Call Delta
First derivative of the Black-Scholes call price with respect to the underlying. Approximates the dollar change in call value per $1 change in the stock, and is the model's implied probability that the call expires in the money under the risk-neutral measure.
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
Strictly between 0 and 1. Deep OTM calls approach 0; deep ITM calls approach 1. ATM calls are slightly above 0.50 (driven up by the positive drift in the risk-neutral measure when r > 0).
Approximately, but not exactly. N(d1) is the call delta; N(d2) is the risk-neutral probability of finishing ITM. The two differ by a function of volatility and time but are close for short-dated options.
For ATM options, delta drifts toward 0.5 as expiration approaches but is otherwise relatively stable. For OTM options it drifts toward 0; for ITM options it drifts toward 1. The rate of change is gamma.