Dual Gamma
Calculate the dual gamma 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 gamma is the second derivative of the option price with respect to the strike price, i.e. the rate of change of the dual delta as the strike price changes. It is the strike-space counterpart of gamma and describes the (discounted) risk-neutral probability density of the underlying finishing at the strike.
The dual gamma calculation is the theoretical value of the dual gamma. The actual dual gamma 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:
\[d_{1} = (\ln(S / K) + (r - q + (\sigma ^{2}) / 2) \cdot t) / (\sigma \cdot \sqrt{t})\] \[d_{2} = d_{1} - \sigma \cdot \sqrt{t}\] \[\text{Dual Gamma} = e ^{- r \cdot t} \cdot N'(d_{2}) / (K \cdot \sigma \cdot \sqrt{t})\]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’(d2) is the standard normal probability density at d2 and N(d1) is the cumulative normal distribution of d1. Note that Dual Gamma is a second derivative with respect to the strike price, so it is the strike and not the stock price that appears in the denominator.
Note that the dual gamma of a call option and put option are equal to each other.
Also known as: strike gamma, binary option gamma.
Dual Gamma in Python or via the MCP
Install the Finance Toolkit with:
pip install financetoolkit -U
Then call get_dual_gamma as shown below.
from financetoolkit import Toolkit
toolkit = Toolkit(
["T"],
api_key="FINANCIAL_MODELING_PREP_KEY",
start_date="2025-01-01",
end_date="2025-12-31",
)
toolkit.options.get_dual_gamma(
strike_price_range=0.2, strike_step_size=1
).loc["T", ["2026-01-07", "2026-01-14", "2026-01-28"]]
Or, after setting up the Finance Toolkit MCP, ask:
What was the dual gamma of AT&T's call options at the end of 2025 for the expiries of 7, 14 and 28 January 2026, for strikes within 20% of the share price in $1 steps?
Which returns:
| Strike Price | 2026-01-07 | 2026-01-14 | 2026-01-28 |
|---|---|---|---|
| 19 | 0 | 0 | 0 |
| 20 | 0 | 0 | 0.0007 |
| 21 | 0 | 0.0003 | 0.008 |
| 22 | 0.0003 | 0.0089 | 0.0445 |
| 23 | 0.0254 | 0.0864 | 0.1339 |
| 24 | 0.2943 | 0.2846 | 0.235 |
| 25 | 0.5063 | 0.3616 | 0.2566 |
| 26 | 0.1612 | 0.198 | 0.1842 |
| 27 | 0.0115 | 0.0513 | 0.0911 |
| 28 | 0.0002 | 0.0068 | 0.0323 |
Parameters
get_dual_gamma 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 means it will use the current dividend yield. | 0.02 |
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 |