Dual Delta
Calculate the dual 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 dual delta is the actual probability of an option finishing in the money which is the first derivative of option price with respect to strike.
The dual delta calculation is the theoretical value of the dual delta. The actual dual 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:
\[d_{1} = (\ln(S / K) + (r - q + (\sigma ^{2}) / 2) \cdot t) / (\sigma \cdot \sqrt{t})\] \[d_{2} = d_{1} - \sigma \cdot \sqrt{t}\] \[\text{Call Dual} \Delta = - e ^{- r \cdot t} \cdot N(d_{2})\] \[\text{Put Dual} \Delta = e ^{- r \cdot t} \cdot N(- d_{2})\]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 Dual Delta is the sensitivity of the option value to the strike price rather than to the underlying price. Up to the discount factor and a sign it is the risk-neutral probability that the option finishes in the money: a call Dual Delta of -0.5 corresponds to a roughly 50% chance of finishing in the money. It is negative for a call, since raising the strike lowers the call’s value, and positive for a put.
Also known as: cash delta, binary option delta.
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Calculate the Dual Delta in Python
The Dual Delta is available in the Options module of the open-source Finance Toolkit. Install it with:
pip install financetoolkit -U
Then call get_dual_delta as shown below.
from financetoolkit import Toolkit
toolkit = Toolkit(["AAPL", "ASML"], api_key="FINANCIAL_MODELING_PREP_KEY")
toolkit.options.get_dual_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 | -0.1071 | -0.1122 | -0.1172 | -0.122 | -0.1267 | -0.1312 | -0.1356 | -0.1398 |
| 340 | -0.0711 | -0.0756 | -0.0801 | -0.0844 | -0.0886 | -0.0928 | -0.0968 | -0.1008 |
| 345 | -0.0456 | -0.0493 | -0.0529 | -0.0566 | -0.0602 | -0.0637 | -0.0673 | -0.0707 |
| 350 | -0.0282 | -0.031 | -0.0339 | -0.0368 | -0.0397 | -0.0426 | -0.0455 | -0.0484 |
| 355 | -0.0168 | -0.0189 | -0.021 | -0.0232 | -0.0254 | -0.0277 | -0.0299 | -0.0322 |
| 360 | -0.0097 | -0.0112 | -0.0126 | -0.0142 | -0.0158 | -0.0175 | -0.0192 | -0.021 |
| 365 | -0.0054 | -0.0064 | -0.0074 | -0.0085 | -0.0096 | -0.0108 | -0.012 | -0.0133 |
| 370 | -0.0029 | -0.0035 | -0.0042 | -0.0049 | -0.0057 | -0.0065 | -0.0074 | -0.0083 |
| 375 | -0.0015 | -0.0019 | -0.0023 | -0.0028 | -0.0033 | -0.0038 | -0.0044 | -0.005 |
| 380 | -0.0008 | -0.001 | -0.0012 | -0.0015 | -0.0018 | -0.0022 | -0.0026 | -0.003 |
Parameters
get_dual_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.
- 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 dual delta. Defaults to False which means it will calculate the call option dual 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.
Related First-Order Greeks
The Options module page introduces the module, and the sidebar lists all of its functions.