This Demonstration uses the multi-period (seven time periods for seven games) setting of the Baseball World Series of 2001and 2009. In such a series, the first two games are played at the original host team's city and the third game is played at the original guest team's city. The original host and original guest teams were the Arizona Diamondbacks (AZD) versus the New York Yankees (NYY) in 2001 and the New York Yankees (NYY) versus the Philadelphia Phillies (PHI) in 2009.

We use these series to illustrate an implementation of Basis Instruments Contracts (BICs) insights for odds pricing purposes in a manner that reduces the curse of dimensionality and yields flexibility in the choice of dynamics of underlyings (i.e., outcome of a game), and help identify potential mispricings or arbitrage opportunities. The example also serves as a ready-made template for the analysis of odds in seven-game contests in baseball or other sports, such as basketball.

General BICs and Demonstration Computations Considerations

In this instance, a BIC is an agreement between one or more buyers and one or more sellers.

i) The agreement is entered into at a time prior to the time at which any of the seven games conclude ().

ii) The agreement stipulates that at time , , one or more buyers shall pay one or more sellers a premium amount that is a deterministic function .

iii) The agreement also stipulates that at time , , one or more sellers shall pay one or more buyers a payout amount that is a deterministic function or .

In a frictionless market where there is no bid/offer spread and no sensitivity to notional amounts of contracts, reduces to the discounted conditional probability of or knowing , …, and reduces to the normal multiplication operator.

We denote the discount factor at each time by . We set to be the price at time of a payoff . When this price is a function of other variables, these may be added; for instance is the price at time of the payoff , which is a priori a function of the values of . We simply note that is the price of the bet that NYY will win game .

In order to price a derivatives contract paying out at time , in this Demonstration we use the simplified BICs iterative backward replication strategy to combine the premium amounts of BICs with the payout as follows:

;

...

....

.

We seek to compute the odds for the NYY to win the World Series while taking into account the changing dynamics of incremental odds adjustments as the outcome of games is revealed. The odds of victory or defeat in each game appropriately incorporate the impact of the outcome of previous games and the location of the game in predicting the results of a game. The means of incorporation are not necessarily Markovian, as the conditional probabilities may incorporate the outcome of all earlier games, and not just the last one.

We implement and extend here to 2009 both the literal (example II) and actual translation (example I) of the baseball example described in the introduction to chapter IV of the book *BICs 4 Derivatives Volume I: Theory*.

Example I:

The numbers used in example I above are actually those later used in the book to provide and discuss numeric results, in particular the 8.3% profit that is based on an actual fair price of the New York Yankees 2001 World Series victory computed price of 48.3% versus 40% odds of winning the first game using different assumptions. They are based on the composition of multi-period BICs that are in reality derived from the following assumptions:

(the probability that the original guest team will win the first game in the original host team home stadium = );

(the probability that the original guest team will win the third game in the original guest team home stadium; the plot shows results for alternative choices for this number);

(in the sixth game, which is in the original host team's home stadium, the the original guest team will have a 40% probability of winning unless they won at more than four of the earlier five games or won less than two);

for and , (the team that wins the first or third game increases their probability of winning the next game by 10%; in this Demonstration, the first slider enables the user to change this number);

for and , (the team that wins the fourth or sixth game increases their probability of winning the next game by the slider-adjustable 10% unless one of the teams has already won four games, in which case the remaining games are cancelled and we assume for convenience that the team that is short will certainly win the remaining games);

and the payoff whose price is computed in this Demonstration, .

Example II:

Example II implements the literal description made in the book of the different odds of victory or defeat in each game and how these are influenced by the outcome of previous games and the location of the game.

We suppose each team has a 60% versus 40% probability of winning its first match in its home stadium and that this probability is increased by 10% after each game won if the next game is at home and decreases by 10% after each game lost at home if the next game is also at home. We also assume that once a team wins four games, its probability of winning the remaining games is zero. These translate algebraically into:

(this is the probability that the original guest team will win the first game in the original host team home stadium );

(this is the probability that the original guest team will win the third game in the original guest team home stadium; the plot shows results for alternative choices for this number);

(the fifth game in in the original host team has a chance of being won by the original guest team unless the four previous games were all won by them, in which case there is no fifth game and we assume for convenience that the team that is short will certainly win the remaining games);

for and , (the team that wins the first or third game increases their probability of winning the next game by 10%; in this Demonstration, the first slider enables the user to change this number);

for and , (the team that wins the fifth or sixth game increases their probability of winning the next game by the slider-adjustable 10% unless one of the teams has already won four games, in which case the remaining games are cancelled and we assume for convenience that the team that is short will certainly win the remaining games);

and the payoff whose price is computed in this Demonstration is .

This Demonstration computes the odds of winning in the original guest team under a variety of scenarios, namely varying odds of winning the first game at home or varying odds of winning the next game in the same stadium after a victory. The price is then obtained by multiplying with an interest rate discount factor.

We provide sliders to allow changes in the discount factor that results in a present price as well as an increase in the probability of winning the next game after winning a game in the same location.

We plot the price of a contract that pays one if the original guest team wins the World Series against the probability of each team winning the first game in their home stadium and contrast it against the discounted probability of the original guest team winning the first game.

Curse of Dimensionality Elimination Illustration

The example also illustrates how to implement the BICs functional compression insights to reduce the computational cost of multiperiod expectations from a priori exponential (here ) to a polynomial of low order (here 40 or 42). This gain is obtained by noticing that the functional dependence on , …, is in fact merely a functional dependence on , which is an integer varying between 0 and and .

Note that in the implementation of this Demonstration the discount factors , …, are compounded and applied at once at the end as the slider-adjustable discount factor (in the code named DF) and what is iteratively processed are merely conditional probabilities.

The straight line corresponds to the odds of the original guest team winning the World Series taken as the odds of the original guest team winning the first game (for example 40% as assumed at first in the examples).

In the case of the 2001 contest, since the odds of winning the first game at home have to be 1 minus the odds of NYY winning the first game that is in Arizona, the straight line is a descending line anchored at 1 at the origin. For the 2009 contest, the line is ascending.

The red area corresponds to the odds of the original guest team winning the first game and overstates the true odds of winning the World Series, while the green area corresponds to the odds of the original host team winning the first game and understates the odds of winning the World Series, such as is the case when the 40% odds of winning the first game understate the actual odds of winning the World Series (48.3%).

Putting these two graphs in contrast is to underscore how BICs, by taking into account a more refined level of information, may help correct more biased superficial estimates (such as equating the odds of winning the first game to the odds of winning the World Series). Note that the odds of winning the first game used here is one among many possible choices and may be replaced by straight market-based estimates that come from sports betting markets.

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