Creating A Replicating Portfolio

The objective in creating a replicating portfolio is to use a combination of risk-free borrowing/lending and the underlying asset to create the same cash flows as the option being valued. In the case of the general formulation above, where stock prices can either move up to Su or down to Sd in any time period, the replicating portfolio for a call with a given strike price will involve borrowing $B and acquiring A of the underlying asset. Of course, this formulation is of no use if we cannot determine how much we need to borrow and what A is. There is a way, however, of identifying both variables. To do this, note that the value of this position has to be same as the value of the call no matter what the stock price does. Let us assume that the value of the call is Cu if the stock price goes to Su, and Cd if the stock price goes down to Sd. If we had borrowed $B and bought A shares of stock with the money, the value of this position under the two scenarios would have been as follows:

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