# py_vollib.ref_python.black_scholes_merton package¶

## py_vollib.ref_python.black_scholes_merton.implied_volatility module¶

### py_vollib.ref_python.black_scholes_merton.implied_volatility¶

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

py_vollib.ref_python is a pure python version of py_vollib without any dependence on LetsBeRational. It is provided purely as a reference implementation for sanity checking. It is not recommended for industrial use.

`py_vollib.ref_python.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
```
```>>> price = black_scholes_merton(flag, S, K, t, r, sigma, q)
>>> implied_volatility(price, S, K, t, r, q, flag)
0.20000000000000018
```
```>>> flac = 'p'
>>> sigma = 0.3
>>> price = black_scholes_merton(flag, S, K, t, r, sigma, q)
>>> price
8.138101080183894
>>> implied_volatility(price, S, K, t, r, q, flag)
0.30000000000000027
```

## Module contents¶

### py_vollib.ref_python.black_scholes_merton¶

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

py_vollib.ref_python is a pure python version of py_vollib without any dependence on LetsBeRational. It is provided purely as a reference implementation for sanity checking. It is not recommended for industrial use.

`py_vollib.ref_python.black_scholes_merton.``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 = black_scholes_merton('p', S, K, t, r, sigma, q)
>>> abs(p_published_value - p_calc) < 0.0001
True
```
`py_vollib.ref_python.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
`py_vollib.ref_python.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
```
`py_vollib.ref_python.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
```
`py_vollib.ref_python.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
```