No Arbitrage
The reason that finance could make progress from such minimal economic foundations is that for finance the empirical relevance of no arbitrage is high, for two simple reasons: there are many actors (speculators) actively searching for arbitrage opportunities, and it is relatively costless for them to exploit any opportunities they find.
It follows that, for finance, the principle of no arbitrage is not merely a convenient assumption that makes it possible to derive clean theoretical results, but even more an idealization of observable empirical reality, and a characterization of the deep and simple structure underlying multifarious surface phenomena.Crucially, it is the principle of no arbitrage that lies behind the fundamental theorem of asset pricing which asserts the existence of a positive linear pricing rule (Ross 1976b, 1978). The most important practical implication of this theorem is that assets that are not yet traded can be priced simply by reference to the price of assets that are already traded, and without the need to invoke any particular theory of asset pricing. This feature opened the possibility of creating new assets, such as options but also other financial derivatives, that would in practical terms “complete” markets, and so help move the economy closer to the ideal characterized by Arrow (1953). This possibility subsequently served as the legitimation for the entire field of financial engineering (Ross 1976a). However, the first use of the principle of no arbitrage, and to this day its most resoundingly successful use, was to solve the previously intractable problem of pricing options. The story of how that happened is instructive.
A call option, the right to purchase a given security at a given exercise price on or before a given maturity date, is clearly a risky security, and so in principle should be amenable to pricing by means of the CAPM.
Such was the idea of Fischer Black when he first started to work on the problem, having been initially introduced to CAPM through the work of his colleague Jack Treynor. What made the problem hard was that the amount of risk in the option changed over time, and in a non-linear way, as the price of the underlying referenced security changed. Nevertheless, by characterizing these price dynamics, Black was able to obtain a differential equation and then, working with Myron Scholes, solve it for the famous formula:
where x is the stock price, N(d) the cumulative normal density function that describes the distribution of the stock price, c the exercise price, r the rate of interest, t* the expiration date, and d1 and d2 are functions of these data and the variance of the stock return (Black and Scholes 1973).
Black and Scholes came to the correct formula using CAPM, but in the end it turned out that the formula did not depend on CAPM at all, or indeed on any other theory of asset pricing either. Robert Merton (1973) came to the same formula in a different way, by thinking about the option as a portfolio of stock and riskless debt, with portfolio proportions that changed over time as the price of the stock changed. (That proportion is called the hedge ratio; it is a number indicating how many shares of the referenced security should be held to hedge the risk in the option.) Assuming continuous and costless trading, the portfolio could exactly replicate the payoff from the option. Since we know the price of the replicating portfolio, it follows that we also know the price of the option. One problem with Merton’s formulation was the crucial but implausible assumption of continuous and costless trading; in practice exact replication was simply impossible. To the rescue came the principle of no arbitrage, since it allows us to price the option even if replication is not possible in practice.
The solution of the option pricing problem was just the beginning. An entire field of financial engineering was the result. Notwithstanding Merton (1990), instead of calling that field “continuous time finance” and so conceptualizing it as a field of applied mathematics, as an economist I would rather emphasize the principle of no arbitrage as the essential foundation for the field, and the source of its continuing link to economics. Crucially, it was the principle of no arbitrage that made it possible for developments in finance to re-enter economics in the 1980s, through the unlikely mechanism of the martingale equivalence theorem (Harrison and Kreps 1979). The story of how that happened is also instructive.
In Samuelson’s original work on the connection between no arbitrage and the martingale property, he assumed an exogenously fixed discount rate, and limited his attention to asset prices. As a consequence, it was not at all clear how general his result was, since in a more general economic model the discount rate should be endogenous and should fluctuate with the economy as a whole. That means, as LeRoy (1973) and Lucas (1978) pointed out, that there is no reason to suppose that the martingale property will necessarily be a general feature of efficient asset market prices; in other words, Samuelson’s results were a special case. From this point of view, Harrison and Kreps (1979) was crucial, rehabilitating the foundational place of the martingale property, by shifting attention to risk-neutral pricing under a martingale equivalent probability measure. In doing so, they also made it possible to treat no arbitrage and the martingale property as foundational for economics, as well as finance.