Black-Scholes-Merton option model on non-dividend paying stocks

The Black-Scholes-Merton option model is a closed-form solution to price a 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...