Python for Finance

Black-Scholes-Merton option model on
non-dividend paying stocks
The Black-Scholes-Merton option model is a closed-form solution to price a
Chapter 10
European option on a stock which does not pay any dividends before its maturity
date. If we use or the price today, X for the exercise price, r for the continuously
compounded risk-free rate, T for the maturity in years, for the volatility of the
stock, the closed-form formulae for a European call (c) and put (p) are:
Here, N() is the cumulative standard normal distribution. The following Python
codes represent the preceding equations to evaluate a European call:
from scipy import log,exp,sqrt,stats
def bs_call(S,X,T,r,sigma):
d1=(log(S/X)+(r+sigma*sigma/2.)*T)/(sigma*sqrt(T))
d2 = d1-sigma*sqrt(T)
return S*stats.norm.cdf(d1)-X*exp(-r*T)*stats.norm.cdf(d2)
In the preceding program, the stats.norm.cdf() is the cumulative normal
distribution, that is, N() in the Black-Scholes-Merton option model. The current stock
price is $40, the strike price is $42, the time to maturity is six months, the risk-free
rate is 1.5% compounded continuously, and the volatility of the underlying stock is
20% (compounded continuously). Based on the preceding codes, the European call is
worth $1.56:
>>>c=bs_call(40.,42.,0.5,0.015,0.2)
>>>round(c,2)
1.56
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