What Is Theta? Options Time Decay Explained

Theta is the change in an option's value when one day passes and nothing else moves: the same spot price, the same implied volatility, the same interest rate. It is the clock term in an option's price, and it acts on time value rather than on intrinsic value. The shorthand that options always decay is not quite right, and the exception is a specific, checkable case.

Part of what an option is worth comes purely from the time left on the contract, and that part is repriced every day whether or not the underlying moves. This page covers the split between intrinsic and time value, how theta is defined and quoted, why decay speeds up into expiration for at-the-money contracts, the case where theta turns positive, how it differs from charm, and how it reads once it is aggregated across a listed chain. It is descriptive throughout.

Intrinsic value, time value, and which one theta touches

If calls and puts are new to you, start with what a call option and a put option are. An option's price divides into two pieces. Intrinsic value is what the contract would be worth if it were exercised right now, floored at zero: for a call, spot minus strike; for a put, strike minus spot. Time value is everything the option is worth above that floor.

Intrinsic value changes only when the relationship between spot and the strike changes. Time value is the part that answers "what else could still happen before expiry", and it is the part a passing day acts on. Theta is the rate at which the whole option value changes as the calendar advances, and in most states that change is dominated by time value shrinking toward zero.

How theta is defined

Kresmion computes greeks in closed form Black-Scholes with a zero dividend yield. Writing S for spot, K for the strike, s for volatility, r for the risk free rate, T for time to expiry in years, phi for the standard normal density and N for its cumulative function:

``` d1 = (ln(S/K) + (r + s^2/2)T) / (s sqrt(T)) d2 = d1 - s sqrt(T)

theta_call = -S phi(d1) s / (2 sqrt(T)) - r K e^{-rT} N(d2) theta_put = -S phi(d1) s / (2 sqrt(T)) + r K e^{-rT} N(-d2) ```

Two things read straight off that. First, theta in this form comes out per YEAR, so a per day figure is theta divided by 365. Quoting conventions differ between platforms, which is why one contract can be quoted with two theta numbers a factor of 365 apart.

Second, theta is the greek whose two sides differ in substance rather than by a constant. Delta differs between a call and a put by exactly 1, and gamma and vega are identical on both sides because neither formula knows which side it is on. Theta carries a term that flips sign.

Each theta above is a sum of two pieces. The first piece, `-S phi(d1) s / (2 sqrt(T))`, is the decay term. It is identical for calls and puts and it is always negative. It is large when phi(d1) is large, which happens when the option sits near the money, and it grows as T shrinks because of the sqrt(T) in the denominator.

The second piece is the rate term, and it is the only place the two sides part company: negative for the call, positive for the put. Subtract one theta from the other and everything but that term cancels:

``` theta_call - theta_put = -r K e^{-rT} ```

The identity is exact. The entire call versus put theta difference is discounted financing on the strike, and nothing else.

Why decay accelerates into expiration for at-the-money options

Hold a strike at the money and let T run down. phi(d1) stays near its peak, so the decay term behaves like a constant divided by sqrt(T), which grows without bound as T approaches zero. Per calendar day, that contract gives up more of its value in its last week than in any earlier week, and more on its last day than on any earlier day.

The acceleration belongs to at-the-money contracts specifically. Move far from the strike in either direction and phi(d1) collapses toward zero, taking the decay term with it. A far out-of-the-money option has very little time value left to lose, and a deep in-the-money one is mostly intrinsic value, which the calendar does not touch. Drawn against strike, the decay term is a hump centered near spot that grows taller and narrower as expiry approaches.

The case where theta is positive

This is where the common shorthand fails. It is not true that every option decays, and it is not true that calls and puts both always carry negative theta.

For a call with no dividend, the decay term is negative and the rate term is never positive at a nonnegative rate, so call theta is always negative. There is no state in this model where a European call gains value from a day passing on its own.

A put behaves differently. Take a European put that is deep in the money, with spot far below the strike. phi(d1) has collapsed, so the decay term has effectively vanished. At the same time N(-d2) has approached 1, so the rate term approaches `+r K e^{-rT}`, which is positive. The sum is positive, and theta on that put is positive.

The reason is easier to see than the algebra. A deep in-the-money European put behaves much like a claim on a fixed cash amount, the strike, payable only at expiry, and European exercise means that cash cannot be collected before then. The contract is worth the discounted value of the strike today, and every day that passes leaves one day less of discounting between the holder and the full amount. Value accrues from the calendar itself. The same effect is what produces an early exercise premium on an American put, where the holder can take the cash now instead of waiting for the discount to unwind.

Two boundaries on the claim, both worth stating. It is a European-exercise result inside Black-Scholes, and it needs a positive rate: set r to zero and the rate term disappears from both sides, leaving only the shared negative decay term.

Theta and charm answer different questions

Theta and charm both describe what one day does to an option while spot and volatility hold still. They measure different things. Theta is the change in the option's value. Charm is the change in its delta. In the same house convention charm is also quoted per year, and dividing by 365 gives a per day figure:

``` charm = -phi(d1) (2rT - d2 s sqrt(T)) / (2 T s sqrt(T)) ```

A hedged book reads both. Theta says what the position is worth after a day has passed. Charm says how much hedge has to be added or removed overnight because delta moved on its own while the market stood still. Treating one as the other puts a valuation number where a hedging number belongs.

Reading theta as an exposure

Per contract theta is a small number. Added up across every strike and expiry listed on one underlying it becomes an aggregate: dollars of option premium decaying per calendar day. That aggregate is theta exposure, or TEX, built from listed open interest as `theta/365 * OI * 100`, where 100 is the contract multiplier, and netted as calls minus puts under the standard dealer sign convention. Like dealer gamma exposure, it is a modeled positioning assumption rather than a measured dealer book, because the open interest published at each strike does not say which side opened it.

That family is what the live SPY reading further down this page shows. Kresmion computes modeled dealer greeks exposure nightly for a curated universe of US names from listed option open interest, using closed form Black-Scholes with its assumptions stated, and the full suite including the implied volatility tab sits in Kresmion's greeks tool with a free account.

Key takeaways

Frequently asked questions

Is theta quoted per day or per year?

Both conventions are in circulation. The closed form used here produces theta per year, and Kresmion divides by 365 to publish a per calendar day figure. A theta that looks 365 times off from a number seen elsewhere is nearly always this convention rather than a mistake, so check the units before checking the arithmetic.

Do all options lose value as time passes?

No. A call with no dividend always carries negative theta in this model, but a deep in-the-money European put can carry positive theta, because its decay term has collapsed while the interest term on the strike stays positive. The shorthand that everything decays describes near-the-money contracts, not every contract.

Does a theta reading say where the market is going?

No. Theta describes what the calendar does to option values while everything else holds still, so on its own it carries no direction at all. Aggregated into an exposure it describes a backdrop: how much premium runs off per day, and which side of the chain it sits on.

What is the difference between theta and charm?

Theta is the change in an option's value per day. Charm is the change in its delta per day. Same passage of time, different quantity: theta answers what the position is worth after a day, charm answers how much hedging that day created or removed.

This page is information, not investment advice.

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Source: closed form Black-Scholes greeks, with the modeling assumptions stated on Kresmion's greeks exposure tool; Kresmion modeled dealer positioning data. Kresmion Research.