Call Theta

Rate of change of call price with respect to the passage of time, expressed per year. Almost always negative — calls lose value as time passes (time decay).

When to use: Use to size time-decay exposure on long or short call positions. Convert to a per-day figure by dividing by 365 (or 252 for trading-day-based brokers) — the "$/day" decay most platforms display.

Calculator

Formula

\Theta_{\text{call}} = -\frac{S \varphi(d_1) \sigma}{2\sqrt{T}} - r K e^{-rT} N(d_2)

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 Call Theta, 30% Vol, 90 Days

Stock at $100, $100-strike 90-day call, r = 4%, σ = 30%.

Given

S = $100.00Strike = $100.00Rf = 4.0000%Sigma = 30.0000%T = 0.25 years

Step-by-Step

1.d1 = 0.1417, d2 = −0.0083

2.φ(d1) ≈ 0.3950, N(d2) ≈ 0.4967

3.First term = −(100 × 0.3950 × 0.30) / (2 × √0.25) = −11.851 / 1 = −11.851

4.Second term = −0.04 × 100 × e^(−0.01) × 0.4967 = −1.967

5.Θ = −11.851 − 1.967 = −13.82 per year

Result:-$13.82

The call loses about $13.82 per year of time, or roughly $0.038 per calendar day. Theta accelerates as expiration nears — the per-day decay grows even though the per-year figure shrinks.

Frequently Asked Questions

For this non-dividend-paying European call model with nonnegative interest rates, theta is negative in the interior. Negative rates can produce positive call theta in some cases.

The output is price change per year as calendar time passes, holding other inputs fixed. Divide by 365 for a calendar-day approximation; this is not an exact forecast of tomorrow’s change.

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