Arbitrage

Abstract

The absence of arbitrage is the unifying concept for much of finance. Absence of arbitrage is more general than equilibrium because it does not require all agents to be rational. The Fundamental Theorem of Asset Pricing asserts the equivalence of absence of arbitrage, existence of a positive linear pricing rule, and existence of some hypothetical agent who prefers more to less and has an optimum. Equivalent representations of the pricing rule are the martingale measure (risk-neutral pricing), and a positive state price density. Applications of no arbitrage and these representations include Modigliani–Miller theory, option pricing, investments, and forward exchange parity.

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, which requires production, and trading under the price system, which does not. 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 article, 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 (for example, 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, euros, in terms of dollars, and let f denote the forward price of euros one year in the future. The forward price is the price at which agreements can be struck currently for the future delivery of euros with no money changing hands today. Also, let r_s and r_m denote the one year dollar and euro 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 r_s one year from now. Alternatively, the investor can purchase euros on the spot market, lend for one year in the German market, and convert the euros 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

$$ f\left(1+{r}_m\right)/s $$

dollars one year from now. If this exceeds 1 + r_s, then the foreign route offers a sure higher return than domestic lending. By borrowing dollars at the domestic rate r_s and lending them in the foreign market, a sure profit at the rate

$$ f\left(1+{r}_m\right)/s-\left(1+{r}_s\right) $$

can be made with no net investment of funds. Alternatively, if

$$ f\left(1+{r}_m\right)/s-\left(1+{r}_s\right)<0, $$

the arbitrage works in reverse. By borrowing in euros, investing in dollars, and buying euros forward, a sure profit at the rate

$$ \left(1+{r}_s\right)-f\left(1+{\mathrm{r}}_m\right)/s $$

can be made with no investment in funds.

Thus, the prevention of arbitrage will enforce the forward parity result,

$$ \left(1+{r}_s\right)/\left(1+{r}_m\right)\ \ \ =\ \ \ f/s. $$

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:

$$ f\le s\left(1+r\right). $$

In this equation, f is the currently quoted forward rate for the purchase of the commodity – for example, 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

$$ f\le s\left(1+c\right)\left(1+r\right). $$

(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 one 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. Traditionally, the emphasis has been on the linear pricing rule as an implication of the absence of arbitrage. Including the existence of an optimum (introduced in the version of this article in the first edition of The New Palgrave) is useful both because it reminds us why we are interested in arbitrage, and because the converse tells us that, absent other restrictions, consistency with equilibrium is equivalent to the absence of arbitrage. 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 non-negative 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 satisfies 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

$$ q\left( ax+ by\right)\ \ \ =\ \ \ aq(x)+ bq(y), $$

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 article, 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 Θ denote the state space,

$$ \varTheta =\left\{1,\dots, m\right\}, $$

where there are m states and the state of nature θ occurs with probability π_θ. Applying q to the ‘indicator’ asset e_θ whose payoff is 1 in state θ and 0 otherwise, we can define a price q_θ for each state θ as the value of e_θ;

$$ {q}_{\theta}\ \ \ =\ \ \ q\left({e}_{\theta}\right). $$

Now, if there were linearity, the value of any payoff, x, could be written as

$$ q(x)\ \ \ =\ \ \ \sum_{\theta }{q}_{\theta }{x}_{\theta }. $$

Of course, this argument presupposes that q(e_θ) is well defined, which is a strong assumption if e_θ 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 ${X}_{\theta_i}$ (inclusive of dividends, and so on) in state of nature θ. The matrix X ≡ [${X}_{\theta_i}$] denotes the state space tableau whose columns correspond to assets and whose rows correspond to states. Lower-case x represents the random vector of terminal payoffs to the various securities. An arbitrage opportunity is a portfolio (vector) η with two properties. It does not cost anything today or in a 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,

$$ p\eta \le 0, $$

(1)

and its random payoff later is never negative,

$$ X\eta \ge 0. $$

(2)

(We use the notation that ≥ denotes greater or equal in each component, > denotes ≥ and greater in some component, and ≫ denotes greater in all components. Note that writing the price of Xη as pη for arbitrary η embodies an assumption that investment in marketed assets is divisible.) The second property says that the arbitrage portfolio η has a strict inequality, either in (1) or in some component of (2). We can express both properties together as

$$ {X}_{\ast}\eta \ \ \ =\ \ \ \left[\begin{array}{l}-p\\ {}\ X\end{array}\right]\eta >0. $$

(3)

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 η satisfies (3). A consistent positive linear pricing rule is a vector of state prices q ≫ 0 that correctly prices all marketed assets, that is, such that

$$ p\ \ \ =\ \ \ qX. $$

(4)

We have now collected enough definitions to prove the first half (that (i) ⇔ (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 η be an arbitrage opportunity. By (4),

$$ p\eta \ \ \ =\ \ \ qX\eta, $$

or equivalently

$$ 0\ \ \ =\ \ \ - p\eta +q\left( X\eta \right)\ \ \ =\ \ \ \left[1q\right]{X}_{\ast}\eta . $$

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).

The absence of arbitrage is equivalent to the requirement that the linear space of net trades defined by

$$ s\ \ \ \equiv \ \ \ \left\{y\left|\ \ \ \mathrm{for}\ \ \ \mathrm{some}\ \ \ \eta, y\ \ \ =\ \ \ {X}_{\ast}\eta \right.\right\}, $$

(5)

does not intersect the positive orthant ${\mathrm{R}}_{+}^m+1=\left\{y\left|y\right.\ge 0\right\}$ except at the origin, that is $S\cap {\mathrm{R}}_{+}^m+1=\left\{0\right\}.$

Since S is a subspace (and is therefore a convex closed cone), a simple separation theorem (Karlin 1959, Theorem B3.5) implies that there exists a nonzero vector q_* such that for all y ∈S and all $z\in {\mathrm{R}}_{+}^m+1$,z ≠ 0, we must have

$$ {q}_{\ast }z>0\ge {q}_{\ast }y. $$

(6)

Letting z be each of the unit vectors in turn, the first inequality in (6) implies that q_∗ is a strictly positive vector.

Since S is a subspace, the second inequality in (6) must hold with equality for all y ∈ S Define

$$ q\ \ \ \equiv \ \ \ \left({q}_{{}^{\ast }2},{q}_{{}^{\ast }3},\dots, {q}_{{}^{\ast}\mathrm{n}}\right)/{q}_{{}^{\ast }1}. $$

Since q_∗ ≫ 0, likewise q ≫ 0.

Dividing the second equality in (6) (which we now know to be an equality) by q_∗1 and expanding using the definition of X_* [from (3)], we have that

$$ 0=-p+ qX, $$

or

$$ p\ \ \ =\ \ \ qX, $$

which shows that q is a consistent positive linear pricing rule.

Before we can prove the second half of the pricing theorem, we need to define the maximization problem faced by a typical investor. In this problem, all we really need to assume is that more is preferred (strictly) to less, that is, that increasing initial consumption or random consumption later in one or more states always leads to a preferred outcome. In fact, this is literally all we need: we do not need completeness or even transitivity of preferences, let alone a utility function representation or any restriction to a functional form. However, for concreteness, we will write down preferences using a state-dependent utility function of consumption now and in the future. The assumption that the investor prefers more to less is satisfied if the utility function in each state is increasing in consumption at both dates.

The state-dependent restriction implies that the maximization problem faced by a particular agent is the maximization of the expectation of the state-dependent utility function u₀ (·, ·) of initial wealth and terminal wealth, given initial wealth w₀ and the possibility of trading in the security market. Then the maximization problem faced by a typical agent is the unconstrained choice of a vector α of portfolio weights to maximize

$$ \sum_0{\pi}_0{u}_0\left[{w}_0- p\alpha, {\left( X\alpha \right)}_0\right]. $$

The quantity pα is the price of the portfolio, and therefore w₀ − pα is the residual amount of the initial wealth available for initial consumption. The preferences of the agent are said to be increasing if each u₀ (·, ·) is (strictly) increasing in both arguments. Saying the agent prefers more to less is just another way of saying that preferences are increasing.

Here is the rest of the proof of the Fundamental Theorem of Asset Pricing.

Theorem

(second half of the Fundamental Theorem of Asset Pricing) There is no arbitrage if and only if there exists some (at least hypothetical) agent with increasing preferences whose choice problem has a maximum.

Proof

If there is an arbitrage opportunity, η, then clearly the choice problem for an agent with increasing preferences cannot have a maximum, since for every α,

$$ \sum_{\theta }{\pi}_{\theta }{u}_{\theta}\left\{{w}_0-p\left(\alpha + k\eta \right),{\left[X\left(\alpha + k\eta \right)\right]}_{\theta}\right\} $$

increases as k increases.

Conversely, if there is no arbitrage, by the first half of the Fundamental Theorem of Asset Pricing (proven earlier), there exist a consistent positive linear pricing rule q. Let w₀ = 0 and α = 0. Consider the particular utility function

$$ {u}_{\ast \theta}\left({c}_0,{c}_1\right)\ \ \ \equiv \ \ \ -\exp \left[-\left({c}_0-{w}_0\right)\right]-\left({q}_{\theta }/{\pi}_{\theta}\right)\exp \left(-{c}_1\right). $$

(7)

Each function u_∗θ is strictly increasing and also happens to be strictly concave, infinitely differentiable, and additively separable over time. Using p = qX, it is easy to show that this utility function satisfies the first-order conditions for a maximum, which are necessary and sufficient by concavity. (Note: by a more complicated argument, it can be shown that the von Neumann–Morgenstern ‘state independent’ utility function − exp (− c₀) − exp (− c₁) has a maximum, but the maximum will not necessarily be achieved at α = 0).

As should be clear from the proof, it is not really important what class of preference we use, so long as all agents having preferences in the class prefer more to less and the class includes the particular preferences used in the proof (which are additive over states and time, increasing, concave, and infinitely differentiable).

Recent research on arbitrage, starting with Ross (1978) and Harrison and Kreps (1979), has focused on extending these results to more general state spaces in which there are many time periods and, more importantly, infinitely many states. In these spaces, deriving a positive linear pricing rule for marketed claims is still straightforward (one can prove the algebraic linearity condition and positivity directly from the no-arbitrage condition), but extending the pricing rule from the priced claims to all non-marketed claims requires some sort of extension theorem, such as a Hahn–Banach theorem. Obtaining a truly general result is complicated by the fact that the positive orthant is not typically an open set in these general spaces, and openness is a condition of the Hahn–Banach theorems. One part of the result that goes through in general is the implication that existence of an optimum implies existence of a linear pricing rule: so long as preferences are continuous in our topology, the preferred set will be open, and the linear pricing rule will be a hyperplane that separates the optimum from the preferred set.

Alternative Representations of Linear Pricing Rules

There are many equivalent ways of representing a linear pricing rule. Which representation is simplest depends on the context. In one representation, the price is the expected value under artificial ‘risk-neutral’ probabilities discounted at the riskless rate. (The risk-neutral probability measure is also referred to as an equivalent martingale measure.) In another representation, the price is the expectation of the quantity times the state price density, which is the state price per unit probability. In yet another representation, the price is the expected value discounted at a risk- adjusted rate. The purpose of this section is to show the fundamental equivalence of these representations.

The motive for using a particular representation is usually found in the study of intertemporal models or models with a continuum of states. Nonetheless, we will continue our formal analysis of the single-period model with finitely many states, leaving the more general discussion of the merits of the various approaches until afterwards. Now, we have already seen the basic linear pricing rule representation. For any portfolio α,

$$ p\alpha = qX\alpha =\sum_{\theta }{q}_{\theta }{\left( X\alpha \right)}_{\theta }, $$

(8)

that is, the sum across states of state price times the payoff.

The risk-neutral or martingale representation asserts the existence of a vector Π of artificial probabilities and a shadow riskless rate r such that

$$ p\alpha \ \ \ =\ \ \ {\left(1+r\right)}^{-1}\Pi X\alpha \ \ \ =\ \ \ {\left(1+r\right)}^{-1}{E}_n\left( x\alpha \right), $$

(9)

that is, the expectation E_Π of the payoff under the risk-neutral (martingale) probabilities Π, discounted at the riskless rate. It is easy to see the shadow riskless rate is equal to the riskless rate if one exists. The risk neutral approach is trivially equivalent to the positive linear pricing rule approach. Simply let

$$ \Pi \ \ \ =\ \ \ q/\sum_{\theta }{q}_{\theta } $$

(10)

and

$$ {\left(1+r\right)}^{-1}\ \ \ =\ \ \ \sum_{\theta }{q}_{\theta } $$

(11)

For the converse, let

$$ q\ \ \ =\ \ \ {\left(1+r\right)}^{-1}\ \ \ =\ \ \ \sum_{\theta }{q}_{\theta } $$

(12)

Therefore, the existence of a positive linear pricing rule is the same as the existence of positive risk-neutral probabilities. (The risk-neutral measure is equivalent to the original probability measure, that is, Π has the same null sets as α. Here, that is simply the requirement that the list of states with positive probability is the same for both measures.)

A third approach emphasizes the role of the state price density, ρ₀. In this case, the price is given by

$$ p\alpha \ \ \ =\ \ \ \sum_{\theta }{\pi}_{\theta }{\rho}_{\theta }{\left( X\alpha \right)}_{\theta}\ \ \ =\ \ \ E\left(\rho x\alpha \right). $$

(13)

To see that this is equivalent to the linear pricing rule, simply let

$$ {\rho}_{\theta}\ \ \ =\ \ \ {q}_{\theta }/{\pi}_{\theta }, $$

(14)

or, conversely, let

$$ {q}_{\theta}\ \ \ =\ \ \ {\rho}_{\theta }{\pi}_{\theta }. $$

(15)

Clearly, p is positive in all states if and only if q is.

We have shown the equivalence of these three approaches. This equivalence is stated in the following theorem.

Theorem

(Pricing Rule Representation Theorem) The following are equivalent:

• existence of a positive linear pricing rule;

• existence of positive risk-neutral probabilities and an associated riskless rate (the martingale property);

• existence of a positive state price density.

The remaining representation is that the value is equal to the terminal value discounted at a risk-adjusted interest rate r_a.

$$ p\alpha \ \ \ =\ \ \ {\left(1+{r}_a\right)}^{-1}E\left( x\alpha \right) $$

(16)

While this might at first appear to be inconsistent with the other representations, the risk-adjusted rate r_a is typically proportional to the covariance of return (=xα/pα) with some random variable, and consequently solving this equation for px yields a linear rule. (See Beja 1971; Rubinstein 1976, for general results concerning pricing rules using covariances.) For example, in the capital asset pricing model,

$$ {r}_a=r+\lambda \operatorname{cov}\left( x\alpha / p\alpha, {r}_m\right), $$

(17)

where r_m is the random return on the market and λ is the market price of risk. Solving these two equations for px, we obtain

$$ p\alpha \ \ \ =\ \ \ {\left(1+r\right)}^{-1}E\left[ x\alpha \left\{1-\lambda \left[{r}_m-E\left({\mathrm{r}}_m\right)\right]\right\}\right], $$

(18)

which is certainly linear in xα. The subtle question is whether or not this is positive, and this hinges on whether the market return can get larger than E(r_m) + 1/λ (Dybvig and Ingersoll 1982). In any case, the important observation is that the basic form of the representation is linear even if verification of positivity depends on the exact form of the risk premium.

Now we return to the question of the comparative advantages of the various representations. The risk-neutral or martingale representation was first employed by Cox and Ross (1976a) for use in option pricing problems and was later developed more formally by Harrison and Kreps (1979) and a number of others. The risk- neutral representation is particularly useful for problems of valuation or optimization without reference to individual preferences, since under the martingale probabilities we can ignore risk altogether and maximize discounted expected value. In fact, for some problems this approach tells us that risk-neutral results generalize immediately to worlds where risk is priced. However, this approach tends to be complicated when preferences are introduced, since von Neumann–Morgenstern (state independent) preferences under ordinary probabilities become state dependent under the martingale probabilities. As an aside, we note that, in intertemporal contexts in which the interest rate is stochastic, the price is the risk-neutral expectation of the future value discounted by the rolled-over spot rate (which is stochastic).

The state price density representation (Cox and Leland 2000; Dybvig 1980, 1988) is most useful when we want to look at choice problems. Samuelson (1947) emphasized the value of deriving equilibrium conditions from first- and second-order conditions for optimization. In asset pricing the first-order condition for an agent with von Neumann–Morgenstern preferences is that the agent’s marginal utility of consumption is proportional to a consistent state price density (not necessarily unique) for the security market (Dybvig and Ross 1982). (Note that if there is a non- atomic continuum of states, the state price density will typically be well-defined even though all primitive states have probability zero and state price zero.) For the CAPM, this fact was used implicitly by Sharpe (1964) and Lintner (1965), and was made explicit by Dybvig and Ingersoll (1982).

The representation of discounting expected returns using a risk-adjusted rate is most useful when we can get some independent assessment of the risk premium involved. Otherwise, it is needlessly complicated, since the price appears not only on the left-hand side of the equation but also in the denominator on the right-hand side. Discounting using a risk-adjusted rate is usually the method of choice for capital budgeting, since the risk adjustment is usually determined from comparables (for example, from past returns on assets in similar firms). For capital budgeting, there may also be a pedagogical advantage that (so far) it has been easier to communicate to practitioners than the other methods. Furthermore, focusing on the risk-adjusted discount rate sharpens the comparison of competing approaches (such as the capital asset pricing model and the dividend discount model).

It is useful to note how the various representations evolve over time. State prices are simply the product of state prices over sub-periods. For example, for t < s < T, the state price of a state at T given the state at t is equal to the state price of the state at T given the state at s times the state price of the state at s given the state at t. (The state at s is determined by the state at T given the pervasive assumption of perfect recall, that is, the assumption that the family of sigma-algebras is increasing. If we use some reduced specification of the state – as when looking at Markov processes – the state price is the product of the two, summed over all possible intermediate states.)

The martingale representation yields a price equal to the expected value under the martingale measure of the product of the terminal value times a discount factor that corresponds to rolling over shortest maturity default-free bonds. This representation makes particularly clear the interaction between term structure effects and other effects. If there is a significant term structure, the discount factor is random, and we cannot ignore the interplay between term structure risk and random terminal value unless the terminal value of the asset under consideration is independent of interest rates (under the martingale measure). If the terminal value is independent of interest rate movements, then the value of the asset today is the risk- neutral expected terminal value of the asset discounted at the riskless discount factor (which equals the risk-neutral expected discount factor from rolling over shorts).

The state price density has an evolution over time similar to that of the state price, namely, the state price density over a long interval is the product of the state price density over short intervals. Since the state price density equals the state price divided by the probability, the ratio of the two evolutions gives us a relation involving only probabilities, which is Bayes’ law.

Finally, the discounted expected value approach is more complicated than the others. The exact evolution over time depends on whether uncertainty is multiplicative, linear, a distributed lag, or whatever. This difficulty is usually overlooked in capital budgeting applications, which is probably not so bad in practice, given the imprecision of our estimates of risk premia and future cash flows.

Modern Results Based on the Absence of Arbitrage

Most of modern finance is based on either the intuitive or the actual theory of the absence of arbitrage. In fact, it is possible to view absence of arbitrage as the one concept that unifies all of finance (Ross 1978). In this section, we will try to provide a sample of how arbitrage arguments are used in diverse areas in finance. We will touch on applications in option pricing, corporate finance, asset pricing and efficient markets.

The efficient market hypothesis says that the price of an asset should fully reflect all available information. The intuition behind this hypothesis is that, if the price does not fully reflect available information, then there is a profit opportunity available from buying the asset if the asset is underpriced or from selling it if it is overpriced. Clearly this is consistent with the intuition of the absence of arbitrage, even if what we have here is only an approximate arbitrage possibility, that is, a large profit at little risk. Approximate arbitrage is always profitable to a risk-neutral investor. More generally, the issue is clouded somewhat by qsts of risk tolerance and what is the appropriate risk premium. Happily, empirical violation of efficiency of the market (for example, in event studies) is not significantly affected by the procedure for measuring the risk premium (Brown and Warner 1980, 1985). Therefore, an empirical violation of efficiency is an approximate arbitrage opportunity that presumably would be attractive at large scale to many investors.

The Modigliani–Miller propositions tell us that, in perfect capital markets, changing capital structure or dividend policy without changing investment is a matter of irrelevance to the shareholders. The original proofs of the Modigliani–Miller propositions used the law of one price and assumed the presence of a perfect substitute for the firm that was altering its capital structure. As an illustration of the Fundamental Theorem of Asset Pricing, Ross (1978) demonstrated that these propositions could be derived directly from the existence of a positive linear pricing rule.

To illustrate this argument, consider the proposition that the total value of the firm does not depend on the capital structure. The original argument assumed that there is another identical firm. If we change the financing of our firm, then the value of holding a portfolio of all the parts will give a final payoff equal to that of the identical firm, and must therefore have the same value under the law of one price. Alternatively, suppose that there exists a positive linear pricing rule q. Let x represent the total terminal value of a firm in a one-period model and xi the payoff to financial claim i on the assets of the firm. Then the sum of all the payoffs must add up to the total terminal value.

$$ x\ \ \ =\ \ \ \sum_i{x}_i $$

(19)

Using the positive linear operator, q, which values assets, we have that the value of the firm,

$$ v\ \ \ \equiv \ \ \ {\sum}_iq\left({x}_i\right)\ \ \ =\ \ \ q\left({\sum}_i{x}_i\right)\ \ \ =\ \ \ q\left({x}_i\right), $$

(20)

which is independent of the number of structure of the financial claims.

Note that both proofs make an implicit assumption that goes beyond what absence of arbitrage promises, namely, that changing the capital structure of the firm does not change the way in which prices are formed in the economy. In the original proof this is the assumption that the other firm’s price will not change when the firm changes its capital structure. In the linear pricing rule proof this is the assumption that the state price vector q does not change.

Another application of the absence of arbitrage is to asset pricing. The most obvious application is the derivation of the arbitrage pricing theory (Ross 1976a, b). We will consider the special case without asset-specific noise. Assume that the mechanism generating the per dollar investment rates of return for a set of assets is given by

$$ {R}_i={E}_i+{\beta}_{i1}{f}_1+\dots +{\beta}_{ik}{f}_k,\ \ \ \ \ i=1,\dots, n. $$

(21)

where E_i is the expected rate of return on asset i per dollar invested and f_i is an exogenous factor. This form is an exact factor generating mechanism (as opposed to an approximate one with an additional asset specific mean zero term).

Applying the pricing operator, q, to Eq. (21) we have that

$$ {\displaystyle \begin{array}{c}1=q\left(1+{R}_i\right)\\ {}=q\left(1+{E}_i+{\beta}_{i1}\;{f}_1+\dots +{\beta}_{ik}\ {f}_k\right)\\ {}=q\left(1+{E}_i\right)+{\beta}_{i1}q\left({f}_1\right)+\dots +{\beta}_{ik}\ q\left({f}_k\right)\\ {}=\left(1+{E}_i\right)/\left(1+r\right)+{\beta}_{ik}q\left({f}_1\right)+\dots +{\beta}_{ik}\ q\left({f}_k\right),\end{array}} $$

which implies that

$$ {E}_i-r\ \ \ =\ \ \ {\lambda}_1{\beta}_{i1}\ \ \ +\dots +\ \ \ {\lambda}_k{\beta}_{ik}, $$

(22)

where λ_j  ≡   − (1 + r)q(f_i) is the risk premium associated with factor j. Equation (22) is the basic equation of the arbitrage pricing theory. We have derived it using absence of exact arbitrage in the absence of asset-specific noise. More general derivations account for asset-specific noise and use absence of approximate arbitrage.

The most important paper in option pricing, Black and Scholes (1973), is based on the absence of arbitrage, as is the whole literature it has generated. At any point in time, the option is priced by duplicating the value one period later using a portfolio of other assets, and assigning a value using the law of one price. We will illustrate this procedure using the binomial process studies by Cox et al. (1979). During each period, the stock price either goes up by 20 per cent or it goes down by 10 per cent, and for simplicity we take the riskless rate to be zero. Assume that we are one period from the maturity of a call option with an exercise price of $100, and that the stock price is now $100 (the call is at the money).

How much is the option worth? To figure this out, we must find a portfolio of the stock and the bond that gives the same terminal value. This is the solution of two linear equations (one for each state) in two unknowns (the two portfolio weights). Explicitly, the terminal call value is the larger of 0 and the stock price less 100. In the good state, the stock value will be $120 and the option will be worth $20. In the bad state, the stock price will be $90 and the option will be worthless. If α_S is the amount of stock and α_B the amount of $100 face bond to hold in the duplicating portfolio, then we have that

$$ 20=120{\alpha}_{\mathrm{S}}+100{\alpha}_{\mathrm{B}} $$

to duplicate the option value in the good state, and

$$ 0=90{\alpha}_{\mathrm{S}}+100{\alpha}_{\mathrm{B}} $$

to duplicate the option value in the bad state. The solution to the two equations is given by

$$ {\alpha}_{\mathrm{S}}=2/3{\alpha}_{\mathrm{B}}=-3/5. $$

Therefore, each option is equivalent to holding 2/3 shares of stock and shorting (borrowing) 3/5 bonds. By the law of one price, the option value is the value of this portfolio, or 100α_S + 100α_B = 6 2/3. In this context, we used arbitrage to value the option exactly. More generally, if less is known about the form of the stock price process, absence of arbitrage still places useful restrictions on the option price (Merton 1973; Cox and Ross 1976b). For example, the price of a call option is less than the current stock price, and the price of a European put option is no smaller than the present value of the stock price less the current stock price.

Absence of arbitrage also implies a surprising feature of the behaviour of long interest rates in the limit as maturity increases. Let V(t, T) denote the zero-coupon bond price, namely, the price at t of a riskless claim for $1 at T. Equivalently, we can describe bond prices in terms of the zero-coupon rate z(t, T) where V(t, T) = 1/(1 + z (t, T))T − t. Defining the long zero-coupon rate, zL(t) ≡ lim T ↑ ∞z (t, T); absence of arbitrage implies that the probability is zero that this rate will ever fall. This is because the bond price today is an average of bond prices tomorrow weighted by (positive) state prices, and the bond price in any state declines asymptotically at the rate zL(t) in that state tomorrow. Thus, the weighted average of prices today declines at a rate equal to the smallest rate under our maintained assumption of finitely many states (and perhaps more slowly given infinitely many states). As a consequence, zL(t) at time t is always less than or equal to its value zL(t) at any future date, s > t, in every realization (with probability one). For details see Dybvig et al. (1996).

Dominance is a useful concept to combine with the absence of arbitrage. A dominance argument gives features of a strategy that are optimal independent of preferences and, often, independent of distributions as well. For example, when we write the payoff on a call as max (S − X, 0), we are implicitly assuming it is a chosen strategy to exercise the option when it is in the money and not to exercise it when it is out of the money. Absent frictions, this is a dominant strategy and the assumption is without loss of generality. A more subtle dominance argument, relying on the absence of frictions and on a non-negative riskless rate, gives the classical result that an American call option (which can be exercised at or before maturity) has the same value as the corresponding European call option (which can only be exercised at maturity), because waiting to exercise is a dominant strategy (Merton 1973; Cox and Ross 1976b). Another dominance argument can be used to show that it is optimal to exercise certain reload options used in executive compensation again and again, whenever they are in the money (Dybvig and Loewenstein 2003).

An alternative to option pricing by arbitrage is to use a ‘preference-based’ model and price options using the first-order conditions of an agent (Rubinstein 1976). While using this alternative approach is very convenient in some contexts, the Fundamental Theorem of Asset Pricing tells us that we are not really doing anything different, and that the two approaches are simply two different ways of making the same assumption. The same point is true of the distinction some authors have made between the ‘equilibrium’ derivations of the arbitrage pricing theory and the ‘arbitrage’ derivations: there is no substance in this distinction. One derivation may give a tighter approximation than another, but all derivations require similar assumptions in one form or another.

See Also

• Finance

• Modigliani–Miller Theorem

• Options

• Present Value