1

Implied (risk neutral) probabilities, and betting odds

Fabrizio Cacciafesta (University of Rome "Tor Vergata")

ABSTRACT - We show that the well known equivalence between the "fundamental theorem of asset pricing"
and the "dutch book argument" consists, in reality, of two parts: the first concerning the existence of a risk free
investment, and the second the possibility of arbitrages. We show how the "implied probabilities" of an odds
system may be used to rate the bookmaker's fairness.

1.

Reference is to the well known analogy between the "dutch book" argument,

according to which a system of betting odds allows no sure wins if and only if the "implied
probabilities" are formally coherent, and the fundamental theorem of assets pricing (FTAP).
This last states that in a market where one euro may be alternatively invested to give the
random amount S or the certain amount r, no arbitrage is possible if and only if a (risk
neutral

: "r.n.") probability distribution exists for which

(1)

E[S] = r

and for which the non zero probability events are the really possible ones. Moreover, there is
only one such distribution if and only if the market is "complete".

We consider the typical (although not the most general) case, in which an odds system

{q

i

}

corresponding to some "partition of reality" {E

i

} is given: a family of n mutually

exclusive events is defined, one of which must necessarily occur; and someone offers to pay
q

i

(>1) euro if event E

i

occurs to anyone that has bet one euro on this fact. Then, the implied

probabilities

are defined as

(2)

p

i

= Prob(E

i

) = 1/q

i

(i= 1, 2, …, n).

Under our hypothesis every p

i

is in (0, 1), but nothing insures that P = ∑p

i

is 1

1

.

Now, the bets portfolio made up by betting, for every i,

(3)

a

i

=

∑ ∏

∏

≠

≠

h

h

j

j

i

j

j

q

q

euro on the event E

i

, has cost 1, and the winning that it yields is in any case 1/P: so, if it

happens that P ≠ 1 (i.e., the "implied probabilities" are not formally "probabilities") there is
some space for a sure gain for some one. The quantity 1/P has the meaning of a risk-free
accumulation factor (although not necessarily bigger than 1).

If 1/P is bigger than 1, a "dutch book" can be created: whoever bets according to the

proportions (3) will receive more than the amount paid. If, conversely, 1/P is smaller than 1,
the situation is more questionable: by selling that portfolio, an operator receives more than
what he will have to pay. To adopt this strategy, he must be allowed to accept bets at the
given prices, but in the exact quantities he wishes (which is not possible for a professional

1

This is the only condition of coherence we must care about, because we are considering bets on mutually

exclusive events.

2

bookmaker). This reminds one of the possibility of selling the risk free asset short in a
financial market.

If we accept this as true, we conclude that in a bets market, the implied probabilities

exist if and only if there is no possibility for a riskless investment

2

.

2.

Now, let us think of a bookmaker who forecasts that, for any i, c

i

euro will be bet on

the event E

i

. If he wants to keep for himself the fraction 1−k of the total income, his

equilibrium will be safe if he offers odds according to the equalities:

(4)

k

∑ c

i

= q

1

c

1

= ... = q

n

c

n

.

Quantity k is connected to the implied probabilities by the elementary relation

(5)

k

= 1/P.

So, if we normalize the (p

i

) by putting

(6)

p

i

* = kp

i

we get a set of "probabilities" (p

i

*

) (they add up to 1 also when the p

i

don't) for which it

results

(7)

p

i

*q

i

= k for every i.

This shows that the mean value - calculated with respect to the (p

i

*) - of the winning a

bettor that has placed one euro on one (whichever) of the (E

i

) can expect is equal to the risk-

free accumulation factor 1/P. Then the normalized implied probabilities appear to correspond
to the r.n. probabilities of the FTAP.


3.

As for the relations between existence of r.n. probabilities and possibilites of arbitrage

in a bets market: we remember that the FTAP gives a necessary and sufficient condition for
the existence of (at least) one probability distribution satisfying the equation (1), in which the
factor r and the random quantity S are supposed to be given. If, in much the same way, we
look for the existence of a solution for the system of n+1 equations:

(8)







=

=

=

∑

)

,...,

2

,

1

(

*

1

*

n

i

k

q

p

p

i

i

i

i

where (q

i

) and k are given and (p

i

*) are the unknown, we find that a necessary and sufficient

condition for it is given by:

(9)

k

=

∑ ∏

∏

≠

h

h

j

j

j

j

q

q

.

2

We mean: an investment that yields in any case the same non zero rate of return.

3

This relation is also a condition of "no arbitrage". To see this, we must suppose that an

odds system (q

i

) is proposed within a bets market in which the parameter k is given

independently from those odds: imagine a situation in which two different bookmakers
propose their odds for the same partition of the reality, so that one can compare the k given
(via (9)) by one system, with the odds of the other.

In this situation, consider an operator which can freely buy and sell bets according the

two different bet odds: no arbitrage is actually possible for him if and only if (9) holds.

The analogies with the FTAP framework may be completed as follows.

Suppose that, in (8), some (two at least) of the (q

i

) are 0. Then the system will

generally admit more than a solution. This situation corresponds to that of a non complete
financial market.

Suppose now that the system (8) is to be considered in a situation in which the actually

possible events E

i

are in number of m ≠ n. In this case, the implied probabilities are not

"equivalent" to the real ones, and arbitrages are possible (for m>n, in some case the bookie
will pay 0; for m<n, the same bookie will have to pay with a bigger probability than he had
accounted for).


4.

Conclusions are: "implied probabilities" exist if and only if no operator can count on a

sure winning; "normalized implied probabilities" always exist, but the question is, whether
they are or not coherent with the - exogeneously given - parameter k.

When they exist, implied probabilites have the exact meaning of the "risk neutral"

probabilities the FTAP deals about. So, some doubts about their informative content is
legitimate. They are indeed directly connected with how the bettors distribute their
preferences, and not with what would maybe more interesting to know: that is, some kind of
"objective", or "real", or in any case in no way biased by the operators' risk aversion,
probability of the events.

It is maybe more meaningful to use k to rate the bookmaker's fairness. It is a parameter

every one can calculate. The quantity 1−k has the meaning of price at which the bookmaker
sells the risk the bettors buy.

Bibliography

N.H. BINGHAM - R. KIESEL, Risk Neutral Valuation, Springer (1998)
M.L. EATON - D.A. FREEDMAN, Dutch Book against some 'Objective' Prior, Bernoulli
(10) 2004
D.P. ELLERMAN, Arbitrage Theory: a Mathematical Introduction; SIAM Review (26)