Calculate the partial derivative 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 partial derivative is the rate of change of the option price with respect to the strike price.

Note that this uses a single, flat assumed volatility (the same value at every strike price) rather than the market’s actual implied volatility smile. This means it is NOT the Breeden-Litzenberger risk-neutral density – with a flat volatility input the second derivative can only ever recover a lognormal density, regardless of what the real market smile looks like, which defeats the entire purpose of that theorem. For the actual market-implied (smile-consistent) risk-neutral density, see get_risk_neutral_density, which uses this same second-derivative relationship but applied to a volatility surface calibrated to real market option prices instead of a flat assumption.

The formula is as follows:

\[\text{Partial Derivative} (\text{PD}) = e ^{- r \cdot t} \cdot (1 / K) \cdot (1 / \sqrt{2 \cdot \pi \cdot \sigma ^{2} \cdot t}) \cdot e ^{- (1 / (2 \cdot \sigma ^{2} \cdot t)) \cdot (\ln(K / S) - ((r - q) - (0.5 \cdot \sigma ^{2})) \cdot t) ^{2}}\]

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 and t is the time to expiration. This expression is algebraically identical to e^(-r * t) * N’(d2) / (K * σ * sqrt(t)), i.e. to the Dual Gamma, since both are the second derivative of the option price with respect to the strike price.

Also known as: numerical derivative, option sensitivity.

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

Calculate the Partial Derivative in Python

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

pip install financetoolkit -U

Then call get_partial_derivative as shown below.

from financetoolkit import Toolkit

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

toolkit.options.get_partial_derivative().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 0.0084 0.0085 0.0085 0.0086 0.0087 0.0087 0.0088 0.0088
340 0.0061 0.0062 0.0064 0.0065 0.0066 0.0067 0.0068 0.0069
345 0.0042 0.0044 0.0046 0.0047 0.0048 0.005 0.0051 0.0052
350 0.0028 0.003 0.0031 0.0033 0.0034 0.0036 0.0037 0.0038
355 0.0018 0.0019 0.0021 0.0022 0.0023 0.0025 0.0026 0.0027
360 0.0011 0.0012 0.0013 0.0014 0.0015 0.0016 0.0018 0.0019
365 0.0006 0.0007 0.0008 0.0009 0.001 0.0011 0.0012 0.0012
370 0.0004 0.0004 0.0005 0.0005 0.0006 0.0007 0.0007 0.0008
375 0.0002 0.0002 0.0003 0.0003 0.0004 0.0004 0.0005 0.0005
380 0.0001 0.0001 0.0002 0.0002 0.0002 0.0002 0.0003 0.0003

Parameters

get_partial_derivative 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.
Gamma Dual Gamma Vanna Charm Vomma Vera Veta

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

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