Sidebar on Black-Scholes for
The New Risk Management:
the Good, the Bad, and the Ugly

Philip H. Dybvig and William J. Marshall

The Black-Scholes Option Pricing Model

The precursor of all modern option pricing models was developed by Fischer Black and Myron Scholes.//footnote: Black, F. and M. Scholes, "The Pricing of Options and Corporate Liabilities" Journal of Political Economy 81, 1973, 637-654.// The main result is an option-pricing formula based on simple and reasonable assumptions in a continuous-time model. The remarkable thing about the result is that it relies on the absence of arbitrage, and part of the proof is a formula that specifies a trading strategy in the underlying stock and the riskless bond that will replicate the payoff of the option at the end.//footnote: For more discussion of why this makes sense, see Rubinstein, M. and H. Leland, ``Replicating Options with Positions in Stock and Cash,'' Fianancial Analysts Journal, Jan-Feb 1995, 149-160.// If the option is priced differently in the economy, buying or selling the option and following either the trading strategy or the trading strategy in reverse will make money! Using the same sort of analysis gives the trading strategy that will hedge the financial risk in a firm's cash flows.

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 = vT
is 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.