Ver traducción al castellano en formato pdf.
After finishing my Ph.D. in mathematics and spending some years as postdoctoral researcher (amongst others at the CRM in Barcelona), I started working in 2000 at the market risk management department of ING in Brussels. Lately a lot of mathematicians (and physicists) have been hired at ING, and this development can also be observed at other financial institutions. In what follows, I’ll try to point out what kind of mathematics is typically involved in this work, and why mathematics is ever more important. I do not claim to be complete; I’ll limit myself rather to comment on my own work as a quantitative risk manager.
Model validation
Most of you are certainly familiar with the stock market - indeed, the media has devoted quite some attention to its bad news recently! Suppose for a moment that you own a stock XYZ that currently is worth 100 EUR. Due to the recent turmoil on the financial markets, you’d like to protect yourself against a loss in this value. Suppose further that I’m willing to sell you the right (not the obligation) to sell me the stock XYZ in one year for 100 EUR. That is, in one year time from now, you can decide to give me the stock XYZ and I’ll pay you 100 EUR for it, whatever the price of XYZ at that moment is. Clearly if, at that time, it is worth more than 100 EUR, you will not exercise your right to sell it to me for 100 EUR. But, if in one year, XYZ is only worth 60 EUR, you’ll have won 40 EUR by exercising your right, and overall you have assured its current value of 100 EUR… What are you willing to pay for this right to sell me the stock (in financial terminology this is called a put option)? Before you start calculating, you have to know that this problem was solved in the 70’s…. it led to the famous Black-Scholes formula, which today is still one of the cornerstones of mathematical finance. Now let’s complicate things a little… What are you willing to pay to receive in one year time twenty times the highest difference in daily performance of your stock XYZ and the performance of the IBEX35 index? Maybe this payout seems a bit weird, but if you go to your local bank, and look at the investment products they are offering, I’m sure that you’ll see a lot of this stuff!
These financial products are called derivatives (since they are derived from the “basic” securities like stocks). Derivatives originated as hedging instruments: they served companies to cover financial risks they are exposed to in their daily business. But derivatives can also be used for speculation, since their leverage permits to realize bigger gains than possible by investing in the basic securities. To value derivatives one needs mathematical models. These are generally given by stochastic differential equations: stochastic calculus (martingales, Brownian motion, Ito’s lemma, Girsanov’s theorem etc.) is the main mathematical subject involved. Most of the time, there are no analytical solutions for these equations, and one has to use numerical methods (like Monte Carlo simulation, finite difference methods, etc.).
These models are normally developed by the “quants” in the trading room; our task, as quantitative risk manager, is to validate these models and compare them to benchmarks - without our approval the model cannot be used! There are two fundamental aspects in this validation work. On one hand, understanding the financial products is necessary to comprehend the risk factors they are exposed to. On the other hand, one has to understand the mathematical models: the strengths and weaknesses of a model need to be evaluated in judging the appropriateness to price some derivative. However, no model is perfect. Therefore we also have to keep an eye on the so-called model risk: what could go wrong when using a particular model?
Market Risk Management
Apart form the above mentioned model risk, other risks should be monitored too. Once a product is part of the trading portfolio, its financial risks should be monitored on a continuous basis. Due to changing market conditions, the price of the derivative can (and will) change too. Think of your put option we discussed above: if the price of the stock XYZ suddenly drops to 90 EUR, you expect the value of your option to change too, right? One of the central concepts in measuring this market risk is the so-called value-at-risk. This stands for the amount of money that, given a certain confidence level, one can maximally lose over some time horizon (say one day). The mathematics involved here does mainly come from statistics. Note that we do not try to predict the future, e.g. we do not try to tell what stock XYZ will do tomorrow. Based on historical observations, we rather try to give an estimate of the worst case that could happen. These risk figures are really important for a bank: limits for the trading room are based on them, and they serve to determine the regulatory capital needed (these are the reserves imposed by the regulators in order to guarantee the quality of a bank).
What will the future bring?
Derivative pricing has evolved quite a lot of time. Financial innovation (either for speculation or to hedge away new risks) has been a key driving factor: many of the derivatives that are traded today did indeed not even exist a few months ago! Likewise mathematical models have become more and more elaborate, trying to capture new risk factors and to give a financial institution an edge over their competitors.
Risk has changed over time too. Less and less financial crises are the direct result of economic or political events. The origins are rather the activity in financial markets itself: the structure of markets (e.g. globalization), new products (e.g. mortgage-backed securities, which are at the origin of the recent subprime crisis) and new participants (e.g. hedge funds), risk dispersion or concentration, model risk… Another tendency is to integrate various risks into one framework: market risk, credit risk (what if one of your counterparties defaults?), operational risk (including e.g. fraud) etc.
The recent crisis in the financial markets will not lead to less work in the quantitative world. On the contrary, the models used e.g. in the world of credit derivatives (the credit crisis and the huge losses that some investment banks had to admit over the past weeks are related to this) will need further refinement. Furthermore, risk management has become even more important!
The field of quantitative finance is a really fascinating world. Its constant evolution and increasing sophistication create a challenging environment. There are a lot of opportunities for people with a mathematical background here!
Some interesting links:
© 2006-2008 UALMAT. Universidad de Almería
