py_vollib.ref_python.black_scholes_merton.greeks.numerical

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.

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.

Module Contents

Functions

delta(flag, S, K, t, r, sigma, q)	Returns the Black-Scholes-Merton delta of an option.
theta(flag, S, K, t, r, sigma, q)	Returns the Black-Scholes-Merton theta of an option.
vega(flag, S, K, t, r, sigma, q)	Returns the Black-Scholes-Merton vega of an option.
rho(flag, S, K, t, r, sigma, q)	Returns the Black-Scholes-Merton rho of an option.
gamma(flag, S, K, t, r, sigma, q)	Returns the Black-Scholes-Merton gamma of an option.
test_analytical_vs_numerical()	Test by comparing analytical and numerical values.

Attributes

f

f

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

Returns the Black-Scholes-Merton delta of an option.

Parameters:

• flag (str) – ‘c’ or ‘p’ for call or put.

• S (float) – underlying asset price

• K (float) – strike price

• t (float) – time to expiration in years

• r (float) – annual risk-free interest rate

• sigma (float) – volatility

• q (float) – annualized continuous dividend yield

Returns:

float

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

Returns the Black-Scholes-Merton theta of an option.

Parameters:

• flag (str) – ‘c’ or ‘p’ for call or put.

• S (float) – underlying asset price

• K (float) – strike price

• t (float) – time to expiration in years

• r (float) – annual risk-free interest rate

• sigma (float) – volatility

• q (float) – annualized continuous dividend yield

Returns:

float

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

Returns the Black-Scholes-Merton vega of an option.

Parameters:

• flag (str) – ‘c’ or ‘p’ for call or put.

• S (float) – underlying asset price

• K (float) – strike price

• t (float) – time to expiration in years

• r (float) – annual risk-free interest rate

• sigma (float) – volatility

• q (float) – annualized continuous dividend yield

Returns:

float

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

Returns the Black-Scholes-Merton rho of an option.

Parameters:

• flag (str) – ‘c’ or ‘p’ for call or put.

• S (float) – underlying asset price

• K (float) – strike price

• t (float) – time to expiration in years

• r (float) – annual risk-free interest rate

• sigma (float) – volatility

• q (float) – annualized continuous dividend yield

Returns:

float

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

Returns the Black-Scholes-Merton gamma of an option.

Parameters:

• flag (str) – ‘c’ or ‘p’ for call or put.

• S (float) – underlying asset price

• K (float) – strike price

• t (float) – time to expiration in years

• r (float) – annual risk-free interest rate

• sigma (float) – volatility

• q (float) – annualized continuous dividend yield

Returns:

float

test_analytical_vs_numerical()

Test by comparing analytical and numerical values.

>>> S =  49
>>> K = 50
>>> r = .05
>>> q = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'

>>> epsilon = .0001

>>> v1 = delta(flag, S, K, t, r, sigma, q)
>>> v2 = adelta(flag, S, K, t, r, sigma, q)
>>> abs(v1-v2)<epsilon
True

>>> v1 = gamma(flag, S, K, t, r, sigma, q)
>>> v2 = agamma(flag, S, K, t, r, sigma, q)
>>> abs(v1-v2)<epsilon
True

>>> v1 = rho(flag, S, K, t, r, sigma, q)
>>> v2 = arho(flag, S, K, t, r, sigma, q)
>>> abs(v1-v2)<epsilon
True

>>> v1 = vega(flag, S, K, t, r, sigma, q)
>>> v2 = avega(flag, S, K, t, r, sigma, q)
>>> abs(v1-v2)<epsilon
True

>>> v1 = theta(flag, S, K, t, r, sigma, q)
>>> v2 = atheta(flag, S, K, t, r, sigma, q)
>>> abs(v1-v2)<epsilon
True