Lambda
Calculate the lambda 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 lambda, or elasticity, is the percentage change in the option price for a 1% change in the price of the underlying asset, i.e. the leverage the option gives.
The lambda calculation is the theoretical value of the lambda. The actual lambda 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:
\[d_{1} = (\ln(S / K) + (r - q + (\sigma ^{2}) / 2) \cdot t) / (\sigma \cdot \sqrt{t})\] \[d_{2} = d_{1} - \sigma \cdot \sqrt{t}\] \[\text{Call}\; \Delta = e ^{- q \cdot t} \cdot N(d_{1}),\;\; \text{Put}\; \Delta = - e ^{- q \cdot t} \cdot N(- d_{1})\] \[\text{Call Option Price} = S \cdot e ^{- q \cdot t} \cdot N(d_{1}) - K \cdot e ^{- r \cdot t} \cdot N(d_{2})\] \[\text{Put Option Price} = K \cdot e ^{- r \cdot t} \cdot N(- d_{2}) - S \cdot e ^{- q \cdot t} \cdot N(- d_{1})\] \[\text{Lambda} = \Delta \cdot (\text{Stock Price} / \text{Call Option Price or Put Option Price})\]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 Lambda can be interpreted as follows:
- If Lambda is positive, it indicates that the option value will increase as the underlying price increases, and vice versa.
- If Lambda is negative, it implies that the option value will decrease as the underlying price increases, and vice versa.
Also known as: option elasticity, leverage factor.
Lambda in Python or via the MCP
Install the Finance Toolkit with:
pip install financetoolkit -U
Then call get_lambda as shown below.
from financetoolkit import Toolkit
toolkit = Toolkit(
["V"],
api_key="FINANCIAL_MODELING_PREP_KEY",
start_date="2025-01-01",
end_date="2025-12-31",
)
toolkit.options.get_lambda(
strike_price_range=0.1, strike_step_size=5, expiration_time_range=91
).loc["V", ["2026-01-30", "2026-03-01", "2026-03-31"]]
Or, after setting up the Finance Toolkit MCP, ask:
What was the lambda (the leverage) of Visa's call options at the end of 2025 for the expiries of 30 January, 1 March and 31 March 2026, for strikes within 10% of the share price in $5 steps?
Which returns:
| Strike Price | 2026-01-30 | 2026-03-01 | 2026-03-31 |
|---|---|---|---|
| 315 | 9.0898 | 8.0785 | 7.3271 |
| 320 | 10.1189 | 8.7379 | 7.8111 |
| 325 | 11.2877 | 9.4477 | 8.321 |
| 330 | 12.5998 | 10.2062 | 8.8559 |
| 335 | 14.0544 | 11.0116 | 9.4143 |
| 340 | 15.647 | 11.8613 | 9.9949 |
| 345 | 17.37 | 12.7522 | 10.5961 |
| 350 | 19.2134 | 13.6812 | 11.2162 |
| 355 | 21.1656 | 14.6449 | 11.8537 |
| 360 | 23.2145 | 15.6398 | 12.5069 |
| 365 | 25.3476 | 16.6627 | 13.1742 |
| 370 | 27.553 | 17.7102 | 13.8541 |
| 375 | 29.8195 | 18.7791 | 14.5451 |
| 380 | 32.1368 | 19.8667 | 15.2458 |
Parameters
get_lambda 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 uses the dividend yield on the pricing date (the dividends of the 365 days up to it relative to its close, as the continuously compounded ln(1 + q)). A given value is used as a continuous yield. | 0.02 |
put_option | bool | Whether to calculate the put option lambda. Defaults to False which means it will calculate the call option lambda. | True |
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 |