py_vollib.black package

Submodules

py_vollib.black.implied_volatility module

py_vollib.black.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.

copyright:© 2017 Gammon Capital LLC
license:MIT, see LICENSE for more details.

About LetsBeRational:

The source code of LetsBeRational resides at www.jaeckel.org/LetsBeRational.7z .

========================================================================================
Copyright © 2013-2014 Peter Jäckel.

Permission to use, copy, modify, and distribute this software is freely granted,
provided that this notice is preserved.

WARRANTY DISCLAIMER
The Software is provided "as is" without warranty of any kind, either express or implied,
including without limitation any implied warranties of condition, uninterrupted use,
merchantability, fitness for a particular purpose, or non-infringement.
========================================================================================
py_vollib.black.implied_volatility.implied_volatility(discounted_option_price, F, K, r, t, flag)[source]

Calculate the implied volatility of the Black option price

Parameters:
  • discounted_option_price (float) – discounted Black price of a futures option
  • F (float) – underlying futures price
  • K (float) – strike price
  • r (float) – the risk-free interest rate
  • t (float) – time to expiration in years
  • flag (str) – ‘p’ or ‘c’ for put or call
>>> F = 100
>>> K = 100
>>> sigma = .2
>>> flag = 'c'
>>> t = .5
>>> r = .02
>>> discounted_call_price = black(flag, F, K, t, r, sigma)
>>> iv = implied_volatility(
... discounted_call_price, F, K, r, t, flag)
>>> expected_price = 5.5811067246
>>> expected_iv = 0.2
>>> abs(expected_price - discounted_call_price) < 0.00001
True
>>> abs(expected_iv - iv) < 0.00001
True
py_vollib.black.implied_volatility.implied_volatility_of_discounted_option_price(discounted_option_price, F, K, r, t, flag)[source]

Calculate the implied volatility of the Black option price

Parameters:
  • discounted_option_price (float) – discounted Black price of a futures option
  • F (float) – underlying futures price
  • K (float) – strike price
  • r (float) – the risk-free interest rate
  • t (float) – time to expiration in years
  • flag (str) – ‘p’ or ‘c’ for put or call
>>> F = 100
>>> K = 100
>>> sigma = .2
>>> flag = 'c'
>>> t = .5
>>> r = .02
>>> discounted_call_price = black(flag, F, K, t, r, sigma)
>>> iv = implied_volatility_of_discounted_option_price(
... discounted_call_price, F, K, r, t, flag)
>>> expected_price = 5.5811067246
>>> expected_iv = 0.2
>>> abs(expected_price - discounted_call_price) < 0.00001
True
>>> abs(expected_iv - iv) < 0.00001
True
py_vollib.black.implied_volatility.implied_volatility_of_undiscounted_option_price(undiscounted_option_price, F, K, t, flag)[source]

Calculate the implied volatility of the undiscounted Black option price

Parameters:
  • undiscounted_option_price (float) – undiscounted Black price of a futures option
  • F (float) – underlying futures price
  • K (float) – strike price
  • t (float) – time to expiration in years
>>> F = 100
>>> K = 100
>>> sigma = .2
>>> flag = 'c'
>>> t = .5
>>> undiscounted_call_price = undiscounted_black(F, K, sigma, t, flag)
>>> iv = implied_volatility_of_undiscounted_option_price(
... undiscounted_call_price, F, K, t, flag)
>>> expected_price = 5.6371977797
>>> expected_iv = 0.2
>>> abs(expected_price - undiscounted_call_price) < 0.00001
True
>>> abs(expected_iv - iv) < 0.00001
True
py_vollib.black.implied_volatility.implied_volatility_of_undiscounted_option_price_limited_iterations(undiscounted_option_price, F, K, t, flag, N)[source]

Calculate implied volatility of the undiscounted Black option price with limited iterations.

Parameters:
  • undiscounted_option_price (float) – undiscounted Black price of a futures option
  • F (float) – underlying futures price
  • K (float) – strike price
  • t (float) – time to expiration in years
>>> F = 100
>>> K = 100
>>> sigma = .232323232
>>> flag = 'c'
>>> t = .5
>>> price = undiscounted_black(F, K, sigma, t, flag)
>>> iv = implied_volatility_of_undiscounted_option_price_limited_iterations(
... price, F, K, t, flag, 1)
>>> expected_price = 6.54635543387
>>> expected_iv = 0.232323232
>>> abs(expected_price - price) < 0.00001
True
>>> abs(expected_iv - iv) < 0.00001
True
py_vollib.black.implied_volatility.normalised_implied_volatility(beta, x, flag)[source]

Calculate the normalised Black implied volatility, a time invariant transformation of Black implied volatility.

Keyword arguments:

Parameters:
  • x (float) – ln(F/K) where K is the strike price, and F is the futures price
  • beta (float) – the normalized Black price
  • flag (str) – ‘p’ or ‘c’ for put or call
>>> beta_call = normalised_black(0.0, 0.2, 'c')
>>> beta_put = normalised_black(0.1,0.23232323888,'p')
>>> normalized_b76_iv_call = normalised_implied_volatility(beta_call, 0.0, 'c')
>>> normalized_b76_iv_put = normalised_implied_volatility(beta_put, 0.1, 'p')
>>> expected_price = 0.0796556745541
>>> expected_iv = 0.2
>>> abs(expected_price - beta_call) < 0.00001
True
>>> abs(expected_iv - normalized_b76_iv_call) < 0.00001
True
>>> expected_price = 0.0509710222785
>>> expected_iv = 0.23232323888
>>> abs(expected_price - beta_put) < 0.00001
True
>>> abs(expected_iv - normalized_b76_iv_put) < 0.00001
True
py_vollib.black.implied_volatility.normalised_implied_volatility_limited_iterations(beta, x, flag, N)[source]

Calculate the normalised Black implied volatility, with limited iterations.

Parameters:
  • x (float) – ln(F/K) where K is the strike price, and F is the futures price
  • beta (float) – the normalized Black price
  • flag (str) – ‘p’ or ‘c’ for put or call
>>> beta_call = normalised_black(0.0, 0.2, 'c')
>>> beta_put = normalised_black(0.1,0.23232323888,'p')
>>> normalized_b76_iv_call = normalised_implied_volatility_limited_iterations(beta_call, 0.0, 'c',1)
>>> normalized_b76_iv_put = normalised_implied_volatility_limited_iterations(beta_put, 0.1, 'p',1)
>>> expected_price = 0.0796556745541
>>> expected_iv = 0.2
>>> abs(expected_price - beta_call) < 0.00001
True
>>> abs(expected_iv - normalized_b76_iv_call) < 0.00001
True
>>> expected_price = 0.0509710222785
>>> expected_iv = 0.23232323888
>>> abs(expected_price - beta_put) < 0.00001
True
>>> abs(expected_iv - normalized_b76_iv_put) < 0.00001
True

Module contents

py_vollib.black

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.

About LetsBeRational:

The source code of LetsBeRational resides at www.jaeckel.org/LetsBeRational.7z .

========================================================================================
Copyright © 2013-2014 Peter Jäckel.

Permission to use, copy, modify, and distribute this software is freely granted,
provided that this notice is preserved.

WARRANTY DISCLAIMER
The Software is provided "as is" without warranty of any kind, either express or implied,
including without limitation any implied warranties of condition, uninterrupted use,
merchantability, fitness for a particular purpose, or non-infringement.
========================================================================================
py_vollib.black.black(flag, F, K, t, r, sigma)[source]

Calculate the (discounted) Black option price.

Parameters:
  • F (float) – underlying futures price
  • K (float) – strike price
  • sigma (float) – annualized standard deviation, or volatility
  • t (float) – time to expiration in years
>>> F = 100
>>> K = 100
>>> sigma = .2
>>> flag = 'c'
>>> r = .02
>>> t = .5
>>> black(flag, F, K, t, r, sigma)
5.5811067246048118
py_vollib.black.normalised_black(x, s, flag)[source]

Calculate the normalised Black value, a “time value put-call invariant” transformation of the Black pricing formula. In other words, the amount of time value, or “extrinsic” value of a put and call at the same log-moneyness will be always be identical.

Parameters:
  • x (float) – ln(F/K) where K is the strike price, and F is the futures price
  • s (float) – volatility times the square root of time to expiration
  • flag (str) – ‘p’ or ‘c’ for put or call
>>> def normalised_intrinsic(F, K, flag): 
...     if flag=='c':
...         return max(F-K,0)/(F*K)**0.5
...     else:
...         return max(K-F,0)/(F*K)**0.5    
>>> F = 100.
>>> K = 95.
>>> x = log(F/K)
>>> t = 0.5
>>> v = 0.3
>>> s = v * sqrt(t)    
>>> normalised_black(x,s,'p')
0.061296663817558904
>>> normalised_black(x,s,'c')
0.11259558142181655

‘’’ Here the put is OTM, so has only time value. The call is ITM, having both intrinsic and time value. Since the time value must be equal for both, the call normalised price minus its normalised intrinsic must equal the put normalised price.

>>> (normalised_black(x,s,'p') - (
... normalised_black(x,s,'c') - normalised_intrinsic(F,K,'c')))<1e-12 
True
py_vollib.black.undiscounted_black(F, K, sigma, t, flag)[source]

Calculate the undiscounted Black option price.

Parameters:
  • F (float) – underlying futures price
  • K (float) – strike price
  • sigma (float) – annualized standard deviation, or volatility
  • t (float) – time to expiration in years
>>> F = 100
>>> K = 100
>>> sigma = .2
>>> flag = 'c'
>>> t = .5
>>> undiscounted_black(F, K, sigma, t, flag)
5.637197779701664