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

of possibility.

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

1 + k2 .

Finally, we give applications of the above to portfolio selection theory.