From the previous sections, we know that for a set of input variables—
S (the present stock price),
X (the exercise price),
T (the maturity date in years),
r (the continuously compounded risk-free rate), and
sigma (the volatility of the stock, that is, the annualized standard deviation of its returns)—we could estimate the price of a call option based on the Black-Scholes-Merton option model. Recall that to price a European call option, we have the following Python code of five lines:
def bs_call(S,X,T,r,sigma): from scipy import log,exp,sqrt,stats 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)
After entering a set of five values, we can estimate ...