By Hugh Edmundson, Carnegie Mellon University
Arbitrage-free pricing is the primary means of determining the fair value of a security. The principle of arbitrage-free pricing essentially boils down to the claim that you cannot earn something from nothing without assuming some risk.
The concept of arbitrage is historically associated with price differentials between markets. As a (simplified) example, when the price of a good such as timber was lower in Market A than in Market B, merchants would purchase the timber at the lower price in Market A, transport the goods to Market B, then sell the timber at Market B for a higher price. Since prices were relatively stable, merchants could incur a risk-free profit, less the costs of transport, by simply moving goods from where they were more plentiful to where they were less plentiful. The merchants would then earn a profit based on the price difference. Over the long term, such merchants' actions would cause the two independent market prices to converse to an equilibrium.
In modern financial markets, the historical concept of arbitrage has evolved to reflect the potential for similar risk-free profit. Somewhat more formally, a trade or portfolio is said to be an arbitrage when, starting with 0 initial capital, and trading only in a risk-free account and in tradable securities, we can construct a portfolio such that at some end time, T, the value of our portfolio will be at least 0 with probability 1, and greater than 0 with some positive probability. What this formal definition means is that, starting with nothing, we have the possibility of earning a profit without assuming any risk - we can never lose money.
One of the principles of modern financial theory is that arbitrage opportunities - after accounting for trading costs and other “market frictions”- don’t exist, or are, at best, exceedingly small and exceedingly rare. In practice, many traders and investors believe that arbitrage opportunities exist, and where possible, attempt to find an exploit them.
The utility of the no-arbitrage principle derives from the fact that using the assumption of no-arbitrage - a relativity tame assumption, mathematically - we can often determine a unique price point for traded securities and their derivatives.
For example, a financial institution may be interested in determining the fair market price of a European call option. A European call option gives the holder the right, but not the obligation, to purchase an underlying security at a specified price at some expiration time T. At the expiration time, the price of the call is known to be the greater of 0 and the difference between the price of the stock and the strike price. A question then arises; how do we determine the current value of this call? Using arbitrage-free pricing, the bank can attempt to develop a replicating portfolio, which is a combination of bank loans and deposits, stock or other purchases that will have the same value as the call at the expiration time for all possible values of the stock. Since the value of all of the constituents of this portfolio are already known at the initial time, then using the principle of no arbitrage, we can conclude that the current value of the European call must equal the unique value of this replicating portfolio. Where this not the case, and the price of the call option was higher or lower than this unique value, then by shorting the higher priced of the call option and the replicating portfolio and purchasing the lower priced, we could guarantee ourselves a risk-free profit that is the difference between the prices of the two securities, a.k.a. an arbitrage.
Any rational agent, having discovered such an arbitrage, would then conduct this trade repeatedly, reaping a risk-free profit. However, the very act of conducting this trade would cause the two prices to converge, just as the two prices of timber converged in our earlier example, until no one could earn excess returns. It is this self-correcting mechanism that leads many to believe markets are efficient and reflect the true value of their underlying securities; otherwise, it would be possible to earn excess returns without additional risk. Indeed, demonstrating that arbitrages exist is empirically extremely difficult.
This concludes our discussion of arbitrage free pricing. A-F pricing is a powerful means for determining the true value of a derivative security. The construction of a replicating portfolio remains the gold standard for pricing in the field of mathematical finance and underlies famous formulas such as the Black-Scholes-Merton Option Pricing Formula.
