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.
Theta is the cost of holding an option for one more calendar day, all else equal, and is the other side of gamma: a long at-the-money option close to expiry has the most gamma and pays for it with the most negative theta. Because it is per calendar day, a weekend counts as two days of decay; desks that use a trading-day convention divide the yearly figure by 252 instead, which gives a larger daily number.
Also known as: time decay, option time value erosion.
Theta in Python or via the MCP
Install the Finance Toolkit with:
pip install financetoolkit -U
Then call get_theta as shown below.
from financetoolkit import Toolkit
toolkit = Toolkit(
["AVGO"],
api_key="FINANCIAL_MODELING_PREP_KEY",
start_date="2025-01-01",
end_date="2025-12-31",
)
toolkit.options.get_theta(
strike_price_range=0.15, strike_step_size=10
).loc["AVGO", ["2026-01-07", "2026-01-14", "2026-01-28"]]
Or, after setting up the Finance Toolkit MCP, ask:
What was the theta of Broadcom's call options at the end of 2025 for the expiries of 7, 14 and 28 January 2026, for strikes within 15% of the share price in $10 steps?
Which returns:
| Strike Price | 2026-01-07 | 2026-01-14 | 2026-01-28 |
|---|---|---|---|
| 290 | -0.0622 | -0.1332 | -0.18 |
| 300 | -0.1272 | -0.2073 | -0.2284 |
| 310 | -0.2474 | -0.2972 | -0.2762 |
| 320 | -0.4168 | -0.3897 | -0.3184 |
| 330 | -0.5934 | -0.4679 | -0.3511 |
| 340 | -0.7151 | -0.5165 | -0.371 |
| 350 | -0.7358 | -0.5271 | -0.3771 |
| 360 | -0.6538 | -0.4999 | -0.3696 |
| 370 | -0.5071 | -0.4432 | -0.3502 |
| 380 | -0.347 | -0.369 | -0.3216 |
| 390 | -0.2114 | -0.2901 | -0.287 |
Parameters
get_theta accepts the following parameters.
| Parameter | Type | Description | Example |
|---|---|---|---|
start_date | str | None | The date whose price, volatility and rates the values start from; a date without prices (a weekend or a holiday) uses the last trading day before it, and a ticker without a price on that day is left out with a warning. Defaults to None, which uses the last date of the historical data. | "2021-01-01" |
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. | 0.1 |
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. | 1 |
expiration_time_range | int | The number of days to use for the time to expiration: the columns are the expirations 1 up to expiration_time_range - 1 days after the start date. Defaults to 30, which gives expirations of 1 to 29 days. | 90 |
risk_free_rate | float | The risk free rate to use for the calculation. Defaults to None, which uses the Toolkit’s risk-free rate on the pricing date: the 10-year Treasury yield unless the Toolkit’s risk_free_rate parameter selects ‘13w’, ‘5y’ or ‘30y’, converted from its semi-annual bond-equivalent quote to continuous compounding. A rate passed here is used as given, as a continuously compounded rate. | 0.04 |
dividend_yield | float | The dividend yield to use for the calculation. Defaults to None, which uses the dividend yield on the pricing date (the dividends of the 365 days up to it relative to its close, as the continuously compounded ln(1 + q)). A given value is used as a continuous yield. | 0.02 |
put_option | bool | Whether to calculate the put option theta. Defaults to False which means it will calculate the call option theta. | True |
show_input_info | bool | Whether to show the input information. Defaults to False. | True |
rounding | int | None | The number of decimals to round the results to. Defaults to None, which uses the rounding of the Toolkit (4 unless set otherwise). | 2 |
standardize | bool | Whether to standardize (Z-Score) the result across the time to expiration columns for each ticker and strike price. Defaults to False. | True |