Vera
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.
Vera in Python or via the MCP
Install the Finance Toolkit with:
pip install financetoolkit -U
Then call get_vera as shown below.
from financetoolkit import Toolkit
toolkit = Toolkit(
["LLY"],
api_key="FINANCIAL_MODELING_PREP_KEY",
start_date="2025-01-01",
end_date="2025-12-31",
)
toolkit.options.get_vera(
strike_price_range=0.15, strike_step_size=20, expiration_time_range=181
).loc["LLY", ["2026-01-30", "2026-03-31", "2026-06-29"]]
Or, after setting up the Finance Toolkit MCP, ask:
What was the vera of Eli Lilly's call options at the end of 2025 for the expiries of 30 January, 31 March and 29 June 2026, for strikes within 15% of the share price in $20 steps?
Which returns:
| Strike Price | 2026-01-30 | 2026-03-31 | 2026-06-29 |
|---|---|---|---|
| 920 | -45.0097 | -157.22 | -295.3 |
| 940 | -50.1028 | -152.347 | -280.257 |
| 960 | -52.655 | -143.627 | -261.579 |
| 980 | -52.0124 | -131.274 | -239.616 |
| 1000 | -47.8631 | -115.645 | -214.77 |
| 1020 | -40.3005 | -97.2153 | -187.477 |
| 1040 | -29.8155 | -76.5467 | -158.194 |
| 1060 | -17.222 | -54.2559 | -127.389 |
| 1080 | -3.5353 | -30.9814 | -95.5245 |
| 1100 | 10.1718 | -7.3538 | -63.0514 |
| 1120 | 22.905 | 16.0301 | -30.3978 |
| 1140 | 33.8572 | 38.6281 | 2.0364 |
| 1160 | 42.4753 | 59.9686 | 33.8855 |
| 1180 | 48.4829 | 79.6604 | 64.8218 |
| 1200 | 51.8631 | 97.3981 | 94.5578 |
| 1220 | 52.8138 | 112.963 | 122.847 |
Parameters
get_vera accepts the following parameters.
| Parameter | Type | Description | Example |
|---|---|---|---|
start_date | str | None | The date whose price, volatility and rates the values start from; a date without prices (a weekend or a holiday) uses the last trading day before it, and a ticker without a price on that day is left out with a warning. Defaults to None, which uses the last date of the historical data. | "2021-01-01" |
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. | 0.1 |
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. | 1 |
expiration_time_range | int | The number of days to use for the time to expiration: the columns are the expirations 1 up to expiration_time_range - 1 days after the start date. Defaults to 30, which gives expirations of 1 to 29 days. | 90 |
risk_free_rate | float | The risk free rate to use for the calculation. Defaults to None, which uses the Toolkit’s risk-free rate on the pricing date: the 10-year Treasury yield unless the Toolkit’s risk_free_rate parameter selects ‘13w’, ‘5y’ or ‘30y’, converted from its semi-annual bond-equivalent quote to continuous compounding. A rate passed here is used as given, as a continuously compounded rate. | 0.04 |
dividend_yield | float | The dividend yield to use for the calculation. Defaults to None which means it will use the current dividend yield. | 0.02 |
show_input_info | bool | Whether to show the input information. Defaults to False. | True |
rounding | int | None | The number of decimals to round the results to. Defaults to None, which uses the rounding of the Toolkit (4 unless set otherwise). | 2 |
standardize | bool | Whether to standardize (Z-Score) the result across the time to expiration columns for each ticker and strike price. Defaults to False. | True |