Calculate the rho 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 rho is the rate of change of the option price with respect to the risk free interest rate.

The rho calculation is the theoretical value of the rho. The actual rho 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:

\[d_{1} = (\ln(S / K) + (r - q + (\sigma ^{2}) / 2) \cdot t) / (\sigma \cdot \sqrt{t})\] \[d_{2} = d_{1} - \sigma \cdot \sqrt{t}\] \[\text{Call Rho} = K \cdot t \cdot e ^{- r \cdot t} \cdot N(d_{2})\] \[\text{Put Rho} = - K \cdot t \cdot e ^{- r \cdot t} \cdot N(- d_{2})\]

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 Rho can be interpreted as follows:

  • If Rho is positive, it indicates that the option value will increase as the risk free rate increases, and vice versa.
  • If Rho is negative, it implies that the option value will decrease as the risk free rate increases, and vice versa.

Rho is reported unscaled, as the amount of money per share of the underlying that the value of the option gains or loses per 1.00 change in the risk-free rate. Divide by 100 for the more commonly quoted move per 1.0% per annum (100 basis points). Epsilon and Vera follow the same unscaled convention, while Vega and Veta are already divided by 100.

Also known as: option sensitivity to interest rate.

No programming experience? With the Finance Toolkit MCP server, AI assistants such as Claude and ChatGPT can calculate the Rho for you. Just ask in plain English.

Calculate the Rho in Python

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

pip install financetoolkit -U

Then call get_rho as shown below.

from financetoolkit import Toolkit

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

toolkit.options.get_rho().loc["AAPL"]

Which returns:

Strike Price 2026-07-24 2026-07-25 2026-07-26 2026-07-27 2026-07-28 2026-07-29 2026-07-30 2026-07-31
335 2.162 2.369 2.5819 2.8002 3.0237 3.252 3.485 3.7223
340 1.4575 1.6204 1.7898 1.9653 2.1465 2.3333 2.5254 2.7225
345 0.9474 1.0707 1.2004 1.3365 1.4785 1.6264 1.7798 1.9387
350 0.5944 0.6839 0.7796 0.8812 0.9888 1.102 1.2207 1.3449
355 0.3602 0.4227 0.4906 0.5638 0.6425 0.7263 0.8153 0.9095
360 0.211 0.253 0.2994 0.3504 0.4059 0.466 0.5307 0.5999
365 0.1197 0.1468 0.1774 0.2116 0.2496 0.2913 0.3369 0.3863
370 0.0657 0.0826 0.1021 0.1243 0.1494 0.1775 0.2087 0.243
375 0.035 0.0452 0.0572 0.0711 0.0872 0.1056 0.1263 0.1494
380 0.0181 0.024 0.0311 0.0397 0.0497 0.0613 0.0747 0.0899

Parameters

get_rho 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.
  • put_option (bool, optional): Whether to calculate the put option rho. Defaults to False which means it will calculate the call option rho.
  • 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.
Delta Dual Delta Vega Theta Epsilon Lambda

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

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