San José State University
Department of Economics

applet-magic.com
Thayer Watkins
Silicon Valley
U.S.A.

 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.

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:

#### dC = [(∂C/∂t) + (∂C/∂S) μS + (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 given as:

#### 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.

This means that the change in the value of the portfolio dV over the interval dt is:

#### dV = (∂C/∂S)(µSdt + qSdz) - [(∂C/S)μS + (∂C/∂t) + (1/2)(∂2C/∂S2)q2S2]dt - (∂C/∂S)qSdz.

When terms are combined we find that those involving dz cancel out. Also the terms involving μ cancel out leaving:

#### dV = [ -(∂C/∂t) - (1/2)(∂2C/∂S2) q2S2]dt.

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.,

#### dV = rVdt = r[(∂C/∂S)S - C]dt

For this to hold for all dt requires that:

or

#### (∂C/∂t) + (∂C/∂S)rS + (1/2)(∂2C/∂S2)q2S2 = rC.

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:

• No change in the number of shares of stock outstanding
• The underlying stock pays no dividends during the life of the option.
• The price of the stock one period ahead has a log-normal distribution with mean μ and volatility q, which are both constant over the life of the option.
• There exists a risk-free interest rate which is constant over the life of the option.
• Individuals can borrow as well as lend at the risk-free interest rate. 