# Mathematics in the Financial Markets

In the financial markets the big financial market players are in the business of of trading risk through common ‘asset classes’ traded by employees. Mathematics helps deal with the challenges that arise in these trades.

### Asset Classes

A bank has a risk if it can lose or make money on a ‘thing’ in ‘asset classes’ that falls or rises in value.

Asset classes include:

1. Interest rates
2. Foreign exchange
3. Credit
4. Equities and
5. Commodities

Banks trade for one of three reasons:

1. New business. The bank is helping a client reduce its risk.
2. Hedging. When there’s increased exposure to risk with market moves in interest rates, foreign exchange etc., a trade is made to reduce the risk to an acceptable level.
3. Proprietory. The bank gambles on what is going to happen to interest rates etc.

There are also ‘options’ which are the right, but not the obligation, to enter into some sort of transaction.

### Roles within Financial Markets

Depending on which the asset classes people work in, the four main roles are: Sales, Structuring, Trading and Derivatives Research also known as “quants”. Structurers support sales staff with solutions to client problems and also help in solving regulatory/accounting issues. They price transactions and understand how the trader will manage trade risks.

When dealing, the trader will commit the bank to taking on risk. The trader is responsible for ensuring that the deal is managed throughout its life. He will ensure no financial loss by “hedging”, that is, performing additional deals to minimise the net risks that he is exposed to. The maths skills of the trader depend on what type of risk they are tasked with managing.
The financial community develops complex pricing models to describe how the various assets move and interrelate. This is the job for a quant who will advise on which models are suitable and if there are none, either recommend not getting involved or produce a new model.

### Maths Problems in Finance

Many finance models assume that asset prices go on ‘random walks’. The simplest leads to a normal distribution, the most common leads to a lognormal distribution. The theory associated with these processes is “stochastic calculus” and has some unexpected results. Ito’s lemma is used to correct the results. To price an option on an asset which is evolving lognormally leads you to the “contingent claims equation”. If you want to show that the contingent claims equation leads to the same result as an alternative approach which is intuitive, use Green’s functions. For a problem involving both short term and long term interest rates you need to understand correlations, eigenvectors and eigenvalues. Measure theory, sobol sequences for random numbers, poisson processes, nonlinear PDEs etc. are all used to solve maths problems in finance.

To be a quant, you are most likely to need a PhD in something very mathematical. If degree level maths is as far as you go, then don’t discount the other roles.