py_vollib.ref_python.black_scholes.greeks package

Submodules

py_vollib.ref_python.black_scholes.greeks.analytical module

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:© 2017 Gammon Capital LLC
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.

py_vollib.ref_python.black_scholes.greeks.analytical.delta(flag, S, K, t, r, sigma)[source]

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
py_vollib.ref_python.black_scholes.greeks.analytical.gamma(flag, S, K, t, r, sigma)[source]

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
py_vollib.ref_python.black_scholes.greeks.analytical.rho(flag, S, K, t, r, sigma)[source]

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
py_vollib.ref_python.black_scholes.greeks.analytical.theta(flag, S, K, t, r, sigma)[source]

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

py_vollib.ref_python.black_scholes.greeks.analytical.vega(flag, S, K, t, r, sigma)[source]

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

py_vollib.ref_python.black_scholes.greeks.numerical module

py_vollib.ref_python.black_scholes.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:© 2017 Gammon Capital LLC
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.

py_vollib.ref_python.black_scholes.greeks.numerical.delta(flag, S, K, t, r, sigma)[source]

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.
py_vollib.ref_python.black_scholes.greeks.numerical.f(flag, S, K, t, r, sigma, b)
py_vollib.ref_python.black_scholes.greeks.numerical.gamma(flag, S, K, t, r, sigma)[source]

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.
py_vollib.ref_python.black_scholes.greeks.numerical.hull_book_tests()[source]

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

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

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

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

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
py_vollib.ref_python.black_scholes.greeks.numerical.rho(flag, S, K, t, r, sigma)[source]

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.
py_vollib.ref_python.black_scholes.greeks.numerical.test()[source]

Test by comparing analytical and numerical values.

>>> S =  49
>>> K = 50
>>> r = .05
>>> t = 0.3846
>>> sigma = 0.2
>>> flag = 'c'
>>> epsilon = .0001
>>> v1 = delta(flag, S, K, t, r, sigma)
>>> v2 = adelta(flag, S, K, t, r, sigma)
>>> abs(v1-v2)<epsilon
True
>>> v1 = gamma(flag, S, K, t, r, sigma)
>>> v2 = agamma(flag, S, K, t, r, sigma)
>>> abs(v1-v2)<epsilon
True
>>> v1 = rho(flag, S, K, t, r, sigma)
>>> v2 = arho(flag, S, K, t, r, sigma)
>>> abs(v1-v2)<epsilon
True
>>> v1 = vega(flag, S, K, t, r, sigma)
>>> v2 = avega(flag, S, K, t, r, sigma)
>>> abs(v1-v2)<epsilon
True
>>> v1 = theta(flag, S, K, t, r, sigma)
>>> v2 = atheta(flag, S, K, t, r, sigma)
>>> abs(v1-v2)<epsilon
True
py_vollib.ref_python.black_scholes.greeks.numerical.theta(flag, S, K, t, r, sigma)[source]

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.
py_vollib.ref_python.black_scholes.greeks.numerical.vega(flag, S, K, t, r, sigma)[source]

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.

Module contents