Put-Call Parity
Calculate the Put-Call Parity gap, the amount by which Black-Scholes call and put prices deviate from the no-arbitrage relationship between them.
Put-Call Parity states that, for European options sharing the same strike price and time to expiration, the following relationship must hold in order to prevent arbitrage: C - P = S * e^(-q * t) - K * e^(-r * t).
The formula is as follows:
\[\text{Parity Gap} = (C - P) - (S \cdot e ^{- q \cdot t} - K \cdot e ^{- r \cdot t})\]Where C is the Black-Scholes call option price, P is the Black-Scholes put option price, S is the stock price, K is the strike price, r is the continuously compounded risk-free rate, q is the continuous dividend yield (ln(1 + dividend yield)) and t is the time to expiration in years (days / 365).
This method computes the Black-Scholes call and put price for each ticker, strike price and time to expiration and then calculates the parity gap, i.e. the amount by which (C - P) deviates from S * e^(-qt) - K * e^(-rt). Because both prices come from the same Black-Scholes model and inputs, the gap is (up to floating point precision) always zero - this is a useful diagnostic to confirm that a set of option prices is internally consistent, or, when plugging in externally observed call and put prices, to detect potential arbitrage.
Also known as: Put-Call Parity, PCP, the no-arbitrage relationship between calls and puts.
Put-Call Parity in Python or via the MCP
Install the Finance Toolkit with:
pip install financetoolkit -U
Then call get_put_call_parity as shown below.
from financetoolkit import Toolkit
toolkit = Toolkit(
["JNJ"],
api_key="FINANCIAL_MODELING_PREP_KEY",
start_date="2025-01-01",
end_date="2025-12-31",
)
parity_gap = toolkit.options.get_put_call_parity(
strike_price_range=0.1, strike_step_size=5
)
parity_gap.loc["JNJ", ["2026-01-07", "2026-01-14", "2026-01-28"]]
Or, after setting up the Finance Toolkit MCP, ask:
Did Black-Scholes call and put prices on Johnson & Johnson satisfy put-call parity at the end of 2025 for the expiries of 7, 14 and 28 January 2026, for strikes within 10% of the share price in $5 steps?
Which returns:
| Strike Price | 2026-01-07 | 2026-01-14 | 2026-01-28 |
|---|---|---|---|
| 185 | 0 | 0 | 0 |
| 190 | 0 | 0 | 0 |
| 195 | 0 | 0 | 0 |
| 200 | 0 | 0 | 0 |
| 205 | 0 | 0 | 0 |
| 210 | 0 | 0 | 0 |
| 215 | 0 | 0 | 0 |
| 220 | 0 | 0 | 0 |
Parameters
get_put_call_parity 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 |
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 |