# Options

The Options module is meant to calculate important options metrics such as the First, Second and Third Order Greeks, the Black Scholes Model and the Option Chains as well as Implied Volatilities, Breeden—Litzenberger and more.

To install the FinanceToolkit it simply requires the following:

If you are looking for documentation regarding the toolkit, discovery, ratios, technicals, fixed income, risk, performance and economics, please have a look below:

**init**

Initializes the Options Controller Class. The Options module is meant to calculate important options metrics such as the First, Second and Third Order Greeks, the Black Scholes Model and the Option Chains as well as Implied Volatilities, Breeden-Litzenberger and more.

**Args:**

__tickers (str | list[str]):__The tickers to use.__daily_historical (pd.DataFrame, optional):__The daily historical data. Defaults to pd.DataFrame().__annual_historical (pd.DataFrame, optional):__The annual historical data. Defaults to pd.DataFrame().__risk_free_rate (pd.DataFrame, optional):__The risk free rate. Defaults to pd.DataFrame().__quarterly (bool, optional):__Whether to use quarterly data. Defaults to False.__rounding (int | None, optional):__The number of decimals to round the results to. Defaults to 4.

As an example:

Which returns:

Strike Price | Delta | Dual Delta | Vega | Theta | Rho | Epsilon | Lambda | Gamma | Dual Gamma | Vanna | Charm | Vomma | Vera | Veta | PD | Speed | Zomma | Color | Ultima |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

205 | 1 | -0.9999 | 0 | -0.022 | 0.0056 | -0.6533 | 0.0712 | 0 | 0 | -0 | 0.0005 | 0.0003 | -0 | 0.1236 | 0 | -0 | 0 | 0.0004 | 0.0001 |

210 | 1 | -0.9999 | 0 | -0.0226 | 0.0058 | -0.6533 | 0.0837 | 0 | 0 | -0.0002 | 0.0221 | 0.0119 | -0.0001 | 4.6313 | 0 | -0 | 0.0001 | 0.0132 | 0.0034 |

215 | 0.9998 | -0.9997 | 0.0001 | -0.0254 | 0.0059 | -0.6532 | 0.1016 | 0.0001 | 0.0001 | -0.0044 | 0.4426 | 0.1942 | -0.0029 | 77.6496 | 0.0001 | -0.0001 | 0.0021 | 0.209 | 0.0336 |

220 | 0.9973 | -0.9969 | 0.001 | -0.0526 | 0.006 | -0.6515 | 0.1287 | 0.0012 | 0.0014 | -0.0414 | 4.1955 | 1.4351 | -0.0273 | 600.92 | 0.0014 | -0.0005 | 0.0144 | 1.4569 | 0.1196 |

225 | 0.9777 | -0.976 | 0.0066 | -0.2079 | 0.006 | -0.6387 | 0.1723 | 0.0076 | 0.0086 | -0.1884 | 19.0888 | 4.7244 | -0.1249 | 2187.89 | 0.0086 | -0.0022 | 0.0407 | 4.1228 | 0.0829 |

230 | 0.8953 | -0.8898 | 0.0226 | -0.6528 | 0.0056 | -0.5849 | 0.2419 | 0.0261 | 0.028 | -0.3993 | 40.3564 | 6.2557 | -0.267 | 3816.31 | 0.028 | -0.0048 | 0.0253 | 2.5239 | -0.1641 |

235 | 0.6978 | -0.6874 | 0.0435 | -1.2304 | 0.0044 | -0.4558 | 0.3442 | 0.0502 | 0.0516 | -0.306 | 30.653 | 1.9785 | -0.2119 | 3623.7 | 0.0516 | -0.0039 | -0.0672 | -6.8719 | -0.0977 |

240 | 0.4192 | -0.4078 | 0.0488 | -1.3691 | 0.0027 | -0.2739 | 0.4789 | 0.0562 | 0.0555 | 0.1634 | -17.1438 | 0.4159 | 0.0934 | 3407.79 | 0.0555 | 0.0014 | -0.096 | -9.7512 | -0.0222 |

245 | 0.1812 | -0.1736 | 0.0329 | -0.9207 | 0.0012 | -0.1184 | 0.6396 | 0.0379 | 0.0359 | 0.4445 | -45.5549 | 5.0536 | 0.2814 | 4080.87 | 0.0359 | 0.0048 | -0.0098 | -0.9474 | -0.1945 |

250 | 0.0544 | -0.0513 | 0.0138 | -0.3848 | 0.0004 | -0.0355 | 0.8183 | 0.0159 | 0.0144 | 0.3232 | -33.01 | 6.468 | 0.2073 | 3328.37 | 0.0144 | 0.0036 | 0.0461 | 4.7176 | -0.0443 |

255 | 0.0112 | -0.0104 | 0.0037 | -0.1028 | 0.0001 | -0.0073 | 1.0084 | 0.0042 | 0.0037 | 0.1223 | -12.477 | 3.4845 | 0.0789 | 1542.52 | 0.0037 | 0.0014 | 0.0325 | 3.3216 | 0.1424 |

260 | 0.0016 | -0.0015 | 0.0006 | -0.018 | 0 | -0.001 | 1.205 | 0.0007 | 0.0006 | 0.0276 | -2.8148 | 1.0161 | 0.0179 | 421.028 | 0.0006 | 0.0003 | 0.0104 | 1.0578 | 0.1054 |

265 | 0.0002 | -0.0001 | 0.0001 | -0.0021 | 0 | -0.0001 | 1.4049 | 0.0001 | 0.0001 | 0.004 | -0.4041 | 0.1783 | 0.0026 | 71.3544 | 0.0001 | 0 | 0.0019 | 0.1933 | 0.0322 |

270 | 0 | -0 | 0 | -0.0002 | 0 | -0 | 1.6059 | 0 | 0 | 0.0004 | -0.0385 | 0.02 | 0.0002 | 7.8471 | 0 | 0 | 0.0002 | 0.0222 | 0.0054 |

275 | 0 | -0 | 0 | -0 | 0 | -0 | 1.8068 | 0 | 0 | 0 | -0.0025 | 0.0015 | 0 | 0.5804 | 0 | 0 | 0 | 0.0017 | 0.0006 |

## get_option_chains

Get the Option Chains which gives information about the currently available options as reported by Yahoo Finance. This returns the Contract Symbol, Strike Currency, Last Price, Absolute Change, Percent Change, Volume, Open Interest, Bid Pirce, Ask Price, Expiration, Last Trade Date, Implied Volatility and whether the option is In The Money.

The data comes from Yahoo Finance and is not always available. If the data is not available, it is advised to use the theoretical calculations as provided by the Black Scholes Model as well as the Greeks to get a better understanding of the option prices over time.

**Args:**

__expiration_date (str | None, optional):__The expiration date to use. Defaults to None which means it will use the first available expiration date.__rounding (int | None, optional):__The number of decimals to round the results to. Defaults to 4.

**Returns:**
pd.DataFrame: the option chains containing the tickers and strike prices as
the index and the time to expiration as the columns.

As an example:

Which returns:

Strike Price | Contract Symbol | Currency | Last Price | Change | Percent Change | Volume | Open Interest | Bid | Ask | Expiration | Last Trade Date | Implied Volatility | In The Money |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

155 | AAPL240112C00155000 | USD | 29.75 | 0 | 0 | 9 | 0 | 0 | 0 | 2024-01-12 | 2024-01-11 | 0 | True |

157.5 | AAPL240112C00157500 | USD | 24.15 | 0 | 0 | 1 | 0 | 0 | 0 | 2024-01-12 | 2024-01-05 | 0 | True |

160 | AAPL240112C00160000 | USD | 25.75 | 0 | 0 | 11 | 0 | 0 | 0 | 2024-01-12 | 2024-01-11 | 0 | True |

162.5 | AAPL240112C00162500 | USD | 18.5 | 0 | 0 | 7 | 0 | 0 | 0 | 2024-01-12 | 2024-01-05 | 0 | True |

165 | AAPL240112C00165000 | USD | 20.47 | 0 | 0 | 11 | 0 | 0 | 0 | 2024-01-12 | 2024-01-11 | 0 | True |

167.5 | AAPL240112C00167500 | USD | 17.22 | 0 | 0 | 1 | 0 | 0 | 0 | 2024-01-12 | 2024-01-11 | 0 | True |

170 | AAPL240112C00170000 | USD | 15.65 | 0 | 0 | 176 | 0 | 0 | 0 | 2024-01-12 | 2024-01-11 | 0 | True |

172.5 | AAPL240112C00172500 | USD | 13.19 | 0 | 0 | 34 | 0 | 0 | 0 | 2024-01-12 | 2024-01-11 | 0 | True |

175 | AAPL240112C00175000 | USD | 10.64 | 0 | 0 | 258 | 0 | 0 | 0 | 2024-01-12 | 2024-01-11 | 0 | True |

177.5 | AAPL240112C00177500 | USD | 8.3 | 0 | 0 | 489 | 0 | 0 | 0 | 2024-01-12 | 2024-01-11 | 0 | True |

180 | AAPL240112C00180000 | USD | 5.65 | 0 | 0 | 6152 | 0 | 0 | 0 | 2024-01-12 | 2024-01-11 | 0 | True |

182.5 | AAPL240112C00182500 | USD | 3.25 | 0 | 0 | 14721 | 0 | 0 | 0 | 2024-01-12 | 2024-01-11 | 0 | True |

185 | AAPL240112C00185000 | USD | 1.18 | 0 | 0 | 102803 | 0 | 0 | 0 | 2024-01-12 | 2024-01-11 | 0 | True |

## get_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 most recent risk free rate, dividend yield and stock price is used, you can alter this by changing the start date. The volatility is calculated based on the daily returns of the stock price and the selected period (this can be altered by defining this accordingly when defining the Toolkit class, start_date and end_date).

The formulas are as follows:

- d1 = (ln(S / K) + (r - q + (σ^2) / 2) * t) / (σ * sqrt(t))
- d2 = d1 - σ * sqrt(t)
- Call Option Price = S * e^(-q * t) * N(d1) - K * e^(-r * t) * N(d2)
- Put Option Price = K * e^(-r * t) * N(-d2) - S * e^(-q * t) * N(-d1)

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.

**Args:**

__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.__put_option (bool, optional):__Whether to calculate the put option price. Defaults to False which means it will calculate the call option price.__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.

**Returns:**
pd.DataFrame: Black Scholes values containing the tickers and strike prices as the index and the
time to expiration as the columns.

As an example:

Which returns:

Strike Price | 2024-01-12 | 2024-01-13 | 2024-01-14 | 2024-01-15 | 2024-01-16 | 2024-01-17 | 2024-01-18 | 2024-01-19 | 2024-01-20 | 2024-01-21 | 2024-01-22 | 2024-01-23 | 2024-01-24 | 2024-01-25 | 2024-01-26 | 2024-01-27 | 2024-01-28 | 2024-01-29 | 2024-01-30 | 2024-01-31 | 2024-02-01 | 2024-02-02 | 2024-02-03 | 2024-02-04 | 2024-02-05 | 2024-02-06 | 2024-02-07 | 2024-02-08 | 2024-02-09 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

115 | 40.1925 | 40.2051 | 40.2176 | 40.2301 | 40.2427 | 40.2552 | 40.2677 | 40.2803 | 40.2928 | 40.3053 | 40.3179 | 40.3304 | 40.3429 | 40.3554 | 40.3679 | 40.3805 | 40.393 | 40.4055 | 40.4181 | 40.4306 | 40.4432 | 40.4558 | 40.4684 | 40.4811 | 40.4938 | 40.5065 | 40.5193 | 40.5322 | 40.5451 |

120 | 35.1931 | 35.2062 | 35.2192 | 35.2323 | 35.2454 | 35.2585 | 35.2716 | 35.2846 | 35.2977 | 35.3108 | 35.3239 | 35.3369 | 35.35 | 35.3631 | 35.3762 | 35.3894 | 35.4026 | 35.4158 | 35.4292 | 35.4426 | 35.4561 | 35.4697 | 35.4834 | 35.4973 | 35.5113 | 35.5255 | 35.5399 | 35.5544 | 35.5691 |

125 | 30.1936 | 30.2073 | 30.2209 | 30.2345 | 30.2481 | 30.2618 | 30.2754 | 30.289 | 30.3026 | 30.3163 | 30.33 | 30.3438 | 30.3576 | 30.3716 | 30.3858 | 30.4001 | 30.4147 | 30.4296 | 30.4447 | 30.4602 | 30.476 | 30.4921 | 30.5087 | 30.5256 | 30.5429 | 30.5606 | 30.5787 | 30.5972 | 30.6161 |

130 | 25.1942 | 25.2083 | 25.2225 | 25.2367 | 25.2509 | 25.265 | 25.2793 | 25.2936 | 25.3081 | 25.3229 | 25.3381 | 25.3537 | 25.3699 | 25.3866 | 25.4041 | 25.4223 | 25.4412 | 25.4609 | 25.4814 | 25.5026 | 25.5246 | 25.5474 | 25.571 | 25.5952 | 25.6202 | 25.6459 | 25.6723 | 25.6993 | 25.7269 |

135 | 20.1947 | 20.2094 | 20.2242 | 20.2389 | 20.2538 | 20.2691 | 20.2851 | 20.3022 | 20.3206 | 20.3405 | 20.3619 | 20.385 | 20.4098 | 20.4363 | 20.4643 | 20.4938 | 20.5248 | 20.5572 | 20.5908 | 20.6257 | 20.6617 | 20.6987 | 20.7367 | 20.7756 | 20.8153 | 20.8558 | 20.897 | 20.9388 | 20.9813 |

140 | 15.1953 | 15.2105 | 15.2261 | 15.2432 | 15.2631 | 15.2869 | 15.3149 | 15.3471 | 15.3834 | 15.4233 | 15.4664 | 15.5125 | 15.5611 | 15.6119 | 15.6645 | 15.7189 | 15.7747 | 15.8317 | 15.8898 | 15.9488 | 16.0085 | 16.0689 | 16.1299 | 16.1913 | 16.2531 | 16.3152 | 16.3776 | 16.4402 | 16.5029 |

145 | 10.1958 | 10.2147 | 10.2456 | 10.2916 | 10.3506 | 10.4194 | 10.4956 | 10.5769 | 10.662 | 10.7497 | 10.8392 | 10.9299 | 11.0213 | 11.113 | 11.2049 | 11.2967 | 11.3882 | 11.4794 | 11.5701 | 11.6603 | 11.75 | 11.839 | 11.9275 | 12.0153 | 12.1024 | 12.1889 | 12.2747 | 12.3598 | 12.4443 |

150 | 5.2213 | 5.3527 | 5.52 | 5.6952 | 5.8693 | 6.0395 | 6.2047 | 6.3647 | 6.5198 | 6.6702 | 6.8162 | 6.9581 | 7.0962 | 7.2308 | 7.3621 | 7.4903 | 7.6157 | 7.7384 | 7.8586 | 7.9765 | 8.0921 | 8.2056 | 8.3172 | 8.4268 | 8.5347 | 8.6409 | 8.7455 | 8.8485 | 8.9501 |

155 | 1.1757 | 1.6286 | 1.9783 | 2.2744 | 2.5363 | 2.7739 | 2.9931 | 3.1976 | 3.3902 | 3.5728 | 3.7469 | 3.9136 | 4.0737 | 4.2282 | 4.3775 | 4.5221 | 4.6626 | 4.7991 | 4.9322 | 5.062 | 5.1888 | 5.3128 | 5.4341 | 5.5531 | 5.6697 | 5.7842 | 5.8966 | 6.0071 | 6.1158 |

160 | 0.0437 | 0.2013 | 0.3903 | 0.5823 | 0.77 | 0.9513 | 1.1259 | 1.2942 | 1.4565 | 1.6133 | 1.7651 | 1.9124 | 2.0554 | 2.1946 | 2.3302 | 2.4624 | 2.5916 | 2.7179 | 2.8416 | 2.9627 | 3.0814 | 3.198 | 3.3124 | 3.4249 | 3.5355 | 3.6443 | 3.7514 | 3.8569 | 3.9608 |

165 | 0.0001 | 0.0081 | 0.0378 | 0.0889 | 0.1563 | 0.235 | 0.3213 | 0.413 | 0.5081 | 0.6055 | 0.7043 | 0.804 | 0.9039 | 1.0039 | 1.1036 | 1.2029 | 1.3017 | 1.3999 | 1.4974 | 1.5941 | 1.69 | 1.7852 | 1.8795 | 1.973 | 2.0657 | 2.1576 | 2.2487 | 2.339 | 2.4285 |

170 | 0 | 0.0001 | 0.0017 | 0.0079 | 0.0208 | 0.0412 | 0.0689 | 0.103 | 0.143 | 0.1879 | 0.237 | 0.2897 | 0.3454 | 0.4037 | 0.4641 | 0.5263 | 0.59 | 0.6549 | 0.721 | 0.7878 | 0.8555 | 0.9237 | 0.9923 | 1.0614 | 1.1307 | 1.2003 | 1.27 | 1.3398 | 1.4096 |

## get_implied_volatility

Calculate the Implied Volatility (IV) based on the Black Scholes Model and the actual option prices for any of the available expiration dates.

Implied Volatility (IV) is a measure of how much the market expects the price of the underlying asset to fluctuate in the future. It is a key component of options pricing and can also be used to calculate the theoretical value of an option.

By default the most recent risk free rate, dividend yield and stock price is used, you can alter this by changing the start date. The volatility is calculated based on the daily returns of the stock price and the selected period (this can be altered by defining this accordingly when defining the Toolkit class, start_date and end_date).

The formulas are as follows:

- d1 = (ln(S / K) + (r - q + (σ^2) / 2) * t) / (σ * sqrt(t))
- d2 = d1 - σ * sqrt(t)
- Call Option Price = S * e^(-q * t) * N(d1) - K * e^(-r * t) * N(d2)
- Put Option Price = K * e^(-r * t) * N(-d2) - S * e^(-q * t) * N(-d1)

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.

In which the Implied Volatility is then calculated as follows:

- Implied Volatility = MINIMIZE(Black Scholes Theoretical Price - Actual Option Price)

To determine the Implied Volatility, the Black Scholes Model is used to calculate the theoretical option price in which sigma (σ) is the only unknown variable. The actual option price is then used to determine the implied volatility by minimizing the difference between the theoretical and actual option price.

**Args:**

__expiration_date (str | None, optional):__The expiration date to use for the calculation. Defaults to None which means it will use the most recent expiration date.__put_option (bool, optional):__Whether to calculate the put option price. Defaults to False which means it will calculate the call option price.__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_expiration_dates (bool, optional):__Whether to show the expiration dates. Defaults to False.__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.

**Returns:**
pd.Series | list[str]: Implied Volatility values containing the tickers as the index and the expiration
dates as the columns. If show_expiration_dates is True, it will return a list of expiration dates.

As an example:

Which returns:

2024-02-09 | |
---|---|

162.5 | 0.2828 |

165 | 1.0238 |

167.5 | 0.7867 |

170 | 0.6984 |

172.5 | 0.6796 |

175 | 0.4611 |

177.5 | 0.4423 |

180 | 0.4154 |

182.5 | 0.3541 |

185 | 0.3506 |

187.5 | 0.3331 |

190 | 0.3329 |

192.5 | 0.3411 |

195 | 0.361 |

197.5 | 0.3833 |

200 | 0.4033 |

202.5 | 0.4477 |

205 | 0.4452 |

207.5 | 0.518 |

## objective_function

Calculate the Binomial Option Pricing Model, a mathematical model used to estimate the price of European and American style options. It does so by creating a binomial tree of price paths for the underlying asset, and then working backwards through the tree to determine the price of the option at each node.

By default the most recent risk free rate, dividend yield and stock price is used, you can alter this by changing the start date. The volatility is calculated based on the daily returns of the stock price and the selected period (this can be altered by defining this accordingly when defining the Toolkit class, start_date and end_date).

The formulas are as follows:

- up movement (u) = e^(σ * sqrt(t))
- down movement (d) = 1 / u
- risk neutral probability (p) = (e^((r - q) * t) - d) / (u - d)
- stock price at each node = S * u^j * d^(n - j)
- call option price at expiration date = max(S - K, 0)
- put option price at expiration date = max(K - S, 0)

For European Style options:

- call option price at each node = (p * C_u + (1 - p) * C_d) * e^(-r * t)
- put option price at each node = (p * P_u + (1 - p) * P_d) * e^(-r * t)

For American Style options:

- call option price at each node = max(S - K, (p * C_u + (1 - p) * C_d) * e^(-r * t))
- put option price at each node = max(K - S, (p * P_u + (1 - p) * P_d) * e^(-r * t))

Where S is the stock price, K is the strike price, r is the risk free rate, σ is the volatility, t is the time to expiration, j is the number of up movements, n is the number of time steps, C_u is the call option price at the up movement, C_d is the call option price at the down movement, P_u is the put option price at the up movement and P_d is the put option price at the down movement.

The resulting output is a DataFrame containing the tickers, strike prices and movements as the index and the time to expiration as the columns. The movements index contains the number of up movements and the number of down movements. The output is the binomial tree displayed in a table. E.g. when using 10 time steps, the table for each strike price from each company will contain the actual binomial tree as also depicted in the image found here: https://en.wikipedia.org/wiki/Binomial_options_pricing_model#Method

**Args:**

__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.__put_option (bool, optional):__Whether to calculate the put option price. Defaults to False which means it will calculate the call option price.__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.__time_to_expiration (int):__The number of year to use for the time to expiration. Defaults to 1 which equals one year.__timesteps (int):__The number of time steps to use for the binomial tree. Defaults to 10 which equals 10 time steps. This will be evenly distributed over the time to expiration.__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.

**Returns:**
pd.DataFrame: Binomial Trees values containing the tickers, strike prices and movements as the index and the
timesteps as the columns.

As an example:

Which returns:

Movement | 2024-02-02 | 2024-03-09 | 2024-04-15 | 2024-05-21 | 2024-06-27 | 2024-08-02 | 2024-09-08 | 2024-10-14 | 2024-11-20 | 2024-12-26 | 2025-02-01 |
---|---|---|---|---|---|---|---|---|---|---|---|

UUUUUUUUUU | 53.9518 | 69.0761 | 86.6159 | 106.476 | 128.56 | 152.858 | 179.496 | 208.695 | 240.695 | 275.76 | 314.18 |

UUUUUUUUUD | nan | 39.3262 | 52.173 | 67.5522 | 85.3691 | 105.438 | 127.617 | 151.936 | 178.598 | 207.823 | 239.852 |

UUUUUUUUDD | nan | nan | 26.8446 | 37.2767 | 50.3585 | 66.0853 | 84.2272 | 104.466 | 126.663 | 151.003 | 177.689 |

UUUUUUUDDD | nan | nan | nan | 16.6633 | 24.5421 | 35.0986 | 48.5539 | 64.764 | 83.2271 | 103.482 | 125.698 |

UUUUUUDDDD | nan | nan | nan | nan | 8.9417 | 14.216 | 21.9711 | 32.795 | 46.8998 | 63.7382 | 82.2161 |

UUUUUDDDDD | nan | nan | nan | nan | nan | 3.7526 | 6.596 | 11.3548 | 18.9981 | 30.4984 | 45.85 |

UUUUDDDDDD | nan | nan | nan | nan | nan | nan | 0.9457 | 1.9009 | 3.8207 | 7.6794 | 15.4353 |

UUUDDDDDDD | nan | nan | nan | nan | nan | nan | nan | 0 | 0 | 0 | 0 |

UUDDDDDDDD | nan | nan | nan | nan | nan | nan | nan | nan | 0 | 0 | 0 |

UDDDDDDDDD | nan | nan | nan | nan | nan | nan | nan | nan | nan | 0 | 0 |

DDDDDDDDDD | nan | nan | nan | nan | nan | nan | nan | nan | nan | nan | 0 |

## get_stock_price_simulation

Simulate the Stock Price based on the Binomial Model, a mathematical model used to estimate the price of European and American style options. It does so by creating a binomial tree of price paths for the underlying asset based on the stock price, volatility, risk free rate, dividend yield and time to expiration. The stock price is then simulated based on the up and down movements.

By default the most recent risk free rate and stock price is used, you can alter this by changing the start date. The volatility is calculated based on the daily returns of the stock price and the selected period (this can be altered by defining this accordingly when defining the Toolkit class, start_date and end_date).

The formulas are as follows:

- up movement (u) = e^(σ * sqrt(t))
- down movement (d) = 1 / u
- stock price at each node = S * u^j * d^(n - j)

Where S is the stock price, r is the risk free rate, σ is the volatility, t is the time to expiration, j is the number of up movements, n is the number of time steps.

The resulting output is a DataFrame containing the tickers and movements as the index and the time to expiration as the columns. The movements index contains the number of up movements and the number of down movements. The output is the binomial tree displayed in a table. E.g. when using 10 time steps, the table from each company will contain the actual binomial tree’s stock prices as also depicted in the image found here: https://en.wikipedia.org/wiki/Binomial_options_pricing_model#Method

**Hint:** consider plotting the resulting DataFrame for each company to visualize the binomial tree. For example for below’s example use `stock_price_simulation.loc['AMZN'].T.plot(legend=False)`

**Args:**

__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.__time_to_expiration (int):__The number of year to use for the time to expiration. Defaults to 1 which equals one year.__timesteps (int):__The number of time steps to use for the binomial tree. Defaults to 10 which equals 10 time steps. This will be evenly distributed over the time to expiration.__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.__show_unique_combinations (bool, optional):__Whether to show the unique combinations of the stock prices. Defaults to False.__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.

**Returns:**
pd.DataFrame: Simulated Stock Price values containing the tickers and movements as the index and the
timesteps as the columns.

As an example:

Which returns:

Movement | 2020-06-22 | 2020-09-21 | 2020-12-21 | 2021-03-22 | 2021-06-22 |
---|---|---|---|---|---|

UUUU | 135.69 | 160.047 | 188.776 | 222.663 | 262.632 |

UUUD | 135.69 | 160.047 | 188.776 | 222.663 | 188.776 |

UUDU | 135.69 | 160.047 | 188.776 | 160.047 | 188.776 |

UUDD | 135.69 | 160.047 | 188.776 | 160.047 | 135.69 |

UDUU | 135.69 | 160.047 | 135.69 | 160.047 | 188.776 |

UDUD | 135.69 | 160.047 | 135.69 | 160.047 | 135.69 |

UDDU | 135.69 | 160.047 | 135.69 | 115.04 | 135.69 |

UDDD | 135.69 | 160.047 | 135.69 | 115.04 | 97.5323 |

DUUU | 135.69 | 115.04 | 135.69 | 160.047 | 188.776 |

DUUD | 135.69 | 115.04 | 135.69 | 160.047 | 135.69 |

DUDU | 135.69 | 115.04 | 135.69 | 115.04 | 135.69 |

DUDD | 135.69 | 115.04 | 135.69 | 115.04 | 97.5323 |

DDUU | 135.69 | 115.04 | 97.5323 | 115.04 | 135.69 |

DDUD | 135.69 | 115.04 | 97.5323 | 115.04 | 97.5323 |

DDDU | 135.69 | 115.04 | 97.5323 | 82.6891 | 97.5323 |

DDDD | 135.69 | 115.04 | 97.5323 | 82.6891 | 70.1049 |

## collect_all_greeks

Calculate all Greeks of an option based on the Black Scholes Model. This will return the following Greeks per Strike Price and Expiration Date:

**First Order Greeks:**

- Delta: measures the rate of change of the theoretical option value with respect to changes in the underlying asset’s price.
- Dual Delta: the actual probability of an option finishing in the money which is the first derivative of option price with respect to strike.
- Vega: measures sensitivity to volatility. Vega is the derivative of the option value with respect to the volatility of the underlying asset.
- Theta: measures the sensitivity of the value of the derivative to the passage of time, the “time decay.”
- Rho: measures sensitivity to the interest rate: it is the derivative of the option value with respect to the risk -free interest rate (for the relevant outstanding term).
- Epsilon: measures the percentage change in option value per percentage change in the underlying dividend yield, a measure of the dividend risk.
- Lambda: measures the percentage change in option value per percentage change in the underlying price, a measure of leverage, sometimes called gearing. This greek is also sometimes called Omega or Elasticity.

**Second Order Greeks:**

- Gamma: measures the rate of change in the delta with respect to changes in the underlying price. Gamma is the second derivative of the value function with respect to the underlying price.
- Vanna: also referred to as DvegaDspot and DdeltaDvol, is a second-order derivative of the option value, once to the underlying spot price and once to volatility.
- Charm: Charm or delta decay measures the instantaneous rate of change of delta over the passage of time.
- Vomma: also referred to as volga, vega convexity, or DvegaDvol measures second-order sensitivity to volatility. Vomma is the second derivative of the option value with respect to the volatility, or, stated another way, vomma measures the rate of change to vega as volatility changes.
- Veta: also referred to as DvegaDtime, measures the rate of change in the vega with respect to the passage of time. Veta is the second derivative of the value function; once to volatility and once to time.
- Vera: also referred to as rhova, measures the rate of change in rho with respect to volatility. Vera is the second derivative of the value function; once to volatility and once to interest rate.
- Partial Derivative: measures the rate of change in the option price with respect to the strike price.

**Third Order Greeks:**

- Speed: measures the rate of change in Gamma with respect to changes in the underlying price.
- Zomma: measures the rate of change of Gamma with respect to changes in volatility.
- Color: also referred to as gamma decay or DgammaDtime measures the rate of change of gamma over the passage of time.
- Ultima: measures the sensitivity of the option vomma with respect to change in volatility.

For a deeper explanation, please have a look at: https://en.wikipedia.org/wiki/Greeks_(finance) and the references to the literature as found on this page.

**Args:**

__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 delta. Defaults to False which means it will calculate the call option delta.__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.

**Returns:**
pd.DataFrame: the greeks values containing the tickers and strike prices as the index and the
time to expiration and greeks as the columns.

As an example:

Which returns:

Strike Price | Delta | Dual Delta | Vega | Theta | Rho | Epsilon | Lambda | Gamma | Dual Gamma | Vanna | Charm | Vomma | Vera | Veta | PD | Speed | Zomma | Color | Ultima |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

215 | 0.9998 | -0.9997 | 0.0001 | -0.0254 | 0.0059 | -0.6532 | 0.1016 | 0.0001 | 0.0001 | -0.0044 | 0.4426 | 0.1942 | -0.0029 | 77.6496 | 0.0001 | -0.0001 | 0.0021 | 0.209 | 0.0336 |

220 | 0.9973 | -0.9969 | 0.001 | -0.0526 | 0.006 | -0.6515 | 0.1287 | 0.0012 | 0.0014 | -0.0414 | 4.1955 | 1.4351 | -0.0273 | 600.92 | 0.0014 | -0.0005 | 0.0144 | 1.4569 | 0.1196 |

225 | 0.9777 | -0.976 | 0.0066 | -0.2079 | 0.006 | -0.6387 | 0.1723 | 0.0076 | 0.0086 | -0.1884 | 19.0888 | 4.7244 | -0.1249 | 2187.89 | 0.0086 | -0.0022 | 0.0407 | 4.1228 | 0.0829 |

230 | 0.8953 | -0.8898 | 0.0226 | -0.6528 | 0.0056 | -0.5849 | 0.2419 | 0.0261 | 0.028 | -0.3993 | 40.3564 | 6.2557 | -0.267 | 3816.31 | 0.028 | -0.0048 | 0.0253 | 2.5239 | -0.1641 |

235 | 0.6978 | -0.6874 | 0.0435 | -1.2304 | 0.0044 | -0.4558 | 0.3442 | 0.0502 | 0.0516 | -0.306 | 30.653 | 1.9785 | -0.2119 | 3623.7 | 0.0516 | -0.0039 | -0.0672 | -6.8719 | -0.0977 |

240 | 0.4192 | -0.4078 | 0.0488 | -1.3691 | 0.0027 | -0.2739 | 0.4789 | 0.0562 | 0.0555 | 0.1634 | -17.1438 | 0.4159 | 0.0934 | 3407.79 | 0.0555 | 0.0014 | -0.096 | -9.7512 | -0.0222 |

245 | 0.1812 | -0.1736 | 0.0329 | -0.9207 | 0.0012 | -0.1184 | 0.6396 | 0.0379 | 0.0359 | 0.4445 | -45.5549 | 5.0536 | 0.2814 | 4080.87 | 0.0359 | 0.0048 | -0.0098 | -0.9474 | -0.1945 |

250 | 0.0544 | -0.0513 | 0.0138 | -0.3848 | 0.0004 | -0.0355 | 0.8183 | 0.0159 | 0.0144 | 0.3232 | -33.01 | 6.468 | 0.2073 | 3328.37 | 0.0144 | 0.0036 | 0.0461 | 4.7176 | -0.0443 |

255 | 0.0112 | -0.0104 | 0.0037 | -0.1028 | 0.0001 | -0.0073 | 1.0084 | 0.0042 | 0.0037 | 0.1223 | -12.477 | 3.4845 | 0.0789 | 1542.52 | 0.0037 | 0.0014 | 0.0325 | 3.3216 | 0.1424 |

260 | 0.0016 | -0.0015 | 0.0006 | -0.018 | 0 | -0.001 | 1.205 | 0.0007 | 0.0006 | 0.0276 | -2.8148 | 1.0161 | 0.0179 | 421.028 | 0.0006 | 0.0003 | 0.0104 | 1.0578 | 0.1054 |

265 | 0.0002 | -0.0001 | 0.0001 | -0.0021 | 0 | -0.0001 | 1.4049 | 0.0001 | 0.0001 | 0.004 | -0.4041 | 0.1783 | 0.0026 | 71.3544 | 0.0001 | 0 | 0.0019 | 0.1933 | 0.0322 |

## collect_first_order_greeks

Calculate the first order Greeks of an option based on the Black Scholes Model. This will return the following Greeks per Strike Price and Expiration Date:

- Delta: measures the rate of change of the theoretical option value with respect to changes in the underlying asset’s price.
- Dual Delta: the actual probability of an option finishing in the money which is the first derivative of option price with respect to strike.
- Vega: measures sensitivity to volatility. Vega is the derivative of the option value with respect to the volatility of the underlying asset.
- Theta: measures the sensitivity of the value of the derivative to the passage of time, the “time decay.”
- Rho: measures sensitivity to the interest rate: it is the derivative of the option value with respect to the risk-free interest rate (for the relevant outstanding term).
- Epsilon: measures the percentage change in option value per percentage change in the underlying dividend yield, a measure of the dividend risk.
- Lambda: measures the percentage change in option value per percentage change in the underlying price, a measure of leverage, sometimes called gearing. This greek is also sometimes called Omega or Elasticity.

For a deeper explanation, please have a look at: https://en.wikipedia.org/wiki/Greeks_(finance) and the references to the literature as found on this page.

**Args:**

__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 delta. Defaults to False which means it will calculate the call option delta.__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.

**Returns:**
pd.DataFrame: the first order greek values containing the tickers and strike prices as the index
and the time to expiration and greeks as the columns.

As an example:

## get_delta

Calculate the delta 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 delta is the rate of change of the option price with respect to the price of the underlying asset.

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

- d1 = (ln(S / K) + (r - q + (σ^2) / 2) * t) / (σ * sqrt(t))
- Call Option Delta = N(d1)
- Put Option Delta = N(d1) - 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 Delta can be interpreted as follows:

- For call options, Delta is positive, indicating that the option price tends to move in the same direction as the underlying asset’s price.
- For put options, Delta is negative, indicating that the option price tends to move in the opposite direction to the underlying asset’s price.

Note that the delta of a call option is always between 0 and 1, while the delta of a put option is always between -1 and 0.

**Args:**

__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.__put_option (bool, optional):__Whether to calculate the put option delta. Defaults to False which means it will calculate the call option delta.__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.

**Returns:**
pd.DataFrame: the delta values containing the tickers and strike prices as the index and the
time to expiration as the columns.

As an example:

## get_dual_delta

Calculate the dual delta 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 dual delta is the actual probability of an option finishing in the money which is the first derivative of option price with respect to strike.

The dual delta calculation is the theoretical value of the dual delta. The actual dual delta 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:

- d1 = (ln(S / K) + (r - q + (σ^2) / 2) * t) / (σ * sqrt(t))
- Call Dual Delta = e^(-r * t) * N(d2)
- Put Dual Delta = e^(-r * t) * N(-d2)

The Dual Delta can be interpreted as the probability of an option finishing in the money. For example, if the Dual Delta is 0.5, then the probability of the option finishing in the money is 50%.

**Args:**

__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 dual delta. Defaults to False which means it will calculate the call option dual delta.__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.

**Returns:**
pd.DataFrame: the dual delta values containing the tickers and strike prices as the index and the
time to expiration as the columns.

As an example:

## get_vega

Calculate the vega 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 vega is the rate of change of the option price with respect to the volatility of the underlying asset.

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

- d1 = (ln(S / K) + (r - q + (σ^2) / 2) * t) / (σ * sqrt(t))
- Vega = S * e^(-q * t) * N’(d1) * sqrt(t)

The Vega can be interpreted as follows:

- If Vega is positive, it indicates that the option value will increase as the volatility increases, and vice versa.
- If Vega is negative, it implies that the option value will decrease as the volatility increases, and vice versa.

Note that the vega of a call option and put option are equal to each other.

**Args:**

__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.

**Returns:**
pd.DataFrame: the vega values containing the tickers and strike prices as the index and the
time to expiration as the columns.

As an example:

## get_theta

Calculate the theta 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 theta is the rate of change of the option price with respect to the passage of time.

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

- d1 = (ln(S / K) + (r - q + (σ^2) / 2) * t) / (σ * sqrt(t))
- d2 = d1 - σ * sqrt(t)
- Call Theta = e^(-q * t) * (stock_price * N’(d1) * σ) / (2 * sqrt(t)) - r * K * e^(-r * t) * N(d2) + q * S * e^(-q * t) * N(d1)
- Put Theta = e^(-q * t) * (stock_price * N’(d1) * σ) / (2 * sqrt(t)) + r * K * e^(-r * t) * N(d2) - q * S * e^(-q * t) * N(d1)

The Theta can be interpreted as follows:

- If Theta is positive, it indicates that the option value will increase as the time to expiration increases, and vice versa.
- If Theta is negative, it implies that the option value will decrease as the time to expiration increases, and vice versa.

**Args:**

__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 theta. Defaults to False which means it will calculate the call option theta.__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.

**Returns:**
pd.DataFrame: the theta values containing the tickers and strike prices as the index and the
time to expiration as the columns.

As an example:

## get_rho

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:

- d1 = (ln(S / K) + (r - q + (σ^2) / 2) * t) / (σ * sqrt(t))
- d2 = d1 - σ * sqrt(t)
- Call Rho = K * t * e^(-r * t) * N(d2)
- Put Rho = -K * t * e^(-r * t) * N(-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 typically expressed as the amount of money, per share of the underlying, that the value of the option will gain or lose as the risk-free interest rate rises or falls by 1.0% per annum (100 basis points).

**Args:**

__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.

**Returns:**
pd.DataFrame: the rho values containing the tickers and strike prices as the index and the
time to expiration as the columns.

As an example:

## get_epsilon

Calculate the epsilon 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 epsilon is the rate of change of the option price with respect to the dividend yield.

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

- d1 = (ln(S / K) + (r - q + (σ^2) / 2) * t) / (σ * sqrt(t))
- Call Epsilon = -S * t * e^(-q * t) * N’(d1)
- Put Epislon = S * t * e^(-q * t) * N’(-d1)

The Epsilon can be interpreted as follows:

- If Epislon is positive, it indicates that the option value will increase as the dividend yield increases, and vice versa.
- If Epislon is negative, it implies that the option value will decrease as the dividend yield increases, and vice versa.

**Args:**

__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 epsilon. Defaults to False which means it will calculate the call option epsilon.__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.

**Returns:**
pd.DataFrame: the epsilon values containing the tickers and strike prices as the index and the
time to expiration as the columns.

As an example:

## get_lambda

Calculate the lambda 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 lambda is the rate of change of the option price with respect to the underlying price.

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

- d1 = (ln(S / K) + (r - q + (σ^2) / 2) * t) / (σ * sqrt(t))
- Delta = N(d1)
- Call Option = N’(d1) / (S * σ * sqrt(t))
- Put Option = N’(d1) / (S * σ * sqrt(t))
- Lambda = Delta * (Stock Price / Call Option or Put Option)

The Lambda can be interpreted as follows:

- If Lambda is positive, it indicates that the option value will increase as the underlying price increases, and vice versa.
- If Lambda is negative, it implies that the option value will decrease as the underlying price increases, and vice versa.

**Args:**

__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 lambda. Defaults to False which means it will calculate the call option lambda.__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.

**Returns:**
pd.DataFrame: the lambda values containing the tickers and strike prices as the index and the
time to expiration as the columns.

As an example:

## collect_second_order_greeks

Calculate the second order Greeks of an option based on the Black Scholes Model. This will return the following Greeks per Strike Price and Expiration Date:

- Gamma: measures the rate of change in the delta with respect to changes in the underlying price. Gamma is the second derivative of the value function with respect to the underlying price.
- Vanna: also referred to as DvegaDspot and DdeltaDvol, is a second-order derivative of the option value, once to the underlying spot price and once to volatility.
- Charm: Charm or delta decay measures the instantaneous rate of change of delta over the passage of time.
- Vomma: also referred to as volga, vega convexity, or DvegaDvol measures second-order sensitivity to volatility. Vomma is the second derivative of the option value with respect to the volatility, or, stated another way, vomma measures the rate of change to vega as volatility changes.
- Veta: also referred to as DvegaDtime, measures the rate of change in the vega with respect to the passage of time. Veta is the second derivative of the value function; once to volatility and once to time.
- Vera: also referred to as rhova, measures the rate of change in rho with respect to volatility. Vera is the second derivative of the value function; once to volatility and once to interest rate.
- Partial Derivative: measures the rate of change in the option price with respect to the strike price.

For a deeper explanation, please have a look at: https://en.wikipedia.org/wiki/Greeks_(finance) and the references to the literature as found on this page.

**Args:**

__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 delta. Defaults to False which means it will calculate the call option delta.__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.

**Returns:**
pd.DataFrame: the second order greeks values containing the tickers and strike prices as the index and the
time to expiration and greeks as the columns.

As an example:

## get_gamma

Calculate the gamma 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 gamma is the rate of change of the delta with respect to the price of the underlying asset.

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

- d1 = (ln(S / K) + (r - q + (σ^2) / 2) * t) / (σ * sqrt(t))
- Gamma = N’(d1) / (S * σ * sqrt(t))

The Gamma can be interpreted as follows:

- If Gamma is high, it indicates that the option’s Delta is highly sensitive to changes in the underlying asset’s price. The option’s Delta will change more significantly with small movements in the stock price.
- If Gamma is low, it suggests that the option’s Delta is relatively insensitive to changes in the underlying asset’s price. The option’s Delta changes more gradually with movements in the stock price.

Note that the gamma of a call option and put option are equal to each other.

**Args:**

__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.__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.

**Returns:**
pd.DataFrame: the gamma values containing the tickers and strike prices as the index and the
time to expiration as the columns.

As an example:

## get_dual_gamma

Calculate the gamma 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 gamma is the rate of change of the delta with respect to the price of the underlying asset.

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

- d1 = (ln(S / K) + (r - q + (σ^2) / 2) * t) / (σ * sqrt(t))
- d2 = d1 - σ * sqrt(t)
- Dual Gamma = e^(-r * t) * N’(d2) / (S * σ * sqrt(t))

Note that the dual gamma of a call option and put option are equal to each other.

**Args:**

__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.__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.

**Returns:**
pd.DataFrame: the dual gamma values containing the tickers and strike prices as the index and the
time to expiration as the columns.

As an example:

## get_vanna

Calculate the vanna 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 vanna is the rate of change of the vega with respect to the price of the underlying asset.

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

- d1 = (ln(S / K) + (r - q + (σ^2) / 2) * t) / (σ * sqrt(t))
- d2 = d1 - σ * sqrt(t)
- Vanna = -e^(-q * t) * N’(d1) * (d2 / σ)

The Vanna can be interpreted as follows:

- If Vanna is positive, it indicates that the Delta of the option becomes more positive as both the underlying asset’s price and implied volatility increase, and more negative as they both decrease.
- If Vanna is negative, it suggests that the Delta of the option becomes more negative as both the underlying asset’s price and implied volatility increase, and more positive as they both decrease.

Note that the vanna of a call option and put option are equal to each other.

**Args:**

__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.__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.

**Returns:**
pd.DataFrame: the vanna values containing the tickers and strike prices as the index and the
time to expiration as the columns.

As an example:

## get_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:

- d1 = (ln(S / K) + (r - q + (σ^2) / 2) * t) / (σ * sqrt(t))
- d2 = d1 - σ * sqrt(t)
- Call Charm = q * e^(-q * t) * N’(d1) - e^(-q * t) * N(d1) * (2 * (r - q) * t - d2 * σ * sqrt(t)) / (2 * t * σ * sqrt(t))
- Put Charm = -q * e^(-q * t) * N’(-d1) - e^(-q * t) * N(d1) * (2 * (r - q) * t - d2 * σ * sqrt(t)) / (2 * t * σ * sqrt(t))

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.

**Args:**

__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.

**Returns:**
pd.DataFrame: the charm values containing the tickers and strike prices as the index and the
time to expiration as the columns.

As an example:

## get_vomma

Calculate the vomma 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 vomma is the rate of change of the vega with respect to the volatility of the underlying asset.

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

- d1 = (ln(S / K) + (r - q + (σ^2) / 2) * t) / (σ * sqrt(t))
- d2 = d1 - σ * sqrt(t)
- Vomma = S * e^(-q * t) * N’(d1) * sqrt(t) * (d1 * d2) / σ

The vomma can be interpreted as follows:

- If Vomma is high, it indicates that the option’s Vega is highly sensitive to changes in implied volatility. The option’s value will experience more significant fluctuations with variations in implied volatility.
- If Vomma is low, it suggests that the option’s Vega is relatively less sensitive to changes in implied volatility.

**Args:**

__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.__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.

**Returns:**
pd.DataFrame: the vomma values containing the tickers and strike prices as the index and the
time to expiration as the columns.

As an example:

## get_vera

Calculate the vera 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 vera is the rate of change of the rho with respect to volatility.

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

- d1 = (ln(S / K) + (r - q + (σ^2) / 2) * t) / (σ * sqrt(t))
- d2 = d1 - σ * sqrt(t)
- Vera = -K * t * e^(-r * t) * N’(d2) * (d1 / σ)

The Vera can be interpreted as follows:

- If Vera is positive, it indicates that the option’s Rho is becoming more positive over time. In other words, the option is gaining sensitivity to changes in the risk free rate as time passes.
- If Vera is negative, it suggests that the option’s Rho is becoming more negative over time. The option is losing sensitivity to changes in the risk free rate as time passes.

Note that the vera of a call option and put option are equal to each other.

**Args:**

__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.__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.

**Returns:**
pd.DataFrame: the vera values containing the tickers and strike prices as the index and the
time to expiration as the columns.

As an example:

## get_veta

Calculate the veta 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 veta is the rate of change of the vega with respect to the time to expiration.

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

- d1 = (ln(S / K) + (r - q + (σ^2) / 2) * t) / (σ * sqrt(t))
- d2 = d1 - σ * sqrt(t)
- Veta = -S * e^(-q * t) * N’(d1) * sqrt(t) * (q + ((r - q) * d1) / (σ * sqrt(t)) - (1 + d1 * d2) / (2 * t)

It is common practice to divide the mathematical result of veta by 100 times the number of days per year to reduce the value to the percentage change in vega per one day. This is also done here.

The Veta can be interpreted as follows:

- If Veta is positive, it indicates that the option’s Vega is becoming more positive over time. In other words, the option is gaining sensitivity to changes in implied volatility as time passes.
- If Veta is negative, it suggests that the option’s Vega is becoming more negative over time. The option is losing sensitivity to changes in implied volatility as time passes.

**Args:**

__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.__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.

**Returns:**
pd.DataFrame: the veta values containing the tickers and strike prices as the index and the
time to expiration as the columns.

As an example:

## get_partial_derivative

Calculate the partial derivative 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 partial derivative is the rate of change of the option price with respect to the strike price.

The partial derivative is used in the Breeden -Litzenberger theorem is used for risk -neutral valuation and was developed by Fischer Black and Robert Litzenberger in 1978. The theorem states that the price of any derivative security can be calculated by finding the expected value of the derivative under a risk -neutral measure. The theorem is based on the Black -Scholes model and the assumption that the underlying asset follows a lognormal distribution. See the paper: https://www.jstor.org/stable/2352653

The formula is as follows:

- Partial Derivative (PD) = e^(-r * t) * (1 / K) * (1 sqrt(2 * pi * volatility ** 2 * t)) * e^(-(1 / (2 * volatility ** 2 * t)) * (ln(S / K) - ((r - q) - (0.5 * volatility ** 2)) * t) ** 2

**Args:**

__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.__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.

**Returns:**
pd.DataFrame: the partial derivative values containing the tickers and strike prices as the index and the
time to expiration as the columns.

As an example:

## collect_third_order_greeks

Calculate the third order Greeks of an option based on the Black Scholes Model. This will return the following Greeks per Strike Price and Expiration Date:

- Speed: measures the rate of change in Gamma with respect to changes in the underlying price.
- Zomma: measures the rate of change of gamma with respect to changes in volatility.
- Color: also referred to as gamma decay or DgammaDtime measures the rate of change of gamma over the passage of time.
- Ultima: measures the sensitivity of the option vomma with respect to change in volatility.

**Args:**

__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.__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.

**Returns:**
pd.DataFrame: the third order greeks values containing the tickers and strike prices as the index and the
time to expiration and greeks as the columns.

As an example:

## get_speed

Calculate the speed 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 speed is the rate of change of the gamma with respect to the price of the underlying asset.

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

- d1 = (ln(S / K) + (r - q + (σ^2) / 2) * t) / (σ * sqrt(t))
- Speed = -e^(-q * t) * ((N’(d1) / (S ** 2 * σ * sqrt(t)))) * ((d1 / (σ * sqrt(t))) + 1)

The Speed can be interpreted as follows:

- If Speed is positive, it indicates that the option’s Gamma is becoming more positive over time. In other words, the option is gaining sensitivity to changes in the underlying price as time passes.
- If Speed is negative, it suggests that the option’s Gamma is becoming more negative over time. The option is losing sensitivity to changes in the underlying price as time passes.

Note that the speed of a call option and put option are equal to each other.

**Args:**

__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.__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.

**Returns:**
pd.DataFrame: the speed values containing the tickers and strike prices as the index and the
time to expiration as the columns.

As an example:

## get_zomma

Calculate the zomma 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 zomma is the rate of change of the gamma with respect to volatility.

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

- d1 = (ln(S / K) + (r - q + (σ^2) / 2) * t) / (σ * sqrt(t))
- d2 = d1 - σ * sqrt(t)
- Zomma = e^(-q * t) * (N’(d1) / (d1 * d2 - 1)) / (S * σ **2 * sqrt(t))

The Zomma can be interpreted as follows:

- If Zomma is positive, it indicates that the option’s Gamma is becoming more positive over time. In other words, the option is gaining sensitivity to changes in volatility as time passes.
- If Zomma is negative, it suggests that the option’s Gamma is becoming more negative over time. The option is losing sensitivity to changes in volatility as time passes.

Note that the zomma of a call option and put option are equal to each other.

**Args:**

__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.__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.

**Returns:**
pd.DataFrame: the zomma values containing the tickers and strike prices as the index and the
time to expiration as the columns.

As an example:

## get_color

Calculate the color 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 color is the rate of change of the gamma with respect to time to expiration.

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

- d1 = (ln(S / K) + (r - q + (σ^2) / 2) * t) / (σ * sqrt(t))
- d2 = d1 - σ * sqrt(t)
- Color = -e^(-q * t) * (N’(d1) / (2 * S * t * σ * sqrt(t))) * (2 * q * t + 1 + ((2 * (r - q) * t - d2 * σ * sqrt(t)) / (σ * sqrt(t))) * d1)

The Color can be interpreted as follows:

- If Color is positive, it indicates that the option’s Gamma is becoming more positive over time. In other words, the option is gaining sensitivity to changes in time to expiration as time passes.
- If Color is negative, it suggests that the option’s Gamma is becoming more negative over time. The option is losing sensitivity to changes in time to expiration as time passes.

Note that the color of a call option and put option are equal to each other.

**Args:**

__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.__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.

**Returns:**
pd.DataFrame: the color values containing the tickers and strike prices as the index and the
time to expiration as the columns.

As an example:

## get_ultima

Calculate the ultima 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 ultima is the rate of change of the vomma with respect to volatility.

The ultima calculation is the theoretical value of the ultima. The actual gamma 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:

- d1 = (ln(S / K) + (r - q + (σ^2) / 2) * t) / (σ * sqrt(t))
- d2 = d1 - σ * sqrt(t)
- Ultima = (-vega / volatility ** 2) * (d1 * d2 * (1 - d1 * d2) + d1 ** 2 + d2 ** 2)

The Ultima can be interpreted as follows:

- If Ultima is positive, it indicates that the option’s vomma is becoming more positive over time. In other words, the option is gaining sensitivity to changes in volatility as time passes.
- If Ultima is negative, it suggests that the option’s vomma is becoming more negative over time. The option is losing sensitivity to changes in volatility as time passes.

Note that the ultima of a call option and put option are equal to each other.

**Args:**

__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.__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.

**Returns:**
pd.DataFrame: the ultima values containing the tickers and strike prices as the index and the
time to expiration as the columns.

As an example: