Calculate the speed 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 speed is the rate of change of the gamma with respect to the price of the underlying asset.

The speed calculation is the theoretical value of the speed. The actual speed 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:

\[d_{1} = (\ln(S / K) + (r - q + (\sigma ^{2}) / 2) \cdot t) / (\sigma \cdot \sqrt{t})\] \[\text{Speed} = - e ^{- q \cdot t} \cdot ((N'(d_{1}) / (S ^{2} \cdot \sigma \cdot \sqrt{t}))) \cdot ((d_{1} / (\sigma \cdot \sqrt{t})) + 1)\]

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.

The Speed can be interpreted as follows:

  • If Speed is positive, the option’s Gamma rises as the underlying price rises, so the position’s convexity builds up on the way up.
  • If Speed is negative, the option’s Gamma falls as the underlying price rises, which is the usual case just below the strike where Gamma is already close to its peak.

Note that the speed of a call option and put option are equal to each other.

Also known as: gamma rate of change.

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

Calculate the Speed in Python

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

pip install financetoolkit -U

Then call get_speed as shown below.

from financetoolkit import Toolkit

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

toolkit.options.get_speed().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 0.0005 0.0005 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004
340 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004
345 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004
350 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003
355 0.0002 0.0002 0.0002 0.0002 0.0003 0.0003 0.0003 0.0003
360 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002
365 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0002
370 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
375 0 0 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
380 0 0 0 0 0 0 0 0.0001

Parameters

get_speed 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.
Zomma Color Ultima

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

Share