Put Option Theta

Options
Updated Apr 2026 • Has calculator

The daily time decay of a put option's price, holding all other factors constant.

What is Theta (Put)?

Theta (Θ) for a put option measures its daily time decay. Put theta is usually negative but less negative than call theta with the same inputs, because put buyers benefit from the interest earned on the strike price. Deep in-the-money puts can actually have positive theta — they gain value over time as the interest component outweighs the vol premium. Out-of-the-money puts always have negative theta.

Formula

Θ_put = [−S·φ(d₁)·σ / (2√T) + r·K·e^(−rT)·N(−d₂)] / 365

Worked Example

Worked example — Apple Inc. (AAPL) — ATM put, representative Q1 2024

Representative Q1 2024 market conditions

Step 1  Stock price (S): $185, Strike (K): $185, σ = 28%, T = 0.25 yrs, r = 5.25%
Step 2  Annual theta: −S·φ(d₁)·σ/(2√T) + r·K·e^(−rT)·N(−d₂)
Step 3  = −20.35 + (0.0525×185×0.987×0.435)
Step 4  = −20.35 + 4.17 = −16.18 per year
Step 5  Θ per day = −16.18 / 365 ≈ −$0.0443 per share per day
Step 6  → Put theta is less negative than call theta: interest adds value to put buyers

Source: Hull, J.C. — Options, Futures, and Other Derivatives, 11th ed., Ch. 19 (2024-01-15)

Calculate Theta (Put)

Current market price of the underlying stock

Option strike price

Annual risk-free rate

Time to expiration in years

Annualised implied volatility

Theta (per day)

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Not investment advice.

How to Interpret Theta (Put)

< -0.1
High decay — short-dated ATM put
-0.1 – -0.02
Moderate decay — typical near-term put
-0.02 – 0
Low or minimal decay — long-dated or far OTM/ITM
> 0
Positive theta — deep ITM put (interest > vol decay)
Sources
  1. Hull, J.C. — Options, Futures, and Other Derivatives, 11th ed.
  2. Investopedia — Theta