San José State University
Department of Economics
& Tornado Alley
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.
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 these assumptions will be given later.
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 μ. Let C be the price of the call option. The functional dependence can then be expressed as:
C = C(S, X, t, r, q, μ).
The analysis will reveal that the last variable, μ, plays no role in
determining option value for this case.
The change in stock price dS is assumed to be given by:
By Ito's Lemma
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 given as:
The change in value is then:
If h is equal to ∂C/∂S then
This means that the change in the value of the portfolio dV over the interval
When terms are combined we find that those involving dz cancel out. Also the
terms involving μ cancel out leaving:
Thus V is independent of the random variable dz; i.e., is a risk free
portfolio. Also the value of dV is independent of the expected rate of return μ
(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; i.e.,
For this to hold for all dt requires that:
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 are:
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