Arbitrage: The Key to Pricing Options

Federal Reserve Bank of Cleveland. Economic Commentary, Jan 1, 2004 by Nosal, Ed, Wang, Tan

Black and Scholes used the method of repeated, or "dynamic," portfolio replication, along with an arbitrage argument, to determine the price of the option. At each date, the value of the portfolio is precisely equal to what is needed to buy a new portfolio that can replicate the subsequent period's payoffs. In the end, the date-3 payoffs to the option can be replicated. If, in our example, the price of the option is not equal to $17.76, say it is $20, an infinite rate of return can be made; you sell the option and buy the date-0 portfolio. The returns to the portfolio with dynamic replication will precisely pay off your option obligations, you pocket $2.24, and the desire to capture more of this "free money" will imply that the forces of supply and demand will equate the value of the replicating portfolio and the option.

* Arbitrage and Economic Efficiency

Arbitrageurs help make markets efficient. When, for some reason, prices get out of line with one another, arbitrage will get the prices back in line. As a result, prices will reflect the "true values" of the traded objects. The notion of arbitrage has improved the efficiency of markets in another, and perhaps, unexpected way. The ideas behind arbitrage helped financial market participants price derivative assets, such as options. These derivative assets can be used to manage risk and, when correctly priced, will enhance the efficiency of markets and firms and, as a result, society.

The views expressed here are those of the authors and not necessarily those of the Federal Reserve Bank of Cleveland, the Board of Governors of the Federal Reserve System, or its staff.

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Ed Nosal is a senior economic advisor at the Federal Reserve Bank of Cleveland, and Tan Wang is the Peter Lusztig Associate professor of Finance at the University of British Columbia