Zomma
Calculate the zomma 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 zomma is the rate of change of the gamma with respect to volatility.
The zomma calculation is the theoretical value of the zomma. The actual zomma 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:
\[d_{1} = (\ln(S / K) + (r - q + (\sigma ^{2}) / 2) \cdot t) / (\sigma \cdot \sqrt{t})\] \[d_{2} = d_{1} - \sigma \cdot \sqrt{t}\] \[\text{Zomma} = e ^{- q \cdot t} \cdot (N'(d_{1}) \cdot (d_{1} \cdot d_{2} - 1)) / (S \cdot \sigma ^{2} \cdot \sqrt{t})\]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 and N’(d1) is the standard normal probability density at d1. This is equivalently Gamma * (d1 * d2 - 1) / σ, and it is reported unscaled, per 1.00 of volatility.
The Zomma can be interpreted as follows:
- If Zomma is positive, the option’s Gamma rises as implied volatility rises, which is typical for strikes well away from the money.
- If Zomma is negative, the option’s Gamma falls as implied volatility rises, which is typical for strikes near the money where Gamma is already at its peak.
Note that the zomma of a call option and put option are equal to each other.
Also known as: gamma sensitivity to volatility.
Zomma in Python or via the MCP
Install the Finance Toolkit with:
pip install financetoolkit -U
Then call get_zomma as shown below.
from financetoolkit import Toolkit
toolkit = Toolkit(
["CSCO"],
api_key="FINANCIAL_MODELING_PREP_KEY",
start_date="2025-01-01",
end_date="2025-12-31",
)
toolkit.options.get_zomma(
strike_price_range=0.15, strike_step_size=2
).loc["CSCO", ["2026-01-07", "2026-01-14", "2026-01-28"]]
Or, after setting up the Finance Toolkit MCP, ask:
What was the zomma of Cisco'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 $2 steps?
Which returns:
| Strike Price | 2026-01-07 | 2026-01-14 | 2026-01-28 |
|---|---|---|---|
| 64 | 0 | 0.0022 | 0.0379 |
| 66 | 0.0002 | 0.0168 | 0.085 |
| 68 | 0.0062 | 0.0729 | 0.1319 |
| 70 | 0.0671 | 0.1714 | 0.1246 |
| 72 | 0.2478 | 0.1771 | 0.0182 |
| 74 | 0.156 | -0.0713 | -0.1589 |
| 76 | -0.4947 | -0.4015 | -0.3028 |
| 78 | -0.5373 | -0.4265 | -0.3184 |
| 80 | 0.1031 | -0.1274 | -0.2014 |
| 82 | 0.2943 | 0.1514 | -0.032 |
| 84 | 0.1311 | 0.2138 | 0.0997 |
| 86 | 0.0273 | 0.1413 | 0.1536 |
Parameters
get_zomma 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 means it will use the current dividend yield. | 0.02 |
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 |