# vollib.black_scholes.greeks package¶

## vollib.black_scholes.greeks.analytical module¶

### vollib.black_scholes.greeks.analytical¶

A library for option pricing, implied volatility, and greek calculation. vollib is based on lets_be_rational, a Python wrapper for LetsBeRational by Peter Jaeckel as described below.

The source code of LetsBeRational resides at www.jaeckel.org/LetsBeRational.7z .

```======================================================================================

Permission to use, copy, modify, and distribute this software is freely granted,
provided that this notice is preserved.

WARRANTY DISCLAIMER
The Software is provided "as is" without warranty of any kind, either express or implied,
including without limitation any implied warranties of condition, uninterrupted use,
merchantability, fitness for a particular purpose, or non-infringement.
======================================================================================
```
`vollib.black_scholes.greeks.analytical.``delta`(flag, S, K, t, r, sigma)[source]

Return Black-Scholes delta of an option.

Parameters: S (float) – underlying asset price K (float) – strike price sigma (float) – annualized standard deviation, or volatility t (float) – time to expiration in years r (float) – risk-free interest rate flag (str) – ‘c’ or ‘p’ for call or put.

Example 17.1, page 355, Hull:

```>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> delta_calc = delta(flag, S, K, t, r, sigma)
>>> # 0.521601633972
>>> delta_text_book = 0.522
>>> abs(delta_calc - delta_text_book) < .01
True
```
`vollib.black_scholes.greeks.analytical.``gamma`(flag, S, K, t, r, sigma)[source]

Return Black-Scholes gamma of an option.

Parameters: S (float) – underlying asset price K (float) – strike price sigma (float) – annualized standard deviation, or volatility t (float) – time to expiration in years r (float) – risk-free interest rate flag (str) – ‘c’ or ‘p’ for call or put.

Example 17.4, page 364, Hull:

```>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> gamma_calc = gamma(flag, S, K, t, r, sigma)
>>> # 0.0655453772525
>>> gamma_text_book = 0.066
>>> abs(gamma_calc - gamma_text_book) < .001
True
```
`vollib.black_scholes.greeks.analytical.``rho`(flag, S, K, t, r, sigma)[source]

Return Black-Scholes rho of an option.

Parameters: S (float) – underlying asset price K (float) – strike price sigma (float) – annualized standard deviation, or volatility t (float) – time to expiration in years r (float) – risk-free interest rate flag (str) – ‘c’ or ‘p’ for call or put.

The text book analytical formula does not multiply by .01, but in practice rho is defined as the change in price for each 1 percent change in r, hence we multiply by 0.01.

Example 17.7, page 368, Hull:

```>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> rho_calc = rho(flag, S, K, t, r, sigma)
>>> # 0.089065740988
>>> rho_text_book = 0.0891
>>> abs(rho_calc - rho_text_book) < .0001
True
```
`vollib.black_scholes.greeks.analytical.``theta`(flag, S, K, t, r, sigma)[source]

Return Black-Scholes theta of an option.

Parameters: S (float) – underlying asset price K (float) – strike price sigma (float) – annualized standard deviation, or volatility t (float) – time to expiration in years r (float) – risk-free interest rate flag (str) – ‘c’ or ‘p’ for call or put.

The text book analytical formula does not divide by 365, but in practice theta is defined as the change in price for each day change in t, hence we divide by 365.

Example 17.2, page 359, Hull:

```>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> annual_theta_calc = theta(flag, S, K, t, r, sigma) * 365
>>> # -4.30538996455
>>> annual_theta_text_book = -4.31
>>> abs(annual_theta_calc - annual_theta_text_book) < .01
True
```

Using the same inputs with a put. >>> S = 49 >>> K = 50 >>> r = .05 >>> t = 0.3846 >>> sigma = 0.2 >>> flag = ‘p’ >>> annual_theta_calc = theta(flag, S, K, t, r, sigma) * 365 >>> # -1.8530056722 >>> annual_theta_reference = -1.8530056722 >>> abs(annual_theta_calc - annual_theta_reference) < .000001 True

`vollib.black_scholes.greeks.analytical.``vega`(flag, S, K, t, r, sigma)[source]

Return Black-Scholes vega of an option.

Parameters: S (float) – underlying asset price K (float) – strike price sigma (float) – annualized standard deviation, or volatility t (float) – time to expiration in years r (float) – risk-free interest rate flag (str) – ‘c’ or ‘p’ for call or put.

The text book analytical formula does not multiply by .01, but in practice vega is defined as the change in price for each 1 percent change in IV, hence we multiply by 0.01.

Example 17.6, page 367, Hull:

```>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> vega_calc = vega(flag, S, K, t, r, sigma)
>>> # 0.121052427542
>>> vega_text_book = 0.121
>>> abs(vega_calc - vega_text_book) < .01
True
```

## vollib.black_scholes.greeks.numerical module¶

### vollib.black_scholes.greeks.numerical¶

A library for option pricing, implied volatility, and greek calculation. vollib is based on lets_be_rational, a Python wrapper for LetsBeRational by Peter Jaeckel as described below.

The source code of LetsBeRational resides at www.jaeckel.org/LetsBeRational.7z .

```======================================================================================

Permission to use, copy, modify, and distribute this software is freely granted,
provided that this notice is preserved.

WARRANTY DISCLAIMER
The Software is provided "as is" without warranty of any kind, either express or implied,
including without limitation any implied warranties of condition, uninterrupted use,
merchantability, fitness for a particular purpose, or non-infringement.
======================================================================================
```
`vollib.black_scholes.greeks.numerical.``delta`(flag, S, K, t, r, sigma)[source]

Return Black-Scholes delta of an option.

Parameters: S (float) – underlying asset price K (float) – strike price sigma (float) – annualized standard deviation, or volatility t (float) – time to expiration in years r (float) – risk-free interest rate flag (str) – ‘c’ or ‘p’ for call or put.
`vollib.black_scholes.greeks.numerical.``f`(flag, S, K, t, r, sigma, b)
`vollib.black_scholes.greeks.numerical.``gamma`(flag, S, K, t, r, sigma)[source]

Return Black-Scholes gamma of an option.

Parameters: S (float) – underlying asset price K (float) – strike price sigma (float) – annualized standard deviation, or volatility t (float) – time to expiration in years r (float) – risk-free interest rate flag (str) – ‘c’ or ‘p’ for call or put.
`vollib.black_scholes.greeks.numerical.``hull_book_tests`()[source]

Example 17.1, page 355, Hull:

```>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> delta_calc = delta(flag, S, K, t, r, sigma)
>>> # 0.521601633972
>>> delta_text_book = 0.522
>>> abs(delta_calc - delta_text_book) < .01
True
```

Example 17.2, page 359, Hull:

```>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> annual_theta_calc = theta(flag, S, K, t, r, sigma) * 365
>>> # -4.30538996455
>>> annual_theta_text_book = -4.31
>>> abs(annual_theta_calc - annual_theta_text_book) < .01
True
```

Example 17.4, page 364, Hull:

```>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> gamma_calc = gamma(flag, S, K, t, r, sigma)
>>> # 0.0655453772525
>>> gamma_text_book = 0.066
>>> abs(gamma_calc - gamma_text_book) < .001
True
```

Example 17.6, page 367, Hull:

```>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> vega_calc = vega(flag, S, K, t, r, sigma)
>>> # 0.121052427542
>>> vega_text_book = 0.121
>>> abs(vega_calc - vega_text_book) < .01
True
```

Example 17.7, page 368, Hull:

```>>> S = 49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> rho_calc = rho(flag, S, K, t, r, sigma)
>>> # 0.089065740988
>>> rho_text_book = 0.0891
>>> abs(rho_calc - rho_text_book) < .0001
True
```
`vollib.black_scholes.greeks.numerical.``rho`(flag, S, K, t, r, sigma)[source]

Return Black-Scholes rho of an option.

Parameters: S (float) – underlying asset price K (float) – strike price sigma (float) – annualized standard deviation, or volatility t (float) – time to expiration in years r (float) – risk-free interest rate flag (str) – ‘c’ or ‘p’ for call or put.
`vollib.black_scholes.greeks.numerical.``test`()[source]

Test by comparing analytical and numerical values.

```>>> S =  49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
```
```>>> epsilon = .0001
```
```>>> v1 = delta(flag, S, K, t, r, sigma)
>>> v2 = adelta(flag, S, K, t, r, sigma)
>>> abs(v1-v2)<epsilon
True
```
```>>> v1 = gamma(flag, S, K, t, r, sigma)
>>> v2 = agamma(flag, S, K, t, r, sigma)
>>> abs(v1-v2)<epsilon
True
```
```>>> v1 = rho(flag, S, K, t, r, sigma)
>>> v2 = arho(flag, S, K, t, r, sigma)
>>> abs(v1-v2)<epsilon
True
```
```>>> v1 = vega(flag, S, K, t, r, sigma)
>>> v2 = avega(flag, S, K, t, r, sigma)
>>> abs(v1-v2)<epsilon
True
```
```>>> v1 = theta(flag, S, K, t, r, sigma)
>>> v2 = atheta(flag, S, K, t, r, sigma)
>>> abs(v1-v2)<epsilon
True
```
`vollib.black_scholes.greeks.numerical.``theta`(flag, S, K, t, r, sigma)[source]

Return Black-Scholes theta of an option.

Parameters: S (float) – underlying asset price K (float) – strike price sigma (float) – annualized standard deviation, or volatility t (float) – time to expiration in years r (float) – risk-free interest rate flag (str) – ‘c’ or ‘p’ for call or put.
`vollib.black_scholes.greeks.numerical.``vega`(flag, S, K, t, r, sigma)[source]

Return Black-Scholes vega of an option.

Parameters: S (float) – underlying asset price K (float) – strike price sigma (float) – annualized standard deviation, or volatility t (float) – time to expiration in years r (float) – risk-free interest rate flag (str) – ‘c’ or ‘p’ for call or put.