A black function is given by
where is a normal random variable for some and is either or . One can define function equivalently through its analytic formula, i.e.
This function is said to be Black function, since it is useful to compute European call and put prices in Black-Scholes model in the following sense. BS model usually assumes the lognormal distribution for the stock price at time , that is,
where is short rate, is volatility, is dividend yield, and for
Therefore, if -price of call option with maturity and strike is denoted by , then its risk-neutral pricing is given by
Similarly, if -price of put option with maturity and strike is denoted by , then its risk-neutral pricing is given by
Ex. Suppose stock spot price , volatility , interest rate is , and no dividend. For a put with strike and maturity years, we can use Black function to compute as follows.
See the excel file (download).
Ex. Today’s SPY spot price is , and dividend yield is . The spot price of call with maturity June 2016 () and strike is quoted . 2-year US treasury yields is . We want to compute implied volatility based on the above information. That is to solve for from
This can be computed again by excel file with guess and check (download). As a result the implied volatility is 0.159.