Black-Scholes theory is a mathematical theory for valuing European options when the underlying price follows a geometric
Brownian motion. With constant volatility 
, riskless rate 
, and continuous yield 
, the underlying price 
satisfies
 |
(1)
|
under the valuation probability, where 
is a Wiener process.
A continuously rebalanced portfolio can eliminate the 
term from a sufficiently smooth
function 
giving the option value. The remaining locally riskless
portfolio must earn rate 
, giving the Black-Scholes partial
differential equation
 |
(2)
|
For time to expiration 
, the value of a call option
with strike 
is
 |
(3)
|
and the value of the corresponding put option is
 |
(4)
|
where 
is the normal distribution function
and
 |  |  |
(5)
|
 |  |  |
(6)
|
The formulas depend on 
rather than on the expected return of the underlying because
the expected-return term is removed by the locally riskless portfolio construction.
The case 
is the original formula for a stock option. The Garman-Kohlhagen
formula applies the same mathematics to foreign-currency options.
The Heston model replaces constant variance 
by a stochastic variance
process while retaining a solution based on the characteristic
function. The Cox-Ross-Rubinstein binomial model gives a discrete-time approximation
and can also accommodate American options, for
which the exercise time is part of the valuation problem (Cox et al. 1979).
