term is really interest accruing on the strike, and it can flip positive for puts — which is exactly why deep in-the-money puts occasionally show slightly positive theta, a quirk that surprises people the first time they see it.

Time Decay Is Not Linear

Here's the practical fact that matters more than the formula: time decay speeds up the closer you get to expiration. An at-the-money option doesn't shed a tidy 1/365th of its value each day — it loses far more in its final days than in its first.

The reason traces back to extrinsic value being roughly proportional to σ√T, the "vol-time" product. As T shrinks, √T doesn't shrink at a constant rate — its rate of change accelerates:

d(√T)/dt = 1/(2√T)

and that expression blows up as T approaches zero. An ATM option with 30 days left is decaying meaningfully faster, day for day, than the identical option with 180 days left. As a rough rule of thumb, roughly half of an ATM option's remaining time value disappears in just the final quarter of its life.

That's exactly why option sellers love the last few weeks before expiration — theta is paying out fastest right there. And it's why option buyers feel the clock pressing harder every day: each day of waiting costs more than the one before it.

Theta and Volatility: The ATM Peak

Theta hits its most negative point exactly at-the-money. That might seem backwards at first, but it tracks: ATM options carry the most extrinsic value of any strike, so naturally they have the most extrinsic value left to lose.

Deep ITM options barely have any extrinsic value left to decay. Deep OTM options are cheap for the same reason — barely any extrinsic value to begin with. The ATM strike sits at the peak of extrinsic value, and so it sits at the peak of theta exposure too.

Concretely: with S = K = 100, T = 30 days, r = 5%, σ = 20%, the ATM call's theta comes out to about −$0.045/day. Shift to an OTM call (K = 110) and it drops to roughly −$0.012/day. An ITM call (K = 92) lands around −$0.023/day — smaller than ATM, bigger than the OTM strike, exactly as you'd expect from the shape of extrinsic value.

The Theta-Gamma Tradeoff: The Central Tension

Theta and gamma are effectively two sides of the same coin in options trading — you cannot pick up one without paying for it with the other. That's not a rule of thumb; it falls directly out of the Black-Scholes partial differential equation:

Θ + ½σ²S²Γ + rSΔ − rV = 0

Every option price has to satisfy this equation. Simplify it for a delta-hedged position and you get roughly Θ ≈ −½σ²S²Γ — theta and gamma sit on opposite sides of zero and scale together.

• Long options (long gamma, negative theta): you benefit from a big move in either direction — gamma hands you a favorable delta adjustment "for free" as the stock moves. The cost is theta bleeding out every single day you're waiting. You're implicitly rooting for volatility to actually show up.
• Short options (short gamma, positive theta): theta pays you every day, but a large move in the stock hurts. You're rooting for the opposite — quiet markets and time passing without incident.

This tradeoff is the lens every professional options trader structures positions through — "how much gamma am I paying for with this theta?" is close to the central question of the whole business. A market maker delta-hedging an ATM option is long gamma and paying theta at the same time, which means she needs the stock to actually move enough, and often enough, to earn back what theta is quietly costing her. That hunt for enough movement to cover the decay has its own name: gamma scalping.

Calendar Spreads: Trading Pure Theta

A calendar spread is built specifically to exploit the theta gap between near-term and far-term options. Sell the near-term option, which decays fast and carries high theta, and buy a far-term option at the same strike, which decays much more slowly. Net theta on the position ends up positive: if the stock parks itself near the strike, the short leg decays away faster than the long leg loses value, and the difference is your profit.

It's really a volatility bet dressed up as a time-decay trade — you want near-term realized volatility to stay low, so the short leg expires worthless, while hoping forward-looking volatility stays elevated enough to keep the long leg valuable.