1
Arbitrage
There is an arbitrage opportunity when the law of one price is violated
so that it is possible to get something for nothing (a free lunch). This
cannot be true in conditions of equilibrium.
Definitions. Let:

R R ij  be the  (s  n) matrix of payoffs,
i s 12 , , states of nature and  j n  12 , , financial assets;
x = col. (n  1) vector of the quantities of assets in a portfolio.
x
j
0 denotes a long position and  x
j
0 a short position (the
holder of the portfolio has to pay the payoff);  y= col. (s  1) of the
payoffs of the portfolio in the different states:  y Rx  ;  p= row
(1  n) of the assets’ prices.
An arbitrage portfolio (AP) should have a non-positive cost  px  0
and a semi-positive payoff:  y Rx   0 .
Hence:  Rx px    0 0 is the no-arbitrage condition.
The  i th (Arrow-Debreu) contingent security pays 1 euro in the state  i and
0 in the other states. Its payoff  y is semi-positive  (y i =1, y j  i =0) . Then
its price, denoted by  q i to distinguish it from the  p j prices of the actual
securities, is positive  q i >0 : it is the price of 1 euro in the  i th
contingency. The  row (1s) vector  q is the state prices vector. The
(A-D) securities represent a basis for the payoffs space (they form the
s  s unitary matrix).
The payoff of any  j th security [ R j the  j th  col. (s  1) of  R i j ] can be
represented by a portfolio of A-D securities. Its price in no-arbitrage
conditions is then equal to  p q R qR
j i ij
i
j
   . In general, it
holds the: