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bsm_functions.py
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bsm_functions.py
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#
# Valuation of European call options in Black-Scholes-Merton model
# incl. Vega function and implied volatility estimation
# bsm_functions.py
#
# Analytical Black-Scholes-Merton (BSM) Formula
def bsm_call_value(S0, K, T, r, sigma):
''' Valuation of European call option in BSM model.
Analytical formula.
Parameters
==========
S0 : float
initial stock/index level
K : float
strike price
T : float
maturity date (in year fractions)
r : float
constant risk-free short rate
sigma : float
volatility factor in diffusion term
Returns
=======
value : float
present value of the European call option
'''
from math import log, sqrt, exp
from scipy import stats
S0 = float(S0)
d1 = (log(S0 / K) + (r + 0.5 * sigma ** 2) * T) / (sigma * sqrt(T))
d2 = (log(S0 / K) + (r - 0.5 * sigma ** 2) * T) / (sigma * sqrt(T))
value = (S0 * stats.norm.cdf(d1, 0.0, 1.0)
- K * exp(-r * T) * stats.norm.cdf(d2, 0.0, 1.0))
# stats.norm.cdf --> cumulative distribution function
# for normal distribution
return value
# Vega function
def bsm_vega(S0, K, T, r, sigma):
''' Vega of European option in BSM model.
Parameters
==========
S0 : float
initial stock/index level
K : float
strike price
T : float
maturity date (in year fractions)
r : float
constant risk-free short rate
sigma : float
volatility factor in diffusion term
Returns
=======
vega : float
partial derivative of BSM formula with respect
to sigma, i.e. Vega
'''
from math import log, sqrt
from scipy import stats
S0 = float(S0)
d1 = (log(S0 / K) + (r + 0.5 * sigma ** 2) * T) / (sigma * sqrt(T))
vega = S0 * stats.norm.pdf(d1, 0.0, 1.0) * sqrt(T)
return vega
# Implied volatility function
def bsm_call_imp_vol(S0, K, T, r, C0, sigma_est, it=100):
''' Implied volatility of European call option in BSM model.
Parameters
==========
S0 : float
initial stock/index level
K : float
strike price
T : float
maturity date (in year fractions)
r : float
constant risk-free short rate
sigma_est : float
estimate of impl. volatility
it : integer
number of iterations
Returns
=======
simga_est : float
numerically estimated implied volatility
'''
for i in range(it):
sigma_est -= ((bsm_call_value(S0, K, T, r, sigma_est) - C0)
/ bsm_vega(S0, K, T, r, sigma_est))
return sigma_est