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 is the rate of change of the option price with respect to the underlying price.

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.

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

Calculate the Lambda in Python

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

pip install financetoolkit -U

Then call get_lambda as shown below.

from financetoolkit import Toolkit

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

toolkit.options.get_lambda().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 31.7968 30.7952 29.8705 29.0138 28.2175 27.4752 26.7813 26.1311
340 34.3986 33.273 32.2352 31.2748 30.3833 29.5531 28.778 28.0524
345 37.0389 35.7874 34.6346 33.5689 32.5805 31.661 30.8033 30.001
350 39.7078 38.3291 37.0601 35.8879 34.8017 33.7919 32.8506 31.9708
355 42.397 40.8903 39.5044 38.2251 37.0402 35.9395 34.914 33.9561
360 45.0993 43.4642 41.9611 40.5743 39.2905 38.0985 36.9886 35.9523
365 47.8086 46.0452 44.4248 42.9305 41.5478 40.2644 39.0699 37.955
370 50.5198 48.6284 46.891 45.2893 43.8078 42.4332 41.1542 39.9609
375 53.2287 51.2097 49.3558 47.6471 46.0671 44.6016 43.2383 41.9667
380 55.9315 53.7858 51.8158 50.0008 48.3228 46.7667 45.3195 43.97

Parameters

get_lambda 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 dividend yield as obtained through annual historical data.
  • put_option (bool, optional): Whether to calculate the put option lambda. Defaults to False which means it will calculate the call option lambda.
  • 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.
Delta Dual Delta Vega Theta Rho Epsilon

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

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