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.

copyright:

© 2023 Larry Richards

license:

MIT, see LICENSE for more details.

Subpackages

Submodules

Package Contents

Functions

d1(S, K, t, r, sigma, q)

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

d2(S, K, t, r, sigma, q)

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

bsm_call(S, K, t, r, sigma, q)

Return the Black-Scholes-Merton call price

bsm_put(S, K, t, r, sigma, q)

Return the Black-Scholes-Merton put price

black_scholes_merton(flag, S, K, t, r, sigma, q)

Return the Black-Scholes-Merton call price implemented in

Attributes

N

N
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
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
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

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
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