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 noarbitrage 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
f 1  rm  s
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
f 1  rm  s  1  rs 
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,
1  rs  1  rm   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  s 1  r  .
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
f  s 1  c 1  r  .
(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
q  ax  by   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 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  denote the state space,
  1,, m ,
where there are m states and the state of nature  occurs with probability . Applying q
to the ‘indicator’ asset e whose payoff is 1 in state  and 0 otherwise, we can define a
price q for each state  as the value of e ;
q  q  e 
Now, if there were linearity, the value of any payoff, x, could be written as
q  x    q x .
Of course, this argument presupposes that q(e) is well defined, which is a strong
assumption if e 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i (inclusive of
dividends, etc.) in state of nature . The matrix X  [Xi] 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)  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,
p  0,
(1)
and its random payoff later is never negative,
X  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 X as p for arbitrary  embodies an assumption that investment in marketed
assets is divisible.) The second property says that the arbitrage portfolio  has a strict
inequality, either in (1) or in some component of (2). We can express both properties
together as
 p 
X *      0.
(3)
X 
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  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
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  be an
arbitrage opportunity. By (4),
p  qX  ,
or equivalently
0   p  q  X   1q X *.
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   y | for some , y  X * ,
(5)
does not intersect the positive orthant  m1   y | y  0 except at the origin, i.e.
S   m1  0
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  m1 , z  0, we must have
q* z  0  q* 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  (q*2 , q*3 ,, q*n ) / q*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, i.e. 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
0  0u0[w0  p , ( X )0 ].
The quantity p is the price of the portfolio, and therefore w0 – p 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 , then clearly the choice problem for an agent
with increasing preferences cannot have a maximum, since for every ,
  u {w0  p(  k),[ X (  k)] }
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  = 0. Consider the particular utility function
u* (c0 , c1 )  exp[(c0  w0 )]  (q /  ) exp(  c1 ).
(7)
Each function u* 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  =
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 (you 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
,
p  qX 
(8)
  q ( X  ) ,
i.e., the sum across states of state price times the payoff.
The risk-neutral or martingale representation asserts the existence of a vector  of
artificial probabilities and a shadow riskless rate r such that
p  (1  r )1 X 
 (1  r ) 1 En (x ),
(9)
i.e., the expectation E of the payoff under the risk-neutral (martingale) probabilities ,
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
(10)
  q /  q
and
(1  r )1   q
(10)
q  (1  r )1   q
(11)
For the converse, let
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, i.e.  has the same null sets as . 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,  0 . In this case, the
price is given by
p      ( X  )
(13)
 E (  x ).
To see that this is equivalent to the linear pricing rule, simply let
  q /   ,
(14)
or, conversely, let
q     .
(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:
(i) Existence of a positive linear pricing rule
(ii) Existence of positive risk-neutral probabilities and an associated riskless rate
(the martingale property)
(iii) 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 ra.
p  (1  ra ) 1 E ( x )
(16)
While this might at first appear to be inconsistent with the other representations, the riskadjusted rate ra is typically proportional to the covariance of return (= x/p) with some
random variable, and consequently solving this equation for px yields a linear rule. (See
Beja, 1971, and Rubinstein, 1976, for general results concerning pricing rules using
covariances.) For example, in the Capital Asset Pricing Model,
ra  r   cov( x / p , rm ),
(17)
where rm is the random return on the market and  is the market price of risk. Solving
these two equations for px, we obtain
(18)
p  (1  r )1 E[ x{1  [rm  E (rm )]}],
which is certainly linear in x. The subtle question is whether or not this is positive, and
this hinges on whether the market return can get larger than E (rm )  1/  (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, 1982, and Dybvig, 1980,
1985), is most useful when we want to look at choice problems. For von Neumann–
Morgenstern preferences, the state price density is equal to the marginal utility of
consumption, for some consistent positive state price density (Dybvig and Ross, 1982).
(Note that if there is a non-atomic continum of states, the state price density will typically
be well-defined even though all primitive states have probability zero and state price
zero.)
The representation of discounting expected returns using a risk-adjusted discount
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 (e.g. 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 subperiods. 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, i.e. 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 over-looked 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, i.e. a large profit at little risk. Approximate
arbitrage is always profitable to a risk-neutral investor. More generally, the issue is
clouded somewhat by questions of risk tolerance and what is the appropriate risk
premium. Happily, empirical violation of efficiency of the market (e.g. 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.
(19)
x   i xi
Using the positive linear operator, q, which values assets, we have that the value of the
firm,
v  i q ( xi )
 q (i xi )
(20)
 q ( xi ),
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,
1976b). 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
Ri  Ei  i1 f1  ik f k ,
i  1,, n.
(21)
where Ei is the expected rate of return on asset i per dollar invested and fi 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 equation (21) we have that
1  q(1  Ri )
 q(1  Ei   i1 f1   ik f k )
 q(1  Ei )   i1q( f1 )   ik q( f k )
 (1  Ei ) /(1  r )   i1q( f1 )   ik q( f k ),
which implies that
Ei  r  1i1  k ik ,
(22)
where  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, Ross and Rubenstein (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 S is the amount of stock and B the
amount of $100 face bond to hold in the duplicating portfolio, then we have that
20  120S  100 B
to duplicate the option value in the good state, and
0  90S  100 B
to duplicate the option value in the bad state. The solution to the two equations is given
by
S  2 / 3
 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 100S + 100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).
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 (Rubenstein, 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.
Philip H. Dybvig and Stephen A. Ross
See also finance; option pricing; options; present value; zero profit condition.
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