14-Arbitrage Page 395 Wednesday, February 4, 2004 1:08 PM
Arbitrage Pricing: Finite-State Models 395We also know that wA + wB = 1. If we solved for the weights for wA
and wB that would simultaneously satisfy the above equations, we
would find that the portfolio should have 40% in asset A (i.e., wA = 0.4)
and 60% in asset B (i.e., wB = 0.6). The cost of that portfolio will be
equal to(0.4)($70) + (0.6)($60) = $64Our portfolio (i.e., package of assets) comprised of assets A and B
has the same payoff in State 1 and State 2 as the payoff of asset C. The
cost of asset C is $80 while the cost of the portfolio is only $64. This is
an arbitrage opportunity that can be exploited by buying assets A and B
in the proportions given above and shorting (selling) asset C.
For example, suppose that $1 million is invested to create the port-
folio with assets A and B. The $1 million is obtained by selling short
asset C. The proceeds from the short sale of asset C provide the funds to
purchase assets A and B. Thus, there would be no cash outlay by the
investor. The payoffs for States 1 and 2 are shown below:Asset Investment State 1 State 2A $400,000 $285,715 $571,429
B 600,000 300,000 1,200,000
C –1,000,000 –475,000 –1,400,000
Total 0 110,715 371,429ARBITRAGE PRICING IN A ONE-PERIOD SETTING
We can describe the concepts of arbitrage pricing in a more formal
mathematical context. It is useful to start in a simple one-period, finite-
state setting as in the example of the previous section. This means that
we consider only one period and that there is only a finite number M of
states of the world. In this setting, asset prices can assume only a finite
number of values.
The assumption of finite states is not as restrictive as it might
appear. In practice, security prices can only assume a finite number of
values. Stock prices, for example, are not real numbers but integer frac-
tions of a dollar. In addition, stock prices are nonnegative numbers and
it is conceivable that there is some very high upper level that they can-
not exceed. In addition, whatever simulation we might perform is a
finite-state simulation given that the precision of computers is finite.