Summary: April 12, 2007
The Fundamental Theorems of Prevision
and Asset Pricing
Mark Schervish, Teddy Seidenfeld, Joseph B. Kadane
We explore the connections between the concepts of coherence, as defined
by deFinetti, and arbitrage in financial markets.
1. Introduction. Let be a set of states with a -field of subsets A. Let
X stand for a set of measurable real-valued functions defined on . Whether X
contains unbounded functions will be made clear in each context. The elements of
X will be called gambles, risky assets, or random variables. Functions of elements
of X will also be called by those same names.
DeFinetti took the concept of random variables as gambles very seriously, and
used the concept to motivate the familiar concepts of probability and expectation.
For each gamble X, he assumed that "You" would assign a value P(X), called the
prevision of X so that you would be willing to accept the gamble [X - P(X)]
as fair for all positive and negative values . The only constraint that deFinetti
envisioned for you and your previsions is that you insisted that there be no positive
amount that you had to lose for sure. For example, you would not be allowed to
call a gamble fair if its supremum were negative. On the other hand, the criterion
is weak enough to allow you call a gamble fair if its supremum is 0, even if all of