Put Delta
First derivative of the Black-Scholes put price with respect to the underlying. Equal to call delta minus 1 by put-call parity.
When to use: Use to size or hedge put exposure. Negative delta reflects the put's inverse relationship with the underlying.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| Delta | Delta | Change in option price per $1 change in the underlying | integer |
| S | Stock Price | Current price of the underlying | $ |
| Strike | Strike Price | Exercise price of the option contract | $ |
| Rf | Risk-Free Rate | Continuously compounded annual risk-free rate as a decimal | % |
| Sigma | Volatility | Annualized volatility of the underlying as a decimal (e.g. 0.25 for 25%) | % |
| T | Time to Expiration | Time to expiration in years (e.g. 0.25 for 3 months) | years |
Real-Life Examples
Example 1: ATM Put Delta, 30% Vol, 90 Days
Stock at $100, $100-strike 90-day put, r = 4%, σ = 30%.
Given
Step-by-Step
Each $1 rise in the stock drops the put price by ~$0.44. Equivalently, you can read |Δ_put| = 0.44 as a rough approximation of the chance the put finishes ITM.
Frequently Asked Questions
Strictly between −1 and 0. Deep OTM puts approach 0; deep ITM puts approach −1. ATM puts are slightly less negative than −0.50 (mirror of the call delta drift).
Differentiate put-call parity (C − P = S − K·e^(−rT)) with respect to S: ∂C/∂S − ∂P/∂S = 1, so Δ_call − Δ_put = 1.