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.
where
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
where .
Similarly, if -price of put option with maturity
and strike
is denoted by
, then its risk-neutral pricing is given by
where .
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.