Theta
Calculate the theta of an option based on the Black Scholes Model. The Black Scholes Model is a mathematical model used to estimate the price of European-style options. The theta is the rate of change of the option price with respect to the passage of time.
The theta calculation is the theoretical value of the theta. The actual theta can differ from this value due to several factors such as the volatility of the underlying asset, the time to expiration, the risk free rate and more.
The formula is as follows:
\[d_{1} = (\ln(S / K) + (r - q + (\sigma ^{2}) / 2) \cdot t) / (\sigma \cdot \sqrt{t})\] \[d_{2} = d_{1} - \sigma \cdot \sqrt{t}\] \[\text{Call} \Theta = \left[- e ^{- q \cdot t} \cdot (S \cdot N'(d_{1}) \cdot \sigma) / (2 \cdot \sqrt{t}) - r \cdot K \cdot e ^{- r \cdot t} \cdot N(d_{2}) + q \cdot S \cdot e ^{- q \cdot t} \cdot N(d_{1})\right] / 365\] \[\text{Put} \Theta = \left[- e ^{- q \cdot t} \cdot (S \cdot N'(d_{1}) \cdot \sigma) / (2 \cdot \sqrt{t}) + r \cdot K \cdot e ^{- r \cdot t} \cdot N(- d_{2})\right]\] \[q \cdot S \cdot e ^{- q \cdot t} \cdot N(- d_{1}) \text{]} / 365\]Where S is the stock price, K is the strike price, r is the risk free rate, q is the dividend yield, σ is the volatility, t is the time to expiration, N’(d1) is the standard normal probability density at d1 and N(d2) is the cumulative normal distribution of d2.
Theta is the derivative with respect to calendar time elapsed, not with respect to the remaining time to maturity, and the division by 365 expresses it per calendar day rather than per year. Charm, Veta and Color measure the same passage of time in the same direction.
The Theta can be interpreted as follows:
- If Theta is negative, the option loses value with each day that passes, all else equal. This is the normal case for a long option, whose time value erodes towards expiration.
- If Theta is positive, the option gains value with each day that passes. This happens for instance on a deep in-the-money European put, where the discounting of the strike dominates.
Also known as: time decay, option time value erosion.
No programming experience? With the Finance Toolkit MCP server, AI assistants such as Claude and ChatGPT can calculate the Theta for you. Just ask in plain English.
Calculate the Theta in Python
The Theta is available in the Options module of the open-source Finance Toolkit. Install it with:
pip install financetoolkit -U
Then call get_theta as shown below.
from financetoolkit import Toolkit
toolkit = Toolkit(["AAPL", "ASML"], api_key="FINANCIAL_MODELING_PREP_KEY")
toolkit.options.get_theta().loc["AAPL"]
Which returns:
| Strike Price | 2026-07-24 | 2026-07-25 | 2026-07-26 | 2026-07-27 | 2026-07-28 | 2026-07-29 | 2026-07-30 | 2026-07-31 |
|---|---|---|---|---|---|---|---|---|
| 335 | -0.0965 | -0.0977 | -0.0988 | -0.0998 | -0.1006 | -0.1013 | -0.1019 | -0.1024 |
| 340 | -0.0718 | -0.0738 | -0.0755 | -0.0771 | -0.0785 | -0.0798 | -0.081 | -0.082 |
| 345 | -0.0512 | -0.0533 | -0.0554 | -0.0572 | -0.059 | -0.0606 | -0.0622 | -0.0636 |
| 350 | -0.0349 | -0.037 | -0.039 | -0.0409 | -0.0428 | -0.0445 | -0.0461 | -0.0477 |
| 355 | -0.0229 | -0.0247 | -0.0265 | -0.0283 | -0.0299 | -0.0316 | -0.0332 | -0.0347 |
| 360 | -0.0144 | -0.0159 | -0.0174 | -0.0188 | -0.0203 | -0.0217 | -0.0231 | -0.0245 |
| 365 | -0.0087 | -0.0098 | -0.011 | -0.0121 | -0.0133 | -0.0145 | -0.0156 | -0.0168 |
| 370 | -0.0051 | -0.0059 | -0.0067 | -0.0076 | -0.0085 | -0.0094 | -0.0103 | -0.0112 |
| 375 | -0.0029 | -0.0034 | -0.004 | -0.0046 | -0.0052 | -0.0059 | -0.0066 | -0.0073 |
| 380 | -0.0016 | -0.0019 | -0.0023 | -0.0027 | -0.0031 | -0.0036 | -0.0041 | -0.0046 |
Parameters
get_theta accepts the following parameters:
- start_date (str | None, optional): The start date which determines the stock price. Defaults to None which means it will use the most recent date.
- strike_price_range (float): The percentage range to use for the strike prices. Defaults to 0.25 which equals 25% and thus results in strike prices from 75 to 125 if the current stock price is 100.
- strike_step_size (int): The step size to use for the strike prices. Defaults to 5 which means that the strike prices will be 75, 80, 85, 90, 95, 100, 105, 110, 115 and 120 if the current stock price is 100.
- expiration_time_range (int): The number of days to use for the time to expiration. Defaults to 30 which equals 30 days.
- risk_free_rate (float, optional): The risk free rate to use for the calculation. Defaults to None which means it will use the current risk free rate.
- dividend_yield (float, optional): The dividend yield to use for the calculation. Defaults to None which means it will use the dividend yield as obtained through annual historical data.
- put_option (bool, optional): Whether to calculate the put option theta. Defaults to False which means it will calculate the call option theta.
- show_input_info (bool, optional): Whether to show the input information. Defaults to False.
- rounding (int | None, optional): The number of decimals to round the results to. Defaults to 4.
- standardize (bool, optional): Whether to standardize (Z-Score) the result across the time to expiration columns for each ticker and strike price. Defaults to False.
Related First-Order Greeks
The Options module page introduces the module, and the sidebar lists all of its functions.