or specifically
pne+pws+pbb=x
pne= 3/8 = 0.375
pws= 2/5 = 0.4
pbb= 1/3 = 0.333The value of xfrom 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 ×(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 = .301Thus, 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 probabil-
ities 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.
The Single-Step Model
We are now ready to formulate the single-step model to calculate the prob-
abilities implied by the spread. It is, however, good to remind ourselves of
the implicit assumptions we make during the modeling process. First, put-
ting on a spread involves a short position and we need to borrow stock. At
times it may turn out that a particular stock is unavailable for borrow. How-
ever, for purposes of our model, we will assume that the stock is always
available to borrow. Additionally, we also assume that there are no liquid-
ity issues and that it is possible to put on a spread position in size at the
current spread level observed in the market. This would imply that the trans-
action costs if any are negligible and may be assumed to be zero.
The Market Implied Merger Probability 175