Aurora Borealis
1 July 1981 Financial Analysts Journal Volume 37, Issue 4

Replicating Options with Positions in Stock and Cash

  1. Mark Rubinstein
  2. Hayne E. Leland

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 the replicating portfolio as the stock price changes—selling stock as the share price falls and buying stock as the share price rises. Since we are never fully invested when the stock price rises, nor fully disinvested when the stock price falls, this process will deplete our initial investment. By the call’s expiration date, the accumulated depletion will, in principle, exactly equal the initial value of the call.

This concept permits one to replicate, not only calls, but many other option positions. Using replicating portfolios, institutions can create for themselves protective puts and covered calls on stocks for which there is no options market.

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