# vollib.black_scholes_merton package¶

## vollib.black_scholes_merton.implied_volatility module¶

### vollib.black_scholes_merton.implied_volatility¶

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_merton.implied_volatility.``implied_volatility`(price, S, K, t, r, q, flag)[source]

Calculate the Black-Scholes-Merton implied volatility.

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 q (float) – annualized continuous dividend rate flag (str) – ‘c’ or ‘p’ for call or put.
```>>> S = 100
>>> K = 100
>>> sigma = .2
>>> r = .01
>>> flag = 'c'
>>> t = .5
>>> q = .02
```
```>>> reference_price = python_black_scholes_merton(flag, S, K, t, r, sigma, q)
>>> price = black_scholes_merton(flag, S, K, t, r, sigma, q)
```
```>>> abs(reference_price - price) < .00000001
True
>>> iv = implied_volatility(price, S, K, t, r, q, flag)
```

## Module contents¶

### vollib.black_scholes_merton¶

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_merton.``black_scholes_merton`(flag, S, K, t, r, sigma, q)[source]

Return the Black-Scholes-Merton option price.

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 q (float) – annualized continuous dividend rate

From Espen Haug, The Complete Guide To Option Pricing Formulas Page 4

```>>> S=100
>>> K=95
>>> q=.05
>>> t = 0.5
>>> r = 0.1
>>> sigma = 0.2
```
```>>> p_published_value = 2.4648
>>> p_calc = black_scholes_merton('p', S, K, t, r, sigma, q)
>>> abs(p_published_value - p_calc) < 0.0001
True
```
```>>> c1 = bsm_call(S, K, t, r, sigma, q)
>>> c2 = black_scholes_merton('c', S, K, t, r, sigma, q)
>>> abs(c1-c2) < .0001
True
```
`vollib.black_scholes_merton.``bsm_call`(S, K, t, r, sigma, q)[source]

Return the Black-Scholes-Merton call price implemented in python (for reference).

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 q (float) – annualized continuous dividend rate
`vollib.black_scholes_merton.``bsm_put`(S, K, t, r, sigma, q)[source]

Return the Black-Scholes-Merton put price implemented in python (for reference).

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 q (float) – annualized continuous dividend rate

From Espen Haug, The Complete Guide To Option Pricing Formulas Page 4

```>>> S=100
>>> K=95
>>> q=.05
>>> t = 0.5
>>> r = 0.1
>>> sigma = 0.2
```
```>>> p_published_value = 2.4648
>>> p_calc = bsm_put(S, K, t, r, sigma, q)
>>> abs(p_published_value - p_calc) < 0.0001
True
```
`vollib.black_scholes_merton.``d1`(S, K, t, r, sigma, q)[source]

Calculate the d1 component of the Black-Scholes-Merton PDE.

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 q (float) – annualized continuous dividend rate

From Espen Haug, The Complete Guide To Option Pricing Formulas Page 4

```>>> S=100
>>> K=95
>>> q=.05
>>> t = 0.5
>>> r = 0.1
>>> sigma = 0.2
```
```>>> d1_published_value = 0.6102
>>> d1_calc = d1(S,K,t,r,sigma,q)
>>> abs(d1_published_value - d1_calc) < 0.0001
True
```
`vollib.black_scholes_merton.``d2`(S, K, t, r, sigma, q)[source]

Calculate the d2 component of the Black-Scholes-Merton PDE.

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 q (float) – annualized continuous dividend rate

From Espen Haug, The Complete Guide To Option Pricing Formulas Page 4

```>>> S=100
>>> K=95
>>> q=.05
>>> t = 0.5
>>> r = 0.1
>>> sigma = 0.2
```
```>>> d2_published_value = 0.4688
>>> d2_calc = d2(S,K,t,r,sigma,q)
>>> abs(d2_published_value - d2_calc) < 0.0001
True
```
`vollib.black_scholes_merton.``python_black_scholes_merton`(flag, S, K, t, r, sigma, q)[source]

Return the Black-Scholes-Merton call price implemented in python (for reference).

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 q (float) – annualized continuous dividend rate flag (str) – ‘c’ or ‘p’ for call or put.

From Espen Haug, The Complete Guide To Option Pricing Formulas Page 4

```>>> S=100
>>> K=95
>>> q=.05
>>> t = 0.5
>>> r = 0.1
>>> sigma = 0.2
```
```>>> p_published_value = 2.4648
>>> p_calc = python_black_scholes_merton('p', S, K, t, r, sigma, q)
>>> abs(p_published_value - p_calc) < 0.0001
True
```
```>>> c1 = python_black_scholes_merton('c', S, K, t, r, sigma, q)
>>> c2 = black_scholes_merton('c', S, K, t, r, sigma, q)
>>> abs(c1-c2) < .0001
True
```
```>>> p1 = python_black_scholes_merton('p', S, K, t, r, sigma, q)
>>> p2 = black_scholes_merton('p', S, K, t, r, sigma, q)
>>> abs(p1-p2) < .0001
True
```