money must be the probability weighted payoff (also called the expected
payoff) across all outcomes. Also, if it so happens that a single set of prob-
ability weights is not able to account for the entire set of bets, then arbitrage
opportunities exist.
According to the theory, the probabilities are derived such that any two
bets with the same expected payoff have the same current value. It may be
that one of the two bets yields the expected payoff almost certainly and the
risk associated with it is minimal when compared to the other bet. This
scheme, however, treats both bets on an equal footing; that is, we are neu-
tral to risk. For this reason, the set of probabilities that are implied by the
definition of the bets are called risk neutral probabilities.
Continuing with the horse race example, let us say that the odds given
by the bookie for the horse race is as follows: 3 to 5 in favor of NiceAnd-
Easy, 2 to 3 in favor of WindSlicer, and 1 to 2 in favor of ButterBiscuit. For
instance, a successful bet of one dollar on ButterBiscuit returns the stake plus
two dollars, which is a total of three dollars. The terms of the bets are pre-
sented in Table 11.1.
Note that the first scenario described in Table 11.1 is the risk-free sce-
nario where a deposit of xdollars with the bookie results in a payoff of one
dollar no matter which horse wins. According to Arrow-Debreu theory, the
bet amount must be a weighted combination of the payoffs. If the probabil-
ity weights for each of the three horses winning are denoted as pne,pws,pbb,
then the following equations as given in matrix form must hold.
111
800
050
0033
2
1
⋅
=
p
p
px
ne
ws
bb174 RISK ARBITRAGE PAIRS
TABLE 11.1 Terms of the Bet.
Payoff from the Bets
Bet NiceAndEasy WindSlicer ButterBiscuit
Bet scenario amount wins wins wins
Risk-Free Scenario x 111
Bet On NiceAndEasy 3 8 0 0
Bet on WindSlicer 2 0 5 0
Bet on ButterBiscuit 1 0 0 3