Maths 06 Investment Maths Without Fear

Theta: Time Decay

Separate time passing from a promise of daily price decay.

Learn Options and GreeksAdvanced4 min

At-the-money example; rates and dividends omitted

Plain English

The idea

Theta measures how an option value changes as time passes, with other model inputs held still.

Time matters because an option is partly a claim on future uncertainty. As expiry gets closer, there is less time for the underlying price to move into a valuable range.

The sign convention matters. In the formulas below, T means time to expiry. Calendar-time theta is the negative of the derivative with respect to T because the passage of one day reduces time remaining.

Two clocks can describe the same contract. Calendar time moves forward, while the time remaining to expiry decreases. The derivative with respect to time remaining therefore has the opposite sign from calendar-time theta. A sign convention should be stated before a theta value is interpreted.

This article uses calendar-time theta for a long option. The displayed closed-form expressions use years; a daily quotation requires an additional convention about how to convert or reprice the passage of a day.

Worked Example

Reading time decay without making it mechanical

Suppose a platform reports theta near -0.04 per day for an option. Read that as a local estimate: if one day passes and other inputs are held still, the option value is expected to fall by about 0.04.

That is not a promise that the option will fall by exactly 0.04 tomorrow. The underlying price, implied volatility, and rates can all move at the same time.

Theta is most useful when it stops the reader from treating time as free.

The hypothetical quotation is −£0.04 per option unit for one day under the platform's stated convention. If price, volatility and rates are otherwise unchanged, a first-order estimate for that day is a £0.04 decline. It is not an amount charged to an account or a guaranteed daily loss.

For a separate arithmetic illustration, annual calendar theta of −£14.60 gives −£0.04 per calendar day when divided by 365. That convention is explicitly assumed here. Platforms may use other day counts or finite repricing methods; their definitions should be checked.

Reported theta	-0.04 per day
Small time move	One day passes
First-order effect	About -0.04 before other effects
Main caution	The estimate is local, not a forecast

Reading the result

Theta is a sensitivity, not a timetable

The remaining time and the option's other inputs change, so today's daily theta should not simply be multiplied by all days until expiry. Near the strike close to expiry, the value relationship can change particularly quickly. Market prices also respond to information that arrives during weekends and non-trading periods.

A theta contribution may be outweighed by price, volatility or rate changes. Separating the contributions can explain why an option's price moved differently from a naive decay estimate. It does not establish which contribution will dominate tomorrow or make a short-option position a reliable income source.

Limits and assumptions

Positive theta is possible

Long-option theta is often negative, but it is not universally so. A deep in-the-money European put can have positive calendar theta because the financing effect associated with receiving the strike sooner can outweigh the optionality effect. Negative interest rates can also produce positive call theta in some cases.

The optional formulas assume no dividends, European exercise, positive underlying price, strike, volatility and remaining time, with constant continuously compounded rate and annualised decimal volatility. They are not applied directly at expiry. Exercise rights, dividends and other model choices can change the calculation and the interpretation.

Common Mistake

Ignoring the clock

A common mistake is to focus only on direction and forget expiry. Two options with similar delta can have very different time sensitivity.

Another mistake is to compare theta values without checking units. Some model formulas are annualised, while many platforms convert the number to a daily convention.

Saying that more remaining time always increases every option's value would erase those financing and exercise distinctions. Describe the common tendency together with its assumptions, rather than turning it into a universal law.

Formula Summary

Black-Scholes theta convention

European exercise, no dividends, positive S, K, sigma and T; time T is in years, sigma is an annualised decimal and r is a continuously compounded annual rate. Rates and volatility are constant model inputs; jumps and trading frictions are excluded. Do not apply these expressions directly at expiry or zero volatility.

N is the cumulative standard normal distribution; n is its density. The formulas describe long-option model values; signed positions change the exposure.

Convention

Θ_{calendar} = - \frac{\partial V}{\partial T}

$V$: Option value under the model.
$T$: Time to expiry in years.

The displayed theta uses years. Dividing by 365 gives the illustrative calendar-day approximation used here; platform day counts and repricing conventions can differ.

Call Theta

Θ_{call} = - \frac{S n (d_{1}) σ}{2 \sqrt{T}} - r K e^{- r T} N (d_{2})

$S$: Current underlying price.
$n (d_{1})$: Standard normal density at d1.
$σ$: Annualised volatility assumption.
$r$: Continuously compounded risk-free rate in this convention.
$K$: Strike price.

Put Theta

Θ_{put} = - \frac{S n (d_{1}) σ}{2 \sqrt{T}} + r K e^{- r T} N (- d_{2})

Optional derivation Explore the derivation

The derivation below keeps T as time to expiry, then flips the sign at the end to express calendar-time theta.

01

Set the time convention

When T is time remaining, one day of calendar time passing means T becomes smaller. The sign of the value effect also depends on financing and the contract; more remaining time does not universally raise every European option value.

Calendar-Time Convention

$Θ_{calendar} = - \frac{\partial V}{\partial T}$
02

Differentiate the call price with respect to T

The d1 and d2 terms depend on T, and the discounted strike term also changes with T.

Time-to-Expiry Derivative

$\frac{\partial C}{\partial T} = S n (d_{1}) \frac{\partial d_{1}}{\partial T} - K e^{- r T} n (d_{2}) \frac{\partial d_{2}}{\partial T} + r K e^{- r T} N (d_{2})$
03

Collapse the chain-rule terms

Using the density identity and the relationship between d1 and d2, the chain-rule terms reduce to the volatility-over-time term.

Chain-Rule Collapse

$S n (d_{1}) \frac{\partial d_{1}}{\partial T} - K e^{- r T} n (d_{2}) \frac{\partial d_{2}}{\partial T} = \frac{S n (d_{1}) σ}{2 \sqrt{T}}$
04

Flip the sign for calendar time

The derivative with respect to time remaining is positive for the time-value part. Calendar-time theta flips the sign because time remaining shrinks as calendar time passes.

Call Theta Result

$Θ_{call} = - \frac{S n (d_{1}) σ}{2 \sqrt{T}} - r K e^{- r T} N (d_{2})$

Self-check

Check your understanding

Why does calendar theta have the opposite sign from the derivative with respect to remaining time?

One unit of calendar time passing removes one unit of remaining time. The minus sign expresses that relationship between the two clocks.

Is dividing annual theta by 365 a universal platform rule?

No. It is the explicit calendar-day approximation used in the example. Providers can use other conventions or finite repricing, and their definitions matter.

Must a long European option always have negative theta?

No. Positive theta can occur, including for some deep in-the-money puts. Financing effects and the interest-rate assumption can change the sign.

General learning context

Where this helps a public investor

Theta helps readers see why expiry choice changes the risk profile, even when the directional view looks similar.

Disclaimer

Educational Use Only

This article is for informational and educational purposes only. Options involve risk and are not suitable for every investor. Nothing here is a recommendation to buy, sell, write, or trade an option.

Path

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Terms

Calendar theta: value sensitivity to calendar time passing.Time to expiry: the remaining life of the contract.Day-count convention: the method translating years into daily amounts.In the money: having a positive immediate-exercise amount.

Sources

The idea

Reading time decay without making it mechanical

Theta is a sensitivity, not a timetable

Positive theta is possible

Ignoring the clock

Black-Scholes theta convention

Set the time convention

Differentiate the call price with respect to T

Collapse the chain-rule terms

Flip the sign for calendar time

Check your understanding

Where this helps a public investor

Educational Use Only