Call Delta
Call Delta
Sensitivity of a European non-dividend-paying call price to the underlying price. In Black–Scholes it equals N(d1); the risk-neutral probability of finishing in the money is N(d2).
When to use: Use to size a directional bet in delta-equivalent terms, hedge a position to delta-neutral, or read off the model-implied probability of finishing ITM.
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 Call Delta, 30% Vol, 90 Days
Stock at $100, $100-strike 90-day call, r = 4%, σ = 30%.
Given
Step-by-Step
Each $1 move in the stock changes the call price by ~$0.56. Delta also implies a ~56% risk-neutral probability the call finishes ITM — a useful (if imperfect) intuition for traders.
Frequently Asked Questions
Not exactly. N(d2) is the risk-neutral in-the-money probability in this model. N(d1) is the call’s hedge sensitivity; neither is a real-world probability forecast.
Gamma measures how delta changes with the underlying price. Change in delta as calendar time passes is commonly called charm. At expiry, the sensitivity at the strike is not uniquely defined.