Those contemplating doing research in mathematical finance might like to consider the following.
Nobel Prize-winning `Modern Portfolio Theory' is now 40 years old and suffers from an inappropriate measure of risk. A paper[ 6] which was critical of MPT's use of `variance of return' as the measure of risk, received the ``Outstanding Paper of the Year'' award. It led to my delivering the opening address at the Centre for Investment Research's ``Is Variance Dead?'' conference, held in the USA last year. Even Harry Markowitz was recommending that people attend. Reworking MPT, using a more realistic risk measure and including re-balancing costs, would be a worthy challenge for any aspiring financial mathematician.
Most standard statistical tests prove to be inconclusive, or insufficiently discriminating, when applied in finance. While standard statistical tests usually fail to contradict the random-walk hypothesis for prices--which most practitioners simply do not believe, anyway--a novel `term structure of volatility' model[ 3] shows that Australian 10-year-bond yields clearly do not follow a random walk. More discerning tests and more robust statistical algorithms are urgently required, to handle the very high signal-to-noise ratios that occur in finance.
Whilst financial theory assumes that participants in markets act rationally and aim to maximise returns, the psychological literature[ 12] shows quite clearly that this is not the case. Behavioural Finance is likely to produce very important advances for financial markets. Quite sophisticated mathematics will be required to model groups of real, as distinct from ideal, participants. Thermodynamic Statistical Mechanics might even prove enlightening.
Over recent years, the pricing of options and other contingent claims has been the greatest area of activity for mathematicians in finance. If you like running with 40,000 other people in the City to Surf, you might enjoy it.
In short: ``There is money to be made in finance''[ 9], even by mathematicians. ``...These observations should not obscure the profound scientific challenges posed by the area of finance'', nor their value to the wider community.