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:

\[C - P = S \cdot e ^{- q \cdot t} - K \cdot e ^{- r \cdot t}\]

Where C is the call option price, P is the put option price, S is the stock price, K is the strike price, r is the risk-free rate, q is the dividend yield and t is the time to expiration.

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.

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Calculate the Put-Call Parity in Python

The Put-Call Parity is available in the Options module of the open-source Finance Toolkit. Install it with:

pip install financetoolkit -U

Then call get_put_call_parity as shown below.

from financetoolkit import Toolkit

toolkit = Toolkit(["AMZN", "AAPL"], api_key="FINANCIAL_MODELING_PREP_KEY")

parity_gap = toolkit.options.get_put_call_parity()

parity_gap.loc['AMZN']

Parameters

get_put_call_parity 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.
  • 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.
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The Options module page introduces the module, and the sidebar lists all of its functions.

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