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 first derivative of the option price with respect to the strike price; up to the discount factor and its sign it is the risk-neutral probability of the option finishing in the money.
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: strike delta, binary option delta.
Dual Delta in Python or via the MCP
Install the Finance Toolkit with:
pip install financetoolkit -U
Then call get_dual_delta as shown below.
from financetoolkit import Toolkit
toolkit = Toolkit(
["JPM"],
api_key="FINANCIAL_MODELING_PREP_KEY",
start_date="2025-01-01",
end_date="2025-12-31",
)
toolkit.options.get_dual_delta(
strike_price_range=0.15, strike_step_size=5, expiration_time_range=91
).loc["JPM", ["2026-01-30", "2026-03-01", "2026-03-31"]]
Or, after setting up the Finance Toolkit MCP, ask:
What was the dual delta of JPMorgan Chase's call options at the end of 2025 for the expiries of 30 January, 1 March and 31 March 2026, for strikes within 15% of the share price in $5 steps?
Which returns:
| Strike Price | 2026-01-30 | 2026-03-01 | 2026-03-31 |
|---|---|---|---|
| 275 | -0.9832 | -0.9341 | -0.8885 |
| 280 | -0.9717 | -0.91 | -0.8604 |
| 285 | -0.9533 | -0.8799 | -0.8278 |
| 290 | -0.9258 | -0.8434 | -0.791 |
| 295 | -0.8874 | -0.8005 | -0.7504 |
| 300 | -0.8368 | -0.7519 | -0.7065 |
| 305 | -0.7742 | -0.6982 | -0.6599 |
| 310 | -0.7009 | -0.6406 | -0.6113 |
| 315 | -0.6195 | -0.5805 | -0.5616 |
| 320 | -0.5336 | -0.5193 | -0.5116 |
| 325 | -0.4474 | -0.4585 | -0.4621 |
| 330 | -0.3649 | -0.3995 | -0.4139 |
| 335 | -0.2892 | -0.3435 | -0.3676 |
| 340 | -0.2227 | -0.2915 | -0.3237 |
| 345 | -0.1667 | -0.2441 | -0.2827 |
| 350 | -0.1213 | -0.2018 | -0.2449 |
| 355 | -0.0858 | -0.1646 | -0.2105 |
| 360 | -0.059 | -0.1327 | -0.1794 |
| 365 | -0.0395 | -0.1056 | -0.1518 |
Parameters
get_dual_delta 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 dual delta. Defaults to False which means it will calculate the call option dual delta. | 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 |