context, they reveal nothing about the pattern of equilibrium
prices.
Arbitrage 173
4. If the arbitrage principle is to have predictive power, market
frictions must be absent. Conversely, in the presence of
transaction costs or institutional constraints o
positive; and x=0 states that every element of x is zero. Let V
denote the matrix of asset payoffs. This matrix has $ rows (the
number of states) and n columns (the number of assets). A
typical element of V is vkj, the payoff of asset j in state k. The
pa
; and 2. probabilities,7 11(12(1$, one for each state, such that pj
=NE9vj;j =1(2(n (7.2) The symbol vj denotes the list of
payoffs, one for each state, for asset j. (In this context, vj is a
random variable: a set of outcomes, each with its associated
pr
>
1+r0A in at least one. Finally, the absence of arbitrage
opportunities is characterized by a set of asset prices such that
the rate of return on the initial outlay either (a) equals the riskfree rate in every state, or (b) is greater than the risk-free
, so that q1 = 5 16 4 5 = 1 4 and q2 = 5 16 1 5 = 1 16. 3.
Another way of writing the RNVR condition is E9rj;=r0
j=1(2(n (7.4) This can be demonstrated by recalling that rkj =
vkjpj/pj, substituting into pj = NE9vj;, and rearranging the
result. (Remember
the arbitrage principle implies the RNVR i.e. the existence of a
set of artificial (risk-neutral or martingale-equivalent)
probabilities such that the price of each asset equals the expected
payoff (using these probabilities), discounted by a risk-free
in
Be careful not to read too much significance into the RNVR.
The probabilities do not necessarily describe the beliefs of any
investor who behaves in accordance with some principle e.g.
the EUH, for which probabilities are relevant. Nor are they the
object
Nt+Ndt+N =0. While necessary for convergence, this condition
is not sufficient. Rather than attempt to write down a general
condition to ensure that (10.8) is well defined, it is assumed here
that the sum converges to a finite value; individual cases will
assets, but it is also possible to make sense of prices that are
associated with individual states of the world. Once the
existence of state prices has been established, the risk-neutral
valuation
relationship
providesaconvenientwayofexpressinganyassetpri
YZ
E
Fig. 5.6. Efficient portfolios with different lending and
borrowing rates
If the rate at which funds can be borrowed, rB 0 , exceeds the
rate at which funds can be lent, rL 0 , the frontier comprises
three segments: rL 0 Y (some funds are lent), YZ
made in the holding of asset j, the necessary funds being
borrowed at the risk-free interest rate. The quantity of each of
the other assets remains unchanged. Denote the increase in the
proportion of j in the portfolio by aj (and, by construction, a0
=aj)
The linear pricing rule is an equivalence: if arbitrage
opportunities
are
absent,
statepricesexist;
ifstatepricesexist,arbitrageopportunitiesareabsent. Therefore, if
the arbitrage principle holds, it implies that every asset price can
be written as a weig
1=220%. It is rather far-fetched at such a huge value but this
is only a numerical example.
176 The economics of financial markets
then expressions such as (7.2) or (7.4) or (7.5) express the
fundamental valuation relationship. (Once again, see chapter 4.
<0 for some state(s). Arbitrage principle: the arbitrage principle
asserts
that
arbitrage
opportunities
are
absent.
Market
equilibrium: a set of asset prices and an allocation of asset
holdings across investors such that the demand to hold assets is
no gr
(where 5 and 9 correspond to any pair of assets). In practice,
the two estimates always differ as a result of statistical
dispersion;5 they have to be significantly different to warrant a
rejection of the Black CAPM. Complicated though the
hypothesis may
i. Typically, the last period, t+N, denotes the end of the assets
life of N periods, so that pt+N would be a specified value,
possibly zero such as the scrap value for a physical asset or the
maturity value if the asset is a bond. Perhaps the asset has no
=4 > 0
172 The economics of financial markets Hence, pC =4 cannot be
an equilibrium: a portfolio can be found that costs nothing and
yields a positive risk-free payoff.6 What is the equilibrium value
of pC? Consider any portfolio xA(xB(xC of assets for wh
with the sample average of the observed excess return on each
asset j, the average being calculated over some time period
say, a year or several years. Let zj rjr0 denote the sample
average for asset j. For the beta-coefficients, it is natural to
choose
Assets
ABC
State 1 10 8 9 State 2 8 0 12 Price 3 2 pC =?
The absence of arbitrage opportunities implies that the price of
asset C, pC, is not arbitrary. To see this, consider some arbitrary
values of pC and check that risk-free profits can be made
(except