Theta: Time Decay
Why options are a race against the clock · 15 min
Options Are Wasting Assets
Every option has an expiration date. Each day that passes without a favorable move in the underlying, the option loses value — not because the stock moved against you, but simply because there's less time remaining for it to move in your favor. This erosion is called time decay, and it's measured by theta (Θ).
Theta is the partial derivative of the option price with respect to time:
Θ = ∂V / ∂t
By convention, theta is typically quoted as the change in option price per calendar day, with a negative sign — theta is almost always negative for long options because options lose value as time passes.
If you own a call with Θ = −0.05, you lose approximately $0.05 per day from time decay alone, all else equal. Over a three-day weekend, you'd lose ~$0.15 without the stock moving at all.
The Formula
The Black-Scholes theta for a European call is:
Θcall = [−S · n(d₁) · σ / (2√T) − r · K · e−rT · N(d₂)] / 365
And for a put:
Θput = [−S · n(d₁) · σ / (2√T) + r · K · e−rT · N(−d₂)] / 365
where n(·) is the standard normal PDF (not CDF). The division by 365 converts from annualized to daily theta. The first term — involving n(d₁) — is always negative and represents the optionality decaying. The second term involves the interest on the strike and can be positive for puts, which is why deep ITM puts sometimes have slightly positive theta.
Time Decay Is Not Linear
One of the most important practical facts about theta: time decay accelerates as expiration approaches. An ATM option doesn't lose 1/365 of its value each day — it loses much more in its final days.
The reason is that the option's extrinsic value is approximately proportional to σ√T (the "vol-time" product). As T decreases, the rate of decrease of √T accelerates:
d(√T)/dt = 1/(2√T)
This blows up as T → 0. An ATM option with 30 days left decays much faster per day than the same option with 180 days left. Empirically, roughly half of an ATM option's time value is lost in the final 25% of its life.
This is why option sellers love the final weeks before expiration — theta is collecting fastest. It's also why option buyers need their underlying to move quickly — each day of delay costs more than the previous one.
Theta and Volatility: The ATM Peak
Theta is largest (most negative) for at-the-money options. This seems counterintuitive — shouldn't the most expensive options decay fastest in absolute terms? Yes, and ATM options have the most extrinsic value, so they decay the most.
Deep ITM options have almost no extrinsic value left to decay. Deep OTM options are cheap and also have little extrinsic value. The ATM option has the maximum extrinsic value — and therefore the maximum theta exposure.
Numerically: with S = K = 100, T = 30 days, r = 5%, σ = 20%, the ATM call theta is about −$0.055/day. The same call but OTM (K = 120) has theta of about −$0.012/day. The ITM call (K = 80) has theta of about −$0.030/day.
The Theta-Gamma Tradeoff: The Central Tension
Theta and gamma are the yin and yang of options trading. You cannot have one without paying for the other. This relationship comes directly from the Black-Scholes partial differential equation:
Θ + ½σ²S²Γ + rSΔ − rV = 0
This is the Black-Scholes PDE, and it must hold for any option price. Rearranging roughly: Θ ≈ −½σ²S²Γ for delta-hedged positions. Theta and gamma have opposite signs and are proportional to each other.
- Long options (long gamma, negative theta): You benefit from large moves in either direction — gamma gives you a "free" improvement in delta when the stock moves. But you pay for this through daily theta decay. You're rooting for volatility to materialize.
- Short options (short gamma, positive theta): You collect theta every day, but lose when the stock makes a large move. You're rooting for the stock to go nowhere and time to pass quietly.
Professional options traders frame every position around this tradeoff. "How much gamma am I paying for with theta?" is the question that drives structuring decisions. A market maker delta-hedging an ATM option is long gamma (profits from moves) and paying theta (loses each quiet day). They need enough moves to cover the theta cost — this is called gamma scalping.
Calendar Spreads: Trading Pure Theta
A calendar spread exploits the difference in theta between near-term and far-term options. You sell the near-term option (high theta, decays faster) and buy the far-term option (lower theta, decays slower) at the same strike. Your net theta is positive. If the stock stays near the strike, the near-term option decays faster than the long option, and you profit.
This is a pure volatility play: you want near-term realized volatility to be low (so the short option expires worthless) and forward volatility to remain high (keeping the long option valuable).