Law of One Price: Cornerstone for Option Replicating Portfolios

Imagine you are at a farmers market. You see two stalls selling identical apples, same type, same size, same quality. If one stall is selling apples at one dollar each, and the stall right next to it is selling them at five dollars each, what would you do? You would naturally buy all your apples from the one-dollar stall. This simple, everyday scenario illustrates a fundamental economic principle called the Law of One Price. It essentially states that identical assets should trade at the same price in efficient markets.

Now, let’s consider the world of finance, specifically options. An option contract gives you the right, but not the obligation, to buy or sell an underlying asset, like a stock, at a predetermined price on or before a certain date. Valuing these options can seem complex, but one powerful method is the replicating portfolio approach. Think of this method as creating a doppelganger, a perfect twin, of the option’s payoff using other, more basic financial instruments. Typically, this involves combining the underlying stock itself with risk-free borrowing or lending.

The replicating portfolio method works by constructing a portfolio, a specific mix of stock and borrowing, that will generate the exact same cash flows as the option, no matter what happens to the price of the underlying stock. It’s like building a machine that produces the same output as the option contract.

This is where the Law of One Price becomes the cornerstone. If we have successfully created a replicating portfolio that perfectly mirrors the option’s payoff, then according to the Law of One Price, the cost of constructing this portfolio must be equal to the fair price of the option. Think of it like this: if you can bake a cake at home that tastes and looks exactly like a cake from a bakery, then the cost of your homemade cake ingredients should be roughly the same as the price you’d expect to pay for the bakery cake. If your homemade cake costs three dollars in ingredients, then the bakery cake shouldn’t reasonably cost twenty dollars, assuming they are truly identical in value.

The Law of One Price ensures that there are no risk-free profit opportunities, often called arbitrage opportunities, in efficient markets. Imagine a scenario where the replicating portfolio costs six dollars to build, but the option is trading in the market for only four dollars. This situation violates the Law of One Price. Savvy investors would immediately recognize this discrepancy. They would buy the underpriced option for four dollars and simultaneously sell the replicating portfolio for six dollars. Because the replicating portfolio and the option provide identical payoffs, this combination guarantees a risk-free profit of two dollars.

This type of arbitrage activity quickly corrects the price imbalance. As investors buy the option, its price increases due to demand. As they sell the replicating portfolio, the prices of its components adjust until the cost of the replicating portfolio and the option price converge. This process continues until the Law of One Price is restored, meaning the price of the option and the cost of its replicating portfolio are essentially the same.

Conversely, if the option were overpriced compared to the cost of the replicating portfolio, say the option was trading at eight dollars while the replicating portfolio cost six dollars, investors would sell the overpriced option and buy the cheaper replicating portfolio components. This action would also drive prices back towards equilibrium, reinforcing the Law of One Price.

Therefore, the Law of One Price is not just a nice-to-have principle; it is absolutely fundamental to the replicating portfolio method of option valuation. It provides the theoretical justification for equating the option’s price to the cost of its replicating portfolio. Without the Law of One Price, the replicating portfolio method would lose its validity, and there would be no reliable way to determine the fair value of options using this approach. It is the anchor that grounds the entire method and ensures that option prices are rationally related to the prices of the underlying assets and risk-free interest rates. The Law of One Price ultimately ensures that option pricing, when using replicating portfolios, is not arbitrary but reflects the fundamental economics of equivalent assets.