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.

No programming experience? With the Finance Toolkit MCP server, AI assistants such as Claude and ChatGPT can calculate the Zomma for you. Just ask in plain English.

Calculate the Zomma in Python

The Zomma is available in the Options module of the open-source Finance Toolkit. Install it with:

pip install financetoolkit -U

Then call get_zomma as shown below.

from financetoolkit import Toolkit

toolkit = Toolkit(["AAPL", "ASML"], api_key="FINANCIAL_MODELING_PREP_KEY")

toolkit.options.get_zomma().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 0.0168 0.0144 0.0122 0.0102 0.0082 0.0064 0.0047 0.003
340 0.029 0.027 0.0251 0.0233 0.0214 0.0197 0.018 0.0164
345 0.0342 0.0331 0.0318 0.0306 0.0293 0.0279 0.0266 0.0253
350 0.0339 0.0336 0.0332 0.0326 0.032 0.0313 0.0305 0.0297
355 0.0298 0.0303 0.0306 0.0308 0.0308 0.0307 0.0305 0.0303
360 0.0241 0.0251 0.0259 0.0266 0.0271 0.0276 0.0279 0.0281
365 0.0181 0.0193 0.0204 0.0214 0.0222 0.023 0.0237 0.0243
370 0.0128 0.014 0.0151 0.0162 0.0172 0.0181 0.019 0.0198
375 0.0085 0.0096 0.0106 0.0116 0.0126 0.0136 0.0145 0.0154
380 0.0054 0.0063 0.0071 0.008 0.0089 0.0097 0.0106 0.0114

Parameters

get_zomma 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 current dividend yield.
  • 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.
Speed Color Ultima

The Options module page introduces the module, and the sidebar lists all of its functions.

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