arbitrage
A000063
An arbitrage opportunity is an investment strategy that guarantees a positive payoff in some contingency with no possibility of a negative payoff and with no net investment. By assumption, it is possible to run the arbitrage possibility at arbitrary scale; in other words, an arbitrage opportunity represents a money pump. A simple example of arbitrage is the opportunity to borrow and lend costlessly at two different fixed rates of interest. Such a disparity between the two rates cannot persist: arbitrageurs will drive the rate together.
The modern study of arbitrage is the study of the implications of assuming that no arbitrage opportunities are available. Assuming no arbitrage is compelling because the presence of arbitrage is inconsistent with equilibrium when preferences increase with quantity. More fundamentally, the presence of arbitrage is inconsistent with the existence of an optimal portfolio strategy for any competitive agent who prefers more to less, because there is no limit to the scale at which an individual would want to hold the arbitrage position. Therefore, in principle, absence of arbitrage follows from individual rationality of a single agent. One appeal of results based on the absence of arbitrage is the intuition that absence of arbitrage is more primitive than equilibrium, since only relatively few rational agents are needed to bid away arbitrage opportunities, even in the presence of a sea of agents driven by ‘animal spirits’.
The absence of arbitrage is very similar to the zero economic profit condition for a firm with constant returns to scale (and no fixed factors). If such a firm had an activity which yielded positive profits, there would be no limit to the scale at which the firm would want to run the activity and no optimum would exist. The theoretical distinction between a zero profit condition and the absence of arbitrage is the distinction between commerce and simply trading under the price system, namely that commerce requires production. In practice, the distinction blurs. For example, if gold is sold at different prices in two markets, there is an arbitrage opportunity but it requires production (transportation of the gold) to take advantage of the opportunity. Furthermore, there are almost always costs to trading in markets (for example, brokerage fees), and therefore a form of costly production is required to convert cash into a security. For the purposes of this entry, we will tend to ignore production. In practical applications the necessity of production will weaken the implications of absence of arbitrage and may drive a wedge between what the pure absence of arbitrage would predict and what actually occurs.
The assertion that two perfect substitutes (e.g. two shares of stock in the same company) must trade at the same price is an implication of no arbitrage that goes under the name of the law of one price. While the law of one price is an immediate consequence of the absence of arbitrage, it is not equivalent to the absence of arbitrage. An early use of a no arbitrage condition employed the law of one price to help explain the pattern of prices in the foreign exchange and commodities markets.
Many economic arguments use the absence of arbitrage implicitly. In discussions of purchasing power parity in international trade, for example, presumably it is an arbitrage possibility that forces the spot exchange rate between currencies to equal the relative prices of common baskets of (traded) goods. Similarly, the statement that the possibility of repackaging implies linear prices in competitive product markets is essentially a no-arbitrage argument.
Early Uses of the Law of One Price
The parity theory of forward exchange based on the law of one price was first formulated by Keynes (1923) and developed further by Einzig (1937). Let s denote the current spot price of, say, German marks, in terms of dollars, and let f denote the forward price of marks one year in the future. The forward price is the price at which agreements can be struck currently for the future delivery of marks with no money changing hands today. Also, let rs and rm denote the one year dollar and mark interest rates, respectively. To prevent an arbitrage possibility from developing, these four prices must stand in a particular relation.
To see this, consider the choices facing a holder of dollars. The holder can lend the dollars in the domestic market and realize a return of rs one year from now. Alternatively, the investor can purchase marks on the spot market, lend for one year in the German market, and convert the marks back into dollars one year from now at the fixed forward rate. By undertaking the conversion back into dollars in the forward market, the investor locks in the prevailing forward rate, f. The results of this latter path are a return of
dollars one year from now. If this exceeds 1+rs, then the foreign route offers a sure higher return than domestic lending. By borrowing dollars at the domestic rate rs and lending them in the foreign market, a sure profit at the rate
can be made with no net investment of funds. Alternatively, if the foreign route provides a lower return, then by running the arbitrage in reverse, i.e., by selling dollars forward, borrowing against them and converting the resulting marks into dollars on the spot market, the investor will collect an amount which, when lent in the domestic market at the dollar interest rate, rs, will produce more dollars than were sold forward.
Thus, the prevention of arbitrage will enforce the forward parity result,
This result takes on many different forms as we look across different markets. In a commodity market with costless storage, for example, an arbitrage opportunity will arise if the following relation does not hold:
In this equation, f is the currently quoted forward rate for the purchase of the commodity, e.g., silver, one year from now, s is the current spot price, and r is the interest rate. More generally, if c is the up-front proportional carrying cost, including such items as storage costs, spoilage and insurance, absence of arbitrage ensures that
(We normally would expect these relations to hold with equality in a market in which positive stocks are held at all points in time, and perhaps with inequality in a market which may not have positive stocks just before a harvest. However, proving equality is based on equilibrium arguments, not on the absence of arbitrage, since to short the physical commodity you must first own a positive amount.)
The above applications of the absence of arbitrage (via the law of one price) share the common characteristic of the absence of risk. The law of one price is less restrictive than the absence of arbitrage because it deals only with the case in which two assets are identical but have different prices. It does not cover cases in which one asset dominates another but may do so by different amounts in different states. The most interesting applications of the absence of arbitrage are to be found in uncertain situations, where this distinction may be important.
The Fundamental Theorem of Asset Pricing
The absence of arbitrage is implied by the existence of an optimum for any agent who prefers more to less. The most important implication of the absence of arbitrage is the existence of a positive linear pricing rule, which in many spaces including finite state spaces is the same as the existence of positive state prices that correctly price all assets. Taken together with their converses, we refer collectively to these results as the Fundamental Theorem of Asset Pricing. (In the past, the emphasis has been on the linear pricing rule as an implication of the absence of arbitrage. Adding the other result emphasizes why we are concerned with the absence of arbitrage in the first place.) We state the theorem verbally here; the formal meanings of the words and the proof are given later in this section.
Theorem: (Fundamental Theorem of Asset Pricing) The following are equivalent:
(i) Absence of arbitrage
(ii) Existence of a positive linear pricing rule
(iii) Existence of an optimal demand for some agent who prefers more to less.
Beja (1971) was one of the first to emphasize explicitly the linearity of the asset pricing function, but he did not link it to the absence of arbitrage. Beja simply assumed that equilibrium prices existed and observed ‘that equilibrium properties require that the functional q be linear’, where q is a functional that assigns a price or value to a risky cash flow. The first statement and proof that the absence of arbitrage implied the existence of nonnegative state space prices and, more generally, of a positive linear operator that could be used to value risky assets appeared in Ross (1976a, 1978). Besides providing a formal analysis. Ross showed that there was a pricing rule that prices all assets and not just those actually marketed. (In other words, the linear pricing rule could be extended from the marketed assets to all hypothetical assets defined over the same set of states.) The advantage of this extension is that the domain of the pricing function does not depend on the set of marketed assets. We will largely follow Ross’s analysis with some modern improvements.
Linearity for pricing means that the price functional or operator q statisfies the ordinary linear condition of algebra. If we let x and y be two random payoffs and we let q be the operator that assigns values to prospects, then we require that
where a and b are arbitrary constants. Of course, for many spaces (including a finite state space), any linear functional can be represented as a sum or integral across states of state prices times quantities.
To simplify proofs in this essay, we will make the assumption that there are finitely many states, each of which occurs with positive probability, and that all claims purchased today pay off at a single future date. Let Q denote the state space,
where there are m states and the state of nature q occurs with probability pq. Applying q to the ‘indicator’ asset eq whose payoff is 1 in state q and 0 otherwise, we can define a price qq for each state q as the value of eq ;
Now, if there were linearity, the value of any payoff, x, could be written as
Of course, this argument presupposes that q(eq) is well defined, which is a strong assumption if eq is not marketed.
We want to make a statement about the conditions under which all marketed assets can be priced by such a linear pricing rule q. We assume that there is a set of n marketed assets with a corresponding price vector, p. Asset i has a terminal payoff Xqi (inclusive of dividends, etc.) in state of nature q. The matrix X º [Xqi] denotes the state space tableau whose columns correspond to assets and whose rows correspond to states. Lowercase x represents the random vector of terminal payoffs to the various securities. An arbitrage opportunity is a portfolio (vector) h with two properties. It does not cost anything today or in an state in the future. And, it has a positive payoff either today or in some state in the future (or both). We can express the first property as a pair of vector inequalities. The initial cost is not greater than zero, which is to say that it uses no wealth and may actually generate some,
and its random payoff later is never negative,
(We use the notation that ³ denotes greater or equal in each component, > denotes ³ and greater in some component, and » denotes greater in all components. Note that writing the price of Xh as ph for arbitrary h embodies an assumption that investment in marketed assets is divisible.) The second property says that the arbitrage portfolio h has a strict inequality, either in (1) or in some component of (2). We can express both properties together as
Here, we have stacked the net payoff today on top of the vector of payoffs at the future date. This is in the spirit of the Arrow–Debreu model in which consumption in different states, commodities, points of time and so forth, are all considered components of one large consumption vector.
The absence of arbitrage is simply the condition that no h satisfies (3). A consistent positive linear pricing rule is a vector of state prices q » 0 that correctly prices all marketed assets, i.e. such that
We have now collected enough definitions to prove the first half (that (i) Û (ii)) of the Fundamental Theorem of Asset Pricing.
Theorem: (First half of the Fundamental Theorem of Asset Pricing) There is no arbitrage if and only if there exists a consistent positive linear pricing rule.
Proof: The proof that having a consistent positive linear pricing rule precludes arbitrage is simple, since any arbitrage opportunity gives a direct violation of (4). Let h be an arbitrage opportunity. By (4),
or equivalently
By definition of an arbitrage opportunity (3) and positivity of q, we have a contradiction.
The proof that the absence of arbitrage implies the existence of a consistent positive linear pricing rule is more subtle and requires a separation theorem. The mathematical problem is equivalent to Farkas’ Lemma of the alternative and to the basic duality theorem of linear programming. We will adopt an approach that is analogous to the proof of the second theorem of welfare economics that asserts the existence of a price vector which supports any efficient allocation, by separating the aggregate Pareto optimal allocation from all aggregate allocations corresponding to Pareto preferable allocations. Here we will find a price vector that ‘supports’ an arbitrage-free allocation by separating the net trades from the set of free lunches (the positive orthant).