The relatively new discipline of "financial engineering" has developed into an exciting area of applied probability and stocahstic analysis. In this introductory talk I will discuss 4 problems from the world of finance and investment, and show how some suitable mathematics can provide insight into their solution. The problems, in increasing order of difficulty, are
1) If you have some extra disposable income, should you use it to pay down your mortgage?
2) How should you do asset allocation?
3) Given a bunch of random variables, representing the uncertain future returns of some assets (securities), what determines their current prices?
4) In particular, what if one of these assets depends on the price of another? These are called "derivatives", and include stock options as a special case.
The relevant mathematics will be a mix of calculus, linear algebra, and probability.