Now we present the Black-Scholes formula for the price of a call option. Recall that a call option gives the owner the right (at the owner's option) but not the obligation to buy one share of the underlying stock at the strike (or exercise) price X specified in the option contract on or before the maturity date of the option. If the stock price is S and the price of bond promising to pay the amount of the strike price at the maturity date of the option is B, the Black-Scholes price C of the call option is
C = S N(x1) - B N(x2)where
x1 = log(S/B)/s + s/2,
x2 = log(S/B)/s - s/2,s is the standard deviation (or square root of the variance) of the stock price at maturity given the stock price today, and the function N() is the cumulative normal distribution function. If r is a constant continuously-compounded interest rate and T is the time-to-maturity of the option, then B is the discounted exercise price
B = Xexp(-rT).And, if the stock has a variance, v, per unit time, then
2 s = vTis the variance of the final stock price.
In the expression for C, the first term is the stock holding in the hedge strategy, and the second term is the bond holding (which is negative, which is a short sale or borrowing). The main assumptions of the model are absence of arbitrage, a constant riskless rate, continuous stock prices, and a constant variance of returns per unit time for the underlying stock. The intuition is that we can replicate the risk of holding the option by holding just the right portfolio of riskless bonds and the underlying stock. For example, if at a point in time the option moves $.50 for each $1.00 movement in the underlying stock price, then the replicating strategy would hold one share of stock for each two options we are replicating. To hedge the value of the option, we would short (borrow) a share of stock for each option. In that case, the change in value of the stock would neutralize the effect on our wealth of the change in the option price. The hedge's holdings in the stock and bond change over time and in response to the stock price changes, since the sensitivity of the option value is different when the option is in the money than when it is out of the money.