SAN JOSE STATE UNIVERSITY
Derivation of the Black-Scholes
Equation for Option Value
A call option is the right to buy a security at a specified price
(called the exercise or strike price) during a specified period of time.
A put option is the right to sell a security at a specified price during
a specified period of time. American options can be exercised at any time
up to and including the day of expiration of the option. European
options can only be exercised on the day of expiration of the option.
C = C(S, X, t, r, q, ).
Fischer Black and Myron Scholes chose to analyze the simplest case, a
European option on a stock that does not pay a dividend during the life
of the option. They also limited their analysis to conditions which made
the problem simpler mathematically. The list of assumptions will be given
The value of a European call option on a nondividend paying stock could
depend upon a number of factors; the current price of the stock S, the
exercise price X, the time until expiration t, the risk-free interest
rate r, the volatility of the stock price q, and the expected rate of
return on the stock . Let C be the price of the call option.
The functional dependence can then be expressed as:
The analysis will reveal that the last variable, , plays no role
in determining option value for this case.
The change in stock price dS is assumed to be given by:
dS = Sdt + qSdz
By Ito's Lemma
dC = [(C/t) + (C/S)
S + (1/2)(2C/S2)q2S2]dt + (C/S)qSdz.
Now consider a portfolio containing one written call (whose value is -C)
and h shares of the underlying stock. The value V of this portfolio is
V = hS - C
The change in value is then:
dV = hdS - dC
If h is equal to C/S then
dV = (C/S)dS -dC.
that the change in the value of the portfolio dV over the
interval dt is:
dV = (C/S)(µSdt + qSdz)
+ (C/t) + (1/2)(2C/S2)q2S2]dt
When terms are combined we find that those involving dz cancel out. Also
the terms involving cancel out leaving:
dV = [ -(C/t)
Thus V is independent of the random variable dz; i.e., is is a risk free
portfolio. Also the value of dV is independent of the expected rate of
return (which is also the expected rate of growth of stock price S).
Since the value of the portfolio is independent of the random variable it
should increase in value at the same rate as the risk free interest rate;
dV = rVdt = r[(C/S)S - C]dt
For this to hold for all dt requires that:
(C/t) + (1/2)(2C/S2)q2S2
= - r(C/S)S + rC,
(C/t) + (C/S)rS +
This is the Black-Scholes differential equation for call option value.
Had we considered the put value P instead of the call value we would have
come up with the same equation. The solution of the above equation for
C = max(S-X,0) on expiration day gives the Black-Scholes formula
for call option value. The solution of the above equation for
C = max(X-S,0) on expiration day gives the value of a put option.
The assumptions made in deriving the Black-Scholes differential equation