*I was emailed by a reader recently asking about mathematical finance PhD programs and the benefits of such a course. If you are considering gaining a PhD in mathematical finance, this article will be of interest to you.*

If you are currently near the end of your undergraduate studies or are returning to study after some time in industry, you might consider starting a PhD in mathematical finance. This is an alternative to undertaking a Masters in Financial Engineering (MFE), which is another route into a quantitative role. This article will discuss exactly what you will be studying and what you are likely to get out of a PhD program. Clearly there will be differences between studying in the US, UK or elsewhere. I personally went to grad school in the UK, but I will discuss both UK and US programs.

Mathematical finance PhD programs exist because the techniques within the derivatives pricing industry are becoming more mathematical and rigourous with each passing year. In order to develop new exotic derivatives instruments, as well as price and hedge them, the financial industry has turned to academia. This has lead to the formation of mathematical finance research groups - academics who specialise in derivatives pricing models, risk analysis and quantitative trading.

Graduate school, for those unfamiliar with it, is a very different experience to undergraduate. The idea of grad school is to teach you how to effectively research a concept without any guidance and use that research as a basis for developing your own models. Grad school really consists of a transition from the "spoon fed" undergraduate lecture system to independent study and presentation of material. The taught component of grad school is smaller and the thesis component is far larger. In the US, it is not uncommon to have two years of taught courses before embarking on a thesis (and thus finding a supervisor). In the UK, a PhD program is generally 3-4 years long with either a year of taught courses, or none, and then 3 years of research.

A good mathematical finance PhD program will make extensive use of your undergraduate knowledge and put you through graduate level courses on stochastic analysis, statistical theory and financial engineering. It will also allow you to take courses on general finance, particularly on corporate finance and derivative securities. When you finish the program you will have gained a broad knowledge in most areas of mathematical finance, while specialising in one particular area for your thesis. This "broad and deep" level of knowledge is the hallmark of a good PhD program.

Mathematical Finance research groups study a wide variety of topics. Some of the more common areas include:

**Derivative Securities Pricing/Hedging:**The technical term for this is "financial engineering", as "quantitative analysis" now encompasses a wide variety of financial areas. Some of the latest research topics include sophisticated models of options including stochastic volatility models, jump-diffusion models, asymptotic methods as well as investment strategies.**Stochastic Calculus/Analysis:**This is more of a theoretical area, where the basic motivation stems from the need to solve stochastic differential equations. Research groups may look at path-dependent PDEs, functional Ito calculus, measure theory and probability theory.**Fixed Income Modeling:**Research in this area centres on effectively modelling interest rates - such as multi-factor models, multi-curve term structure models as well as interest rate derivatives such as swaptions.**Numerical Methods:**Although not always strictly related to mathematical finance, there is a vast amount of university research carried out to try and develop more effective means of solving equations numerically (i.e. on the computer!). Recent developments include GPU-based Monte Carlo solvers, more efficient matrix solvers as well as Finite Differences on GPUs. These groups will almost certainly possess substantial programming expertise.**Market Microstructure/High-Frequency Modeling:**This type of research is extremely applied and highly valued by funds engaged in this activity. You will find many academics consulting, if not contracting, for specialised hedge funds. Research areas include creating limit order market models, high frequency data statistical modelling, market stability analysis and volatility analysis.**Credit Risk:**Credit risk was a huge concern in the 2007-2008 financial crisis and many research groups are engaged in determining such "counterparty risks". Credit derivatives are still a huge business and so a lot of research goes into collateralisation of securities as well as pricing of exotic credit derivatives.

These are only a fraction of the total areas that are studied within mathematical finance. The best place to find out more about research topics is to visit the websites of all the universities which have a mathematical finance research group, which is typically found within the mathematics, statistics or economics faculty.

The benefits of undertaking a PhD program are numerous:

**Employment Prospects:**A PhD program sets you apart from candidates who only possess an undergraduate or Masters level ability. By successfully defending a thesis, you have shown independence in your research ability, a skill highly valued by numerate employers. Many funds (and to a lesser extent, banks) will only hire PhD level candidates for their mathematical finance positions, so in a pragmatic sense it is often a necessary "rubber stamp". In investment banks, this is not the case so much anymore, as programming ability is generally prized more. However, in funds, it is still often a requirement. Upon being hired you will likely be at "associate" level rather than "analyst" level, which is common of undergraduates. Your starting salary will reflect this too.**Knowledge:**You will spend a large amount of time becoming familiar with many aspects of mathematical finance and derivatives theory. This will give you a holistic view into the industry and a more transferable skill set than an undergraduate degree as you progress up the career ladder. In addition, you will have a great deal of time to learn how to program models effectively (without the day-to-day pressure to get something implemented any way possible!), so by the time you're employed, you will be "ahead of the game" and will know best practices. This aspect is down to you, however!**Intellectual Prospects:**You are far more likely to gain a position at a fund after completing a PhD than without one. Funds are often better environments to work in. There is usually less stress and a more relaxed "collegiate" environment. Compare this to working on a noisy trading floor, where research might be harder to carry out and be perceived as less important.

I would highly recommend a mathematical finance PhD, so long as you are extremely sure that a career in quantitative finance is for you. If you are still unsure of your potential career options, then a more general mathematics, physics or engineering PhD might be a better choice.