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

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

\[d_{1} = (\ln(S / K) + (r - q + (\sigma ^{2}) / 2) \cdot t) / (\sigma \cdot \sqrt{t})\] \[\text{Call Option} \Delta = e ^{- q \cdot t} \cdot N(d_{1})\] \[\text{Put Option} \Delta = - e ^{- q \cdot t} \cdot N(- d_{1})\]

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 Delta can be interpreted as follows:

  • For call options, Delta is positive, indicating that the option price tends to move in the same direction as the underlying asset’s price.
  • For put options, Delta is negative, indicating that the option price tends to move in the opposite direction to the underlying asset’s price.

Note that the delta of a call option is always between 0 and e^(-q * t), while the delta of a put option is always between -e^(-q * t) and 0. Without a dividend yield those bounds collapse to the familiar 0 to 1 and -1 to 0.

Also known as: option price sensitivity to underlying, hedge ratio.

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

Calculate the Delta in Python

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

pip install financetoolkit -U

Then call get_delta as shown below.

from financetoolkit import Toolkit

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

toolkit.options.get_delta().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 0.12 0.1259 0.1316 0.1372 0.1426 0.1478 0.1529 0.1578
340 0.0807 0.0859 0.091 0.096 0.101 0.1058 0.1105 0.1151
345 0.0523 0.0566 0.0609 0.0652 0.0694 0.0736 0.0777 0.0818
350 0.0328 0.0361 0.0395 0.0429 0.0463 0.0497 0.0532 0.0566
355 0.0198 0.0223 0.0248 0.0274 0.03 0.0327 0.0355 0.0382
360 0.0116 0.0133 0.0151 0.017 0.0189 0.021 0.023 0.0252
365 0.0066 0.0077 0.0089 0.0102 0.0116 0.0131 0.0146 0.0162
370 0.0036 0.0043 0.0051 0.006 0.007 0.008 0.009 0.0102
375 0.0019 0.0024 0.0029 0.0034 0.0041 0.0047 0.0055 0.0062
380 0.001 0.0013 0.0016 0.0019 0.0023 0.0027 0.0032 0.0038

Parameters

get_delta 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.
  • put_option (bool, optional): Whether to calculate the put option delta. Defaults to False which means it will calculate the call option delta.
  • 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.
Dual Delta Vega Theta Rho Epsilon Lambda

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

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