Theta (Time Decay)

Theta: Time Decay

Theta (Θ) measures how much the option price decreases as time passes (holding everything else constant):

$\Theta = \frac{\partial V}{\partial t}$

In practice, theta is usually expressed as the daily loss in option value.

Black-Scholes Theta for a Call

The annual theta for a European call is:

$\Theta_{\text{annual}} = -\frac{S \cdot N'(d_1) \cdot \sigma}{2\sqrt{T}} - r \cdot K \cdot e^{-rT} \cdot N(d_2)$

The daily theta divides by 365:

$\Theta_{\text{daily}} = \frac{\Theta_{\text{annual}}}{365}$

Why is Theta Negative?

Options lose value as expiry approaches (for long positions). With less time remaining, there's less opportunity for the stock to move in a favorable direction. This is called time decay or theta decay.

Key Properties

• Theta is most negative for ATM options near expiry
• Deep ITM and deep OTM options have less time value to lose
• Sellers of options profit from theta (positive theta)
• Buyers of options suffer from theta (negative theta)

The Gamma-Theta Trade-off

Long gamma (benefits from large moves) comes with short theta (hurt by time passing). This is the core trade-off in options trading.