Put Theta
Rate of change of put price with respect to the passage of time, expressed per year. Usually negative; can be positive for deep ITM European puts on non-dividend-paying stock when the rate-driven term dominates.
When to use: Use to size time-decay exposure on long or short put positions. Pair with call theta and rho to understand how interest-rate effects shape option values across types.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| Theta | Theta | Change in option price per year of time decay (annualized) | $ |
| 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 Theta, 30% Vol, 90 Days
Stock at $100, $100-strike 90-day put, r = 4%, σ = 30%.
Given
Step-by-Step
The put loses about $9.86 per year of time, less decay than the equivalent call ($13.82). The difference is the rate-effect term: a positive risk-free rate slows put decay because the present value of the strike (which the put pays out) is rising as time passes.
Frequently Asked Questions
Because the put's payout is the strike (paid to the put holder if exercised), and the present value of that fixed strike rises as time-to-expiration shrinks. That is a tailwind for put value, partially offsetting the volatility-driven decay.
Rarely, but possible: deep ITM European puts on non-dividend-paying stock where the rate effect dominates the volatility-decay effect. The setup is unusual because real puts are American and would be exercised early to capture the strike's present value, eliminating the anomaly.