Charm
Calculate the charm 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 charm is the rate of change of the delta with respect to the time to expiration.
The charm calculation is the theoretical value of the charm. The actual charm 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:
\[d_{1} = (\ln(S / K) + (r - q + (\sigma ^{2}) / 2) \cdot t) / (\sigma \cdot \sqrt{t})\] \[d_{2} = d_{1} - \sigma \cdot \sqrt{t}\] \[\text{Call Charm} = q \cdot e ^{- q \cdot t} \cdot N(d_{1}) - e ^{- q \cdot t} \cdot N'(d_{1}) \cdot (2 \cdot (r - q) \cdot t - d_{2} \cdot \sigma \cdot \sqrt{t}) / (2 \cdot t \cdot \sigma \cdot \sqrt{t})\] \[\text{Put Charm} = - q \cdot e ^{- q \cdot t} \cdot N(- d_{1}) - e ^{- q \cdot t} \cdot N'(d_{1}) \cdot (2 \cdot (r - q) \cdot t - d_{2} \cdot \sigma \cdot \sqrt{t}) / (2 \cdot t \cdot \sigma \cdot \sqrt{t})\]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’(d1) is the standard normal probability density at d1 and N(d1) is the cumulative normal distribution of d1.
Charm is the derivative with respect to calendar time elapsed, in the same direction as Theta, but it is reported per year rather than per day. Divide by 365 for delta decay per calendar day. Color follows the same per-year convention.
The Charm can be interpreted as follows:
- If Charm is positive, it suggests that the option’s Delta is becoming more positive over time. In other words, the option is gaining sensitivity to changes in the underlying asset’s price as time passes.
- If Charm is negative, it indicates that the option’s Delta is becoming more negative over time. The option is losing sensitivity to changes in the underlying asset’s price as time passes.
Also known as: delta time decay, delta bleed.
No programming experience? With the Finance Toolkit MCP server, AI assistants such as Claude and ChatGPT can calculate the Charm for you. Just ask in plain English.
Calculate the Charm in Python
The Charm is available in the Options module of the open-source Finance Toolkit. Install it with:
pip install financetoolkit -U
Then call get_charm as shown below.
from financetoolkit import Toolkit
toolkit = Toolkit(["AAPL", "ASML"], api_key="FINANCIAL_MODELING_PREP_KEY")
toolkit.options.get_charm().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 | -2.1899 | -2.1241 | -2.0604 | -1.9988 | -1.9394 | -1.8823 | -1.8274 | -1.7748 |
| 340 | -1.918 | -1.8847 | -1.8499 | -1.8143 | -1.7781 | -1.7418 | -1.7056 | -1.6697 |
| 345 | -1.5682 | -1.5643 | -1.5567 | -1.5461 | -1.533 | -1.5179 | -1.5013 | -1.4833 |
| 350 | -1.2063 | -1.2239 | -1.237 | -1.2462 | -1.2521 | -1.255 | -1.2553 | -1.2534 |
| 355 | -0.8781 | -0.9078 | -0.9335 | -0.9555 | -0.9741 | -0.9896 | -1.0024 | -1.0126 |
| 360 | -0.6076 | -0.6413 | -0.6719 | -0.6998 | -0.7249 | -0.7474 | -0.7675 | -0.7853 |
| 365 | -0.4012 | -0.4329 | -0.463 | -0.4913 | -0.5178 | -0.5425 | -0.5654 | -0.5865 |
| 370 | -0.2536 | -0.2802 | -0.3063 | -0.3316 | -0.356 | -0.3794 | -0.4018 | -0.423 |
| 375 | -0.1538 | -0.1743 | -0.195 | -0.2157 | -0.2362 | -0.2563 | -0.2761 | -0.2953 |
| 380 | -0.0897 | -0.1045 | -0.1198 | -0.1355 | -0.1515 | -0.1676 | -0.1838 | -0.1999 |
Parameters
get_charm 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 current dividend yield.
- put_option (bool, optional): Whether to calculate the put option charm. Defaults to False which means it will calculate the call option charm.
- 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.
Related Second-Order Greeks
The Options module page introduces the module, and the sidebar lists all of its functions.