FRM Exam Part I · Option Sensitivity Measures: The "Greeks"

Theta and Time Decay of Options for FRM Part I

Updated 11 October 2026 · Fact-checked

Theta is the rate of change of an option's value as time passes, with all else fixed. Long options usually have negative theta, so they lose value each day. For a delta-neutral portfolio on a non-dividend stock, the Black-Scholes equation gives Θ + ½σ²S²Γ = rΠ, so theta and gamma trade off against each other.

Understand Theta and Time Decay

Theta (Θ) measures how much an option or portfolio loses or gains in value as time passes, holding the stock price, volatility and interest rate constant. It is the partial derivative of value with respect to time. It is usually quoted per year, then divided by 365 (or 252 trading days, if the question says so) to get a per-day figure.

An option's value has two parts: intrinsic value and time value. Time value is the chance that the price will move in your favour before expiry. As expiry nears, that chance shrinks, and time value falls to zero at maturity. This fall is time decay. For a long option you own that fading chance, so theta is normally negative. For a short option, theta is positive: you earn the decay.

Theta is not always negative. A deep in-the-money European put can have positive theta, because as expiry nears the discounted value of the strike, K e^(−rT), rises. This gain can exceed the loss of time value. A European call on a stock with a high dividend yield can have positive theta when it is deep in the money. For an at-the-money option, decay speeds up as expiry gets close.

Theta links to delta and gamma through the Black-Scholes-Merton differential equation. For a portfolio of derivatives on one stock paying no dividends: Θ + rSΔ + ½σ²S²Γ = rΠ. If the portfolio is delta-neutral (Δ = 0), this becomes Θ + ½σ²S²Γ = rΠ. When rΠ is small or zero, Θ ≈ −½σ²S²Γ. So a delta-neutral position with positive gamma has negative theta: you pay time decay to own convexity. Negative gamma earns positive theta but loses on large moves.

Key formulas to remember

Definition of theta: Θ = ∂f / ∂t; Change in option value per unit of time, other inputs fixed. Quoted per year unless told otherwise; divide by 365 for a per-day figure.
Black-Scholes-Merton equation (no dividends): Θ + r S Δ + ½ σ² S² Γ = r Π; Holds for any portfolio Π of derivatives on the same stock. Π is the portfolio value.
Delta-neutral case: Θ + ½ σ² S² Γ = r Π; If also Π = 0 (or r is negligible), Θ = −½ σ² S² Γ.
With continuous dividend yield q: Θ + (r − q) S Δ + ½ σ² S² Γ = r Π; Use for indices and currencies, where q is the dividend yield or foreign rate.
Theta of a European call (no dividends): Θ(call) = −S₀ N′(d₁) σ ÷ (2√T) − r K e^(−rT) N(d₂); Always negative for a non-dividend stock. N′(x) = e^(−x²÷2) ÷ √(2π).
Theta of a European put (no dividends): Θ(put) = −S₀ N′(d₁) σ ÷ (2√T) + r K e^(−rT) N(−d₂); Can be positive when deep in the money.
d₁ and d₂: d₁ = [ln(S₀ ÷ K) + (r + σ²÷2)T] ÷ (σ√T); d₂ = d₁ − σ√T; Needed for both theta formulas.

How to solve Theta and Time Decay questions

Most theta questions are either a sign and interpretation question, a use of the Black-Scholes equation, or a direct calculation. Use this order.

1Identify what is asked: sign or direction, the link between theta, delta and gamma, or a numerical theta.
2Check the position: long or short, call or put, and whether the portfolio is delta-neutral. Check for a dividend yield q.
3For a link question, write the equation Θ + (r − q)SΔ + ½σ²S²Γ = rΠ. Set Δ = 0 if delta-neutral, and Π = 0 if the portfolio has zero value.
4Solve for the unknown (Θ or Γ), keeping σ as a decimal (20% = 0.20) and squaring both σ and S.
5For a direct calculation, find d₁ and d₂, then N′(d₁), N(d₂) or N(−d₂), and plug into the theta formula.
6Check units: the result is per year. Divide by 365 (or the stated day count) if the question asks for per day.
7Sanity check the sign: a long option on a non-dividend stock should have negative theta, and a delta-neutral long-gamma position should have negative theta.

Quickest way: Delta-neutral shortcut: theta is the price of gamma

When to use it: Use when the question gives gamma, volatility and stock price and asks for theta (or the reverse) on a delta-neutral portfolio.

Confirm the portfolio is delta-neutral. If not, include the rSΔ term.
If the portfolio value is zero, use Θ = −½σ²S²Γ directly.
If the portfolio value is not zero, use Θ = rΠ − ½σ²S²Γ.
Compute ½σ²S² first, then multiply by Γ.
Eliminate options with the wrong sign: positive gamma with Δ = 0 and Π = 0 must give negative theta.

Common mistakes in Theta and Time Decay

Saying theta is always negative
Most examples use long calls, so the pattern feels universal.
Fix: Say 'usually negative for long options'. A deep in-the-money European put can have positive theta. Short options have the opposite sign to long ones.
Forgetting the rΠ term in the Black-Scholes equation
Students memorise Θ = −½σ²S²Γ and apply it to every delta-neutral portfolio.
Fix: That shortcut needs Π = 0 (or r ≈ 0). Otherwise use Θ = rΠ − ½σ²S²Γ.
Using σ in percent instead of decimal
Volatility is quoted as 20%, and the number 20 slips into the formula.
Fix: Convert to 0.20 before squaring. Writing σ² = 0.04 explicitly avoids an error by a factor of 10,000.
Mixing up per-year and per-day theta
The formula gives a per-year figure, but traders quote per day.
Fix: Read the question. Divide by 365 (or 252 if trading days are stated) only when a daily answer is asked for.
Ignoring the sign of the interest term in a put
Call and put theta formulas look alike.
Fix: Call: minus rKe^(−rT)N(d₂). Put: plus rKe^(−rT)N(−d₂). The first term is the same and negative in both.
Thinking long gamma and positive theta go together
Both seem like 'good' features.
Fix: They trade off. A delta-neutral position with positive gamma has negative theta, and negative gamma has positive theta.

Worked examples

Example 1

A delta-neutral portfolio of options on a non-dividend stock has value zero. S = 100, σ = 20% per year, and portfolio gamma Γ = 0.04. Find its theta per year and per calendar day.

Show the solution

Since Δ = 0 and Π = 0, the equation reduces to Θ + ½σ²S²Γ = 0, so Θ = −½σ²S²Γ.
σ² = 0.20² = 0.04. S² = 100² = 10,000.
½ × 0.04 × 10,000 = 200.
200 × 0.04 = 8.
Θ = −8 per year.
Per day: −8 ÷ 365 = −0.0219.

Answer: Θ = −8.00 per year, about −0.0219 per calendar day. The portfolio has positive gamma, so it loses value through time decay.

Example 2

A European call on a non-dividend stock has S₀ = 100, K = 100, r = 5% (continuously compounded), σ = 20% and T = 1 year. Calculate theta per year and per day.

Show the solution

d₁ = [ln(1) + (0.05 + 0.02) × 1] ÷ (0.20 × 1) = 0.07 ÷ 0.20 = 0.35.
d₂ = 0.35 − 0.20 = 0.15.
N′(d₁) = e^(−0.35²÷2) ÷ √(2π) = e^(−0.06125) ÷ 2.5066 = 0.9406 ÷ 2.5066 = 0.3752.
N(d₂) = N(0.15) = 0.5596.
Ke^(−rT) = 100 × e^(−0.05) = 95.12.
First term: −S₀N′(d₁)σ ÷ (2√T) = −100 × 0.3752 × 0.20 ÷ 2 = −3.752.
Second term: −rKe^(−rT)N(d₂) = −0.05 × 95.12 × 0.5596 = −2.662.
Θ = −3.752 − 2.662 = −6.414 per year.
Per day: −6.414 ÷ 365 = −0.0176.

Answer: Θ ≈ −6.41 per year, or about −0.0176 per calendar day. The call loses roughly 1.8 cents of value each day, other inputs unchanged.

Exam tips

Read the sign and the position first. Many questions only test whether you know long options lose value and short options gain it.
Memorise Θ + rSΔ + ½σ²S²Γ = rΠ cold. Most numerical links between theta, delta and gamma start there.
If the question says delta-neutral and zero value, the answer is just −½σ²S²Γ. Do the arithmetic in steps to avoid slips.
Check whether the answer is wanted per year or per day. The wrong option is often the other unit.
On option-value questions, remember that decay is fastest for at-the-money options near expiry, not for deep in- or out-of-the-money ones.

Practice questions from Option Sensitivity Measures: The "Greeks"

Theta and Time Decay: frequently asked questions

Is theta positive or negative for long options?

For most long options it is negative, because time value shrinks as expiry approaches. Short option positions have positive theta. A deep in-the-money European put can have positive theta even when long.

What is the relationship between theta, delta and gamma?

The Black-Scholes-Merton equation gives Θ + rSΔ + ½σ²S²Γ = rΠ for a portfolio on a non-dividend stock. If the portfolio is delta-neutral, theta and gamma offset each other: Θ + ½σ²S²Γ = rΠ. Positive gamma then means negative theta.

How do I calculate theta of a European call?

Find d₁ and d₂, then use Θ = −S₀N′(d₁)σ ÷ (2√T) − rKe^(−rT)N(d₂). This gives a per-year value for a non-dividend stock. Divide by 365 for a per-day figure.

What is time decay in options?

It is the loss of an option's time value as expiry gets closer. Theta measures its speed. At expiry only intrinsic value remains.