There is a great variety of mathematics that has found its way into the world
of finance. Without a doubt a great deal of this work occurs at hedge funds
and investment companies who profit directly from the use of mathematics. One
area of particular interest is the mathematical study of transaction costs – those
costs associate with buying and selling in financial markets. The estimation of
such costs have interesting – and sometimes surprising – mathematical limits
which allow us to illustrate in a general sense the flavor of the mathematical
research that takes place within our group.
When a trader buys or sells in financial markets the trade affects the very
market in which the transaction takes place. Mathematical methods for depend
on our ability to bound what is – and especially what is not – within the realm
In this talk we give an introduction to the mathematics and economics of
transaction costs then show how Cantelli’s lemma may be used to make some
surprising statements about the limits of our ability to estimate transaction
costs. We also give a simple derivation of Cantelli’s lemma, which states that
for a random variable X with mean μ and standard deviation we have
P(X μ − k)
1 + k2 .
Finally, we give applications of the above to portfolio selection theory.