Replicating Options with Positions in Stock and Cash

In most situations of practical relevance, the price behavior of a call option is very similar to a combined position involving the underlying stock and borrowing. The call price and the stock price will change in the same direction. The effect on the call price of a one dollar change in the stock price, however, will depend on the current price of the stock; the number of shares of stock in the replicating portfolio must equal the slope of the call price curve at that price. When the call is deep out of the money—i.e., when the stock price is much lower than the striking price—a one dollar change in the stock price has little effect on the call price. When the stock price is equal to the striking price, a one dollar change in the stock price produces roughly a half-dollar change in the call price. If the stock price rises until the call is deep in the money, a one dollar move in the stock price results in nearly a one dollar move in the call price. Because the call price behaves this way, we must revise ...