Implied Probabilities And Arrowdebreu Theory

The purpose of this section is not so much to provide a formal description of the Arrow-Debreu theory as much as to provide a flavor for it. Let us consider the scenario that involves placing bets on a set of outcomes. Examples of such events could be a boxing match or a horse race. In these cases, the set of outcomes is finite and well defined. We will use the horse race example for purposes of illustration. Important to the discussion is the notion of betting. If, for example, the bet is placed in favor of a horse and it wins the race, then the reward is the payoff from the bet. If it happens to lose, then here, too, the reward is the payoff from the bet, except that the payoff is probably zero dollars. Thus, a bet is completely defined when we specify the payoff for every possible outcome. To place a bet, one has to put up the stake money. This is specified by the bookie.

The Arrow-Debreu theory states that the full and complete specification of bets with the stake money and the payoff for each outcome automatically implies a probability for a particular outcome.1 Additionally, the stake

JThe reasoning stems from the maximization of a linear utility function resulting in a linear program with constraints. The weights happen to be the values of the dual variables in the solution of the linear program. This work by Arrow and Debreu was awarded the Nobel Prize in Economics.

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 probability 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 neutral 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 presented in Table 11.1.

Note that the first scenario described in Table 11.1 is the risk-free scenario where a deposit of x dollars 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 probability 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.

F ne

x 3

Pws

2

Pbb _

1

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

pne = 3/8 = 0.375 pws = 2/5 = 0.4 pbb = 1/3 = 0.333

The value of x from the preceding equations is therefore pne + pws + pbb = 1.108. So, if we deposit close to a dollar and 11 cents with the bookie, we will get back a dollar. The loss for the bettor in this enterprise is therefore 100 x (0.108/1.108), which is approximately 9.7 percent. In other words, on average the bookie gets to keep 10 cents on every dollar that is deposited as stakes. Normalizing the probability weights to add up to one, we now have

Probability of NiceAndEasy winning = 0.375/1.108 = .338 Probability of WindSlicer winning = 0.4/1.108 = .361 Probability of ButterBiscuit winning = 0.333/1.108 = .301

Thus, WindSlicer is favored to win the race, with ButterBiscuit being the underdog.

Note that we started our example with the specification of the bets and their odds and have now derived the probabilities from it. Are the probabilities the true probabilities for the outcome of the race? Maybe, then again maybe not. This is, however, where the bookie will allow the bet to be made. In the case of the markets, unlike the example here, the price/stake amount of a bet is decided by the auctioning process. The price, therefore, represents the consenus opinion of the participants. In such situations it may be argued that the risk neutral probabilities represent the consensus of the market.

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