Black Scholes Model
Calculate the Black Scholes Model, a mathematical model used to estimate the price of European-style options.
The Black Scholes Model is a mathematical model used to estimate the price of European-style options. It is widely used by traders and investors to determine the theoretical value of an option, and to assess the potential risks and rewards of a position.
Within Risk Management, defining the theoretical value of an option is important to assess the potential risk and rewards of an option position. A position that could be used to hedge a portfolio, for example, is a long put option. The theoretical value of this option can be used to determine the potential risk and rewards of this position.
The Black Scholes Model is based on several assumptions, including the following:
- The option is European and can only be exercised at expiration.
- The underlying stock follows a lognormal distribution.
- The risk-free rate and volatility of the underlying stock are known and constant.
- The returns on the underlying stock are normally distributed.
By default the stock price (the traded close), risk free rate, dividend yield and volatility of the most recent date are used; changing the start date prices the option as of that date instead. The volatility is the annualized standard deviation of the daily returns over the year before that date and the dividend yield the dividends of the 365 days up to that date relative to its close, so an earlier start date only uses what was known on that date. The exception is a date in the first year of the price history: without a full year before it, it takes the dividend yield of the first date that has one.
The formulas are as follows:
\[d_{1} = (\ln(S / K) + (r - q + (\sigma ^{2}) / 2) \cdot t) / (\sigma \cdot \sqrt{t})\] \[d_{2} = d_{1} - \sigma \cdot \sqrt{t}\] \[\text{Call Option Price} = S \cdot e ^{- q \cdot t} \cdot N(d_{1}) - K \cdot e ^{- r \cdot t} \cdot N(d_{2})\] \[\text{Put Option Price} = K \cdot e ^{- r \cdot t} \cdot N(- d_{2}) - S \cdot e ^{- q \cdot t} \cdot N(- d_{1})\]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 model price is only as good as its volatility input, and here that input is realised (historical) volatility, while traded options are priced off implied volatility, which usually differs from it and varies by strike and expiry. Practitioners therefore use the Black Scholes value as a benchmark and compare it with market prices from get_option_chains, or back the market’s volatility out with get_implied_volatility. Deep out-of-the-money options are most sensitive to that choice, since the model’s flat volatility ignores the skew.
Also known as: BSM, Black-Scholes-Merton, option pricing model.
Black Scholes Model in Python or via the MCP
Install the Finance Toolkit with:
pip install financetoolkit -U
Then call get_black_scholes_model as shown below.
from financetoolkit import Toolkit
toolkit = Toolkit(
["NVDA"],
api_key="FINANCIAL_MODELING_PREP_KEY",
start_date="2025-01-01",
end_date="2025-12-31",
)
black_scholes = toolkit.options.get_black_scholes_model(
strike_price_range=0.15, strike_step_size=5, expiration_time_range=91
)
black_scholes.loc["NVDA", ["2026-01-30", "2026-03-01", "2026-03-31"]]
Or, after setting up the Finance Toolkit MCP, ask:
What were Nvidia's call options worth under the Black-Scholes model 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 |
|---|---|---|---|
| 160 | 28.7159 | 31.761 | 34.5215 |
| 165 | 24.6249 | 28.1288 | 31.1403 |
| 170 | 20.8373 | 24.7538 | 27.9815 |
| 175 | 17.3911 | 21.6463 | 25.0482 |
| 180 | 14.3123 | 18.8111 | 22.34 |
| 185 | 11.6125 | 16.2474 | 19.8537 |
| 190 | 9.2892 | 13.9492 | 17.5835 |
| 195 | 7.3269 | 11.9066 | 15.5214 |
| 200 | 5.6997 | 10.106 | 13.6576 |
| 205 | 4.3742 | 8.5311 | 11.9811 |
| 210 | 3.313 | 7.1639 | 10.4798 |
Parameters
get_black_scholes_model 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" |
put_option | bool | Whether to calculate the put option price. Defaults to False which means it will calculate the call option price. | True |
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 |
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 |