What Is Financial Mathematics?

TIM JOHNSON

If I tell someone I am a financial mathematician, they often think I am an accountant with pretensions. Since accountants do not like using negative numbers, one of the older mathematical technologies, I find this irritating.

I was drawn into financial maths not because I was interested in finance, but because I was interested in making good decisions in the face of uncertainty. Mathematicians have been interested in the topic of decision-making since Girolamo Cardano explored the ethics of gambling in his Liber de Ludo Aleae of 1564, which contains the first discussion of the idea of mathematical probability. Cardano famously commented that knowing that the chance of a fair dice coming up with a six is one in six is of no use to the gambler since probability does not predict the future. But it is of interest if you are trying to establish whether a gamble is fair or not; it helps in making good decisions.

With the exception of Pascal’s wager (essentially, you’ve got nothing to lose by betting that God exists), the early development of probability, from Cardano, through Galileo and Fermat and Pascal up to Daniel Bernoulli in the 1720s, was driven by considering gambling problems. These ideas about probability were collected by Jacob Bernoulli (Daniel’s uncle), in his work Ars Conjectandi. Jacob introduced the law of large numbers, proving that if you repeat the same experiment (say rolling a dice) a large number of times, then the observed mean (the average of the scores you have rolled) will converge to the expected mean. (For a fair dice each of the six scores is equally likely, so the expected mean is (1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5.)

Building on Jacob Bernoulli’s work, probability theory, in conjunction with statistics, became an essential tool of the scientist and was developed by the likes of Laplace in the eighteenth century and Fisher, Neyman and Pearson in the twentieth. For the first third of the twentieth century, probability was associated with inferring results, such as the life expectancy of a person, from observed data. But as an inductive science (i.e. the results were inspired by experimental observations, rather than the deductive nature of mathematics that builds on axioms), probability was not fully integrated into maths until 1933 when Andrey Kolmogorov identified probability with measure theory, defining probability to be any measure on a collection of events—not necessarily based on the frequency of events.

This idea is counter-intuitive if you have been taught to calculate probabilities by counting events, but can be explained with a simple example. If I want to measure the value of a painting, I can do this by measuring the area that the painting occupies, base it on the price an auctioneer gives it or base it on my own subjective assessment. For Kolmogorov, these are all acceptable measures which could be transformed into probability measures. The measure you choose to help you make decisions will depend on the problem you are addressing, if you want to work out how to cover a wall with pictures, the area measure would be best, if you are speculating, the auctioneer’s would be better.

Kolmogorov formulated the axioms of probability that we now take for granted. Firstly, that the probability of an event happening is a non-negative real number (P(E) > 0). Secondly, that you know all the possible outcomes, and the probability of one of these outcomes occurring is 1 (e.g. for a six-sided dice, the probability of rolling a 1, 2, 3, 4, 5, or 6 is P(1,2,3,4,5,6) = 1). And finally, that you can sum the probability of mutually exclusive events (e.g. the probability of rolling an even number is P(2,4,6) P(2) = P(4) + P(6) = 1/2).

Why is the measure theoretic approach so important in finance? Financial mathematicians investigate markets on the basis of a simple premise; when you price an asset it should be impossible to make money without the risk of losing money, and by symmetry, it should be impossible to lose money without the chance of making money. If you stop and think about this premise you should quickly realise it has little to do with the practicalities of business, where the objective is to make money without the risk of losing it, which is called an arbitrage, and financial institutions invest millions in technology that helps them identify arbitrage opportunities.

Based on the idea that the price of an asset should be such as to prevent arbitrages, financial mathematicians realised that an asset’s price can be represented as an expectation under a special probability measure, a risk-neutral measure, which bears no direct relation to the ‘natural’ probability of the asset price rising or falling based on past observations. The basic arguments are pretty straightforward and can be simply explained using that result from algebra that tells us if we have two equations we can find two unknowns.

However, as with much of probability, what seems simple can be very subtle. A no-arbitrage price is not simply an expectation using a special probability; it is only a price if it is ‘risk neutral’ and will not result in the possibility of making or losing money. You have to undertake an investment strategy, known as hedging, that removes these possibilities. In the real world, which involves awkward things like taxes and transaction costs, it is impossible to find a unique risk-neutral measure that will ensure all these risks can be hedged away. One of the key objectives of financial maths is to understand how to construct the best investment strategies that minimises risks in the real world.

Financial mathematics is interesting because it synthesizes a highly technical and abstract branch of maths, measure theoretic probability, with practical applications that affect people’s everyday lives. Financial mathematics is exciting because, by employing advanced mathematics, we are developing the theoretical foundations of finance and economics. To appreciate the impact of this work, we need to realise that much of modern financial theory, including Nobel prize winning work, is based on assumptions that are imposed, not because they reflect observed phenomena but because they enable mathematical tractability. Just as physics has motivated new maths, financial mathematicians are now developing new maths to model observed economic, rather than physical, phenomena.

Financial innovation currently has a poor reputation and some might feel that mathematicians should think twice before becoming involved with ‘filthy lucre’. However, Aristotle tells us the Thales, the father of western science, became rich by applying his scientific knowledge to speculation, Galileo left the University of Padua to work for Cosimo II de Medici, and wrote On the discoveries of dice, becoming the first “quant”. Around a hundred years after Galileo left Padua, Sir Isaac Newton left Cambridge to become warden of the Royal Mint, and lost the modern equivalent of £2,000,000 in the South Sea Bubble. Personally, what was good enough for Newton is good enough for me.

Moreover, interesting things happen when maths meets finance. In 1202 Fibonacci wrote a book, the Liber abaci, in which he introduced his series and the Hindu-Arabic numbers we use today, in order to help merchants conduct business. At the time merchants had to deal with dozens of different currencies and conduct risky trading expeditions that might last years, to keep on top of this they needed mathematics. There is an argument that the reason western science adopted maths, uniquely, around 1500–1700 was because people realised the importance of maths in commerce. The concept of conditional probability, behind Bayes’s Theorem and the solution to the Monty Hall Problem, emerged when Christian Huygens considered a problem in life-insurance, while the number e was identified when Jacob Bernoulli thought about charging interest on a bank loan. Today, looking at the 23 DARPA Challenges for Mathematics, the first three, the mathematics of the brain, the dynamics of networks and capturing and harnessing stochasticity in nature, and the eighth, beyond convex optimization, are all highly relevant to finance.

The Credit Crisis did not affect all banks in the same way, some banks engaged with mathematics and made good decisions, like J.P. Morgan and described in Gillian Tett’s book Fools’ Gold, while others did not, and caused mayhem. Since Cardano, financial maths has been about understanding how humans make decisions in the face of uncertainty and then establishing how to make good decisions. Making, or at least not losing, money is simply a by-product of this knowledge. As Xunyu Zhou, who is developing the rigorous mathematical basis for behavioural economics at Oxford, recently commented:

financial mathematics needs to tell not only what people ought to do, but also what people actually do. This gives rise to a whole new horizon for mathematical finance research: can we model and analyse (what we are most capable of ) the consistency and predictability in human flaws so that such flaws can be explained, avoided or even exploited for profit?

This is the theory. In practice, in the words of one investment banker, banks need high level maths skills because that is how the bank makes money.