Put Option Delta

Options
Updated Apr 2026 • Has calculator

The rate of change in a put option's price for a $1 move in the underlying stock.

What is Delta (Put)?

Delta (Δ) for a put option measures how much the put price changes when the underlying stock moves by $1. Put delta is always negative, ranging from −1 to 0: a delta of −0.40 means the put gains $0.40 for every $1 fall in the stock. At-the-money puts have delta near −0.50. Deep in-the-money puts approach −1.0; far out-of-the-money puts approach 0. The absolute value approximates the probability that the put expires in the money.

Formula

Δ_put = N(d₁) − 1

Worked Example

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

Representative Q1 2024 market conditions

Step 1  Stock price (S): $185.00, Strike (K): $185.00
Step 2  Risk-free rate: 5.25%, Volatility: 28%, T = 0.25 yrs
Step 3  d₁ = 0.163 (same as call)
Step 4  Δ_put = N(0.163) − 1 = 0.565 − 1 = −0.435
Step 5  → If AAPL falls $1.00, this put gains approximately $0.44
Step 6  → Note: Δ_call + |Δ_put| = 0.565 + 0.435 = 1.00 (put-call parity)

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

Calculate Delta (Put)

Current market price of the underlying stock

Option strike price

Annual risk-free rate

Time to expiration in years

Annualised implied volatility

Put Delta

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Not investment advice.

How to Interpret Delta (Put)

< -0.75
Deep ITM put — acts like short stock
-0.75 – -0.55
In-the-money — strong downside exposure
-0.55 – -0.45
At-the-money — delta ≈ −0.50
> -0.45
Out-of-the-money — limited downside sensitivity
Sources
  1. Hull, J.C. — Options, Futures, and Other Derivatives, 11th ed.
  2. Investopedia — Delta
  3. CBOE Education Center