Implied volatility

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 ...

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