Theta and Time Decay

An option is a wasting asset. Every day that passes, all else equal, it's worth a little less, because there's a little less time left for the price to move your way. That daily leak is theta, and for anyone who owns options it's the rent you pay for the right to profit from movement.

The reason is simple once you see it. An option's price splits into two parts: the intrinsic value (what you'd collect by exercising right now) and the time value (the extra you pay for the chance the price moves further in your favour before expiry). Time value is pure hope, and hope has a shelf life. As expiry approaches, the window for a favourable move shrinks, so the time value drains toward zero. On expiry day, time value is gone entirely and only intrinsic value remains.

Measuring the leak

Theta is the change in the option's value $V$ per unit of time passing:

$$\Theta = \frac{\partial V}{\partial t}.$$

For anything you own, $\Theta$ is negative: value falls as time moves forward. It's usually quoted per calendar day, so a theta of $-0.05$ means the option loses about 5 cents a day just from the clock ticking. The key fact is that this leak is not steady. Time value decays roughly with the square root of time remaining, which means the loss per day accelerates as expiry nears. An at-the-money option loses value slowly with months to go and then falls off a cliff in its final weeks.

Theta is the mirror of gamma

Theta doesn't act alone. For an option you own, negative theta comes bundled with positive gamma, and the two are locked together by the pricing math. For a delta-hedged position with negligible interest rates:

$$\Theta \approx -\tfrac{1}{2}\,\sigma^2 S^2\, \Gamma,$$

where $\sigma$ is volatility, $S$ the spot price, and $\Gamma$ the gamma. In words: the more curvature (gamma) you own, the more theta you pay. You cannot be long the ability to profit from movement without paying rent in time decay. That's the whole trade, movement earns, time bleeds, and they exactly offset when the stock moves at implied volatility (see Gamma Scalping).

Worked example

You own a one-month at-the-money call worth $3.00 today, all of it time value (the stock sits right at the strike, so intrinsic value is zero). Its theta is $-0.05$ per day.

• After one week (7 days), nothing else moving: the call loses roughly $7 \times 0.05 = 0.35$, dropping to about $2.65.
• But theta accelerates. With one week left, its daily theta might have grown to $-0.11$. In the final week it now bleeds around $7 \times 0.11 = 0.77$, more than double the first week's loss, even though the same seven days passed.

So of the $3.00 you paid, the last week alone claws back a quarter of it. For a buyer, this is the enemy: if the stock just sits there, the option melts, fastest right at the end. For a seller, it's the paycheck, they collect that decay as pure income, as long as the stock stays quiet.

Where it catches people out

• Being right but too slow. A call buyer can pick the direction correctly and still lose, if the move arrives after theta has eaten the premium. You need the move to be big enough, soon enough.
• Weekends and holidays. The clock keeps ticking when markets are closed. Options often price in the decay across a weekend before it happens, so a Friday-to-Monday hold can feel like paying three days of theta for two days of trading.
• Theta and gamma flip together for sellers. Selling options to collect theta feels like easy income, small gains, most days. But you're short gamma: the same position that drips in theta can lose violently on a big move. The income is real; so is the tail.
• Deep in- or out-of-the-money options barely decay from time value. Theta is concentrated where there's time value to lose, near the money. A deep option is nearly all intrinsic (or nearly worthless), so its theta is small.

Theta lives inside the wider set of The Option Greeks, is the cost side of Gamma Scalping, and pairs with vega as the two ways an idle option can lose value, time passing and volatility falling.