Calculate the vanna 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 vanna is the rate of change of the vega with respect to the price of the underlying asset.

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

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

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 cumulative normal distribution of d1 and N(d2) is the the cumulative normal distribution of d2.

The Vanna can be interpreted as follows:

  • If Vanna is positive, it indicates that the Delta of the option becomes more positive as both the underlying asset’s price and implied volatility increase, and more negative as they both decrease.
  • If Vanna is negative, it suggests that the Delta of the option becomes more negative as both the underlying asset’s price and implied volatility increase, and more positive as they both decrease.

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

Also known as: delta-vega cross-derivative.

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

Calculate the Vanna in Python

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

pip install financetoolkit -U

Then call get_vanna as shown below.

from financetoolkit import Toolkit

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

toolkit.options.get_vanna().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 0.927 0.9375 0.9463 0.9536 0.9597 0.9647 0.9686 0.9717
340 0.8195 0.8399 0.8582 0.8747 0.8895 0.9027 0.9146 0.9252
345 0.6745 0.702 0.7275 0.7511 0.773 0.7932 0.8119 0.8292
350 0.5215 0.5522 0.5813 0.609 0.6352 0.6599 0.6833 0.7054
355 0.3812 0.4113 0.4406 0.469 0.4965 0.523 0.5484 0.5729
360 0.2646 0.2915 0.3183 0.3448 0.3709 0.3965 0.4217 0.4462
365 0.1752 0.1974 0.22 0.2428 0.2658 0.2888 0.3117 0.3344
370 0.111 0.1281 0.1459 0.1643 0.1832 0.2025 0.2221 0.2419
375 0.0675 0.0798 0.0931 0.1071 0.1218 0.1372 0.153 0.1693
380 0.0394 0.0479 0.0573 0.0674 0.0783 0.0899 0.1021 0.1149

Parameters

get_vanna 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 Charm Vomma Vera Veta Partial Derivative

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

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