# Financial Mathematics

Activity Group leader: Alexander Melnikov, Alberta, melnikov@ualberta.ca

What is Financial Mathematics?

Over the past 20 years a huge number of mathematical scientists, trained at all levels from B.Sc. to Ph.D., have been employed by the world's financial institutions. There they do many things, but chiefly they measure and manage the financial risks of these institutions. The key idea that is used is that the risk of a collection of financial assets is different, and often much smaller , than the sum of the risks of the constituent parts. This central concept is used by pension funds which allocate their plan members retirement savings so as not to put all eggs in a single basket. It is also used by financial institutions with large loan portfolios to manage the many complexities and risks inherent in these instruments. A lender faces many risks. Most obviously, the borrower may not repay the money. But the borrower may also repay too soon, so that a loan, taken out when rates were high is repaid when rates are low! A large body of financial products known as swaps and options have been developed to manage these risks, and other such risks which arise in stock markets, in commodity markets, and in foreign exchange markets. The price, the risk, and any hedging activity required to manage that risk for all these products are quantified by complicated mathematical models. Creating and solving these models requires expertise in probability theory, partial differential equations, and numerical methods, all traditional applied mathematics disciplines. More recently disciplines of machine learning and game theory have been used in areas ranging from the automatic approval of "small" loans like a credit card or a residential mortgage to the development of algorithms for high frequency trading.

Money makes the world go around, and nowadays, mathematicians are the ones making sure that the money goes around. CAIMS/SCMAI is hope to a large and vibrant group of financial mathematicians, the work of which is showcased at our annual conferences.