Calculate the vera 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 vera is the rate of change of the rho with respect to volatility.

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

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

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’(d2) is the standard normal probability density at d2 and N(d1) is the cumulative normal distribution of d1.

Vera is reported unscaled, per 1.00 of volatility and per 1.00 of the risk free rate, matching Rho.

The Vera can be interpreted as follows:

  • If Vera is positive, it indicates that the option’s Rho becomes more positive as implied volatility rises. In other words, the option gains sensitivity to the risk free rate when volatility increases.
  • If Vera is negative, it suggests that the option’s Rho becomes more negative as implied volatility rises. The option loses sensitivity to the risk free rate when volatility increases.

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

Also known as: rho-vega cross-derivative.

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

Calculate the Vera in Python

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

pip install financetoolkit -U

Then call get_vera as shown below.

from financetoolkit import Toolkit

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

toolkit.options.get_vera().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 16.3308 17.2218 18.093 18.9446 19.7767 20.5898 21.3841 22.1601
340 14.5604 15.5679 16.5641 17.5477 18.5181 19.4746 20.4166 21.3439
345 12.0591 13.0965 14.1358 15.1746 16.2109 17.2429 18.269 19.2881
350 9.3674 10.3519 11.3535 12.3687 13.3945 14.4281 15.4673 16.5099
355 6.8714 7.7405 8.64 9.5661 10.5156 11.4853 12.4722 13.4738
360 4.7845 5.5032 6.2612 7.0555 7.8828 8.7404 9.6253 10.5348
365 3.1753 3.7352 4.3381 4.9819 5.6645 6.3834 7.1364 7.9212
370 2.0153 2.4283 2.883 3.3787 3.9142 4.4881 5.0989 5.7452
375 1.2269 1.5164 1.843 2.2068 2.6078 3.0457 3.52 4.0297
380 0.7181 0.9119 1.1359 1.3914 1.6791 1.9996 2.353 2.7394

Parameters

get_vera 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 Veta Partial Derivative

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

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