py_vollib.ref_python.black_scholes.greeks.analytical

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)	Return Black-Scholes delta of an option.
theta(flag, S, K, t, r, sigma)	Return Black-Scholes theta of an option.
gamma(flag, S, K, t, r, sigma)	Return Black-Scholes gamma of an option.
vega(flag, S, K, t, r, sigma)	Return Black-Scholes vega of an option.
rho(flag, S, K, t, r, sigma)	Return Black-Scholes rho of an option.

Attributes

N

N

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

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

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

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

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

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

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

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

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

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