21.1 What has the vet done?

I joined the financial industry in March 1987 with a quant background in engineering and scientific research experience in fluid mechanics. Six months later, the S&P 500 plunged 18% in what has come to be known as Black Monday and have seen long-term interest rate swings of more than 3% in a few days. The following year, I presented an empirical study on the French stock market daily movements – which crashed by an equivalent albeit lesser magnitude (15%) – at the French Finance Association (AFFI) conference, at a time when the stock market had more than rebounded. The French market index crashed by an equivalent albeit lesser magnitude (15%) on the same Black Monday. Sticking to the Gaussian distribution for the stock index variation leads to a roughly 15 standard deviation event, thus a highly unlikely event with an average waiting time period (1/probability) far exceeding the age of the universe … Our universe! This forced me into looking at other more realistic probability distributions.

We all experienced in the following decade a gold rush in quantitative finance stimulated by liberalization of markets, by access to technology and data, as well as by strong interactions between academics and practitioners. As an active member of the European Institute of Quantitative Investment Research, I had access to a large number of empirical studies on all financial markets and also presented my own. Heading a quant team at an innovative French Bank, Crédit Commercial de France, then acquired by HSBC, I looked at the possible use of a variety of these models to create new product and services for the clients of the bank. Our research was partly shared with academics at conferences and also presented in a quarterly review, then called “Quants.” The main applications at that time were pricing, investments, and risk management.

The main competitor to the Gaussian approach to stock price variation in the 1990s was the ARCH (autoregressive conditional heteroscedasticity) model and all its variations. But stable distributions, in particular the Levy stable model, were appealing. For all sorts of reasons, the Gaussian approach was still being used by practitioners in spite of the limitations of the model. As no obvious candidate prevailed at that time (does one today?), the pragmatic approach was to use different models under different circumstances. As the regulation forced the Bank into more accurate risk management and capital allocation, CCF put in place an internal model and had to select stress tests. That is where extreme value theory (EVT) came into play. In the following part, I will explain how and what kind of lessons I learned from that use. I then would like to highlight what I feel are the needs for future research and progress.

Later, during the new century, quant models came to be mainstream, and many people forgot about the inherent limitations of any kind of modeling. The overhang of sophisticated models created a too fuzzy and sometimes unchecked environment. In parallel, as the economy grew and debt developed to unprecedented levels, financial crisis erupted and devastated markets and institutions. It forced everyone to think twice about how to handle risk and modeling, and should one add the risk of modeling. This would lead to a humble conclusion.

21.2 Why Use EVT?

The appetite for more formalized assessment of capital requests for financial institutions has grown dramatically in modern regulation. All started with the Basel 2 drafting, where the internal model concept appeared for the first time. Industry leaders, lured by the potential capital savings but also by the signaling effect attached to this innovative approach, made their best to build such models and to submit them to their regulators. The squadron by the French regulator was involved in the difficult task of checking and agreeing that the internal model was particularly shrewd and experienced. Needless to say, all applicants had their difficult moments. That regulation stipulated that, in addition to the normal model targeted at measuring the value at risk (VaR), there should be a comprehensive set of stress scenarios. VaR was meant to be on a typical horizon of 15 days and a probability of 99%, thus having an average waiting time period of approximately 4 years, an adverse but not a tough event. Stress scenarios, on the other hand, were meant at stressing – a very different game.

How to stress portfolios of mostly liquid assets has been the subject of debates. How to find the right level? Not too large a level that would have destroyed interest in holding any position and thus be a complete deterrent to engage in market activities; not too low because VaR was doing that job. The next question was to have a form of coherence between the shocks on equities, bonds, or currencies. Fairness is the key in people-intensive activities. Then come the difficulties related to the statistical estimation of these stress scenarios. Indeed, if one has the luxury of having data of more than a century, a question remains on the relevance of such old-time data to the present. But fortunately, or unfortunately, the lack of data is often what blocks efforts to estimate these rare quantiles with classic statistics.

The solution we came across at that time was to combine forward-looking stresses and a set of coherent stresses based on EVT. As an example of forward-looking stress, we ended naturally with the things we were afraid of during that period. I remember, for example, the exit from the nascent Euro, which was only used by institutions. Then the difficulty to have the Euro accepted by the general public could have triggered panic and therefore shocks, which we attempted to model with mostly guessed magnitudes. How to describe the perfect storms is indeed a never-ending exercise. Which can prove useful, not really to foster forecasting ability but more to enable a thorough preparation, in a disciplined manner? Let us now come to the historically based stresses.

Generally speaking, statistically the variation of a security price shows a distribution close to a normal one but not identical. The probability of little, almost nil, movements and the probability of sharp disruptions, especially on the downside, are much more important than in the Gaussian case. Instead, the “jaws” of the actual historical distribution are much less pronounced. Price hardly moves or moves hard! The relative difference for the tail of the distribution is the most acute because, as the Black Monday case showed, the probability of an event more than three standard deviations is negligible under the normal law, whereas history shows crises are quite frequent. As mentioned earlier, a stable law tends to offer a more realistic model for stock prices.

As the purpose was to design a coherent and flexible set of shocks, upwards and downwards, we really only cared about the modeling of the tails.

Modeling tails is not such a rare endeavor, and many practitioners use these models. This is particularly true in actuarial sciences, which are used in pricing insurance contracts of various sorts.

A number of empirical studies have been done on a large set of securities, stocks and bonds, indices, and currencies. The Fréchet distribution was the dominant outcome of these studies. Once the tail was modeled and the relevant parameters were estimated, we could come back to the leaders of the bank with a reliable and flexible representation of rare shocks. Depending on the management's appetite for risk, which varied over time and fluctuated with crisis or bank-specific concerns, we could adjust the magnitude of the stress scenarios. The language we used on these scenarios was related to the average waiting time. We talk about a 50-year shock, or a century shock, exactly like in hydrology for floods. All were on a 1-day period. But sometimes we considered longer periods to accommodate liquidity or execution time. The reference point, but not necessarily the one selected, was the probability assigned to the rating of the bank: historical corporate default distribution per rating (better known in the US than in other markets) gives an idea of the average waiting time attached to the rating. Part of the empirical findings were published and presented in many conferences at that time.

21.3 What EVT could additionally bring to the party?

The first major finding highlighted by EVT was that extreme situations demanded another model. It seems to be true in human-related (prices are) and purely natural (e.g., seism) phenomena. There is a need to understand why it is so. Will this research come with results, hopefully yes; but in any case digging into what makes “extreme” different seems useful. Herding, force behavior, and non-rational behavior have to be better forecasted and, in my view, controlled. Regulation should take these behaviors into account, and the side effects of existing regulation in turbulent times should be better mastered. Is it a shame to change the rules in the middle of a crisis? Or, should we not positively think of the necessary changes in regulation in such times? By the way, modern central banking carries out such an experiment with quantitative ease, another word for unorthodox, unconventional, and improbable behavior, just suited to an extreme situation where the ordinary, conventional financing channel does not work. If the normal law does not work in critical situations, why do we try to tweak thing to cope with it? This leaves the unanswered question of what extreme is and how we measure it. Another area where EVT may help, at least in the description of phenomena, is liquidity. A great deal of the huge swings we have experienced during the last crisis was indeed related to liquidity – all sorts of liquidity by the way. My guess is that the many relations of financial agents change radically during a crisis. The consequences in trading and forced sales, like the so-called pro-cyclical capital rules, have been mentioned many times. But we should go beyond and look at many other aspects, such as the intensity in reporting and the widening gap in trust and behaviors, clearly highlighting a special period, like the phase approach in fluid mechanics! But this was another story, for me at least.

21.4 A final thought

To a quant-minded professional, EVT is great but by no means a panacea. It has proven already very useful in modeling and managing extreme events. It has also the appeal to be different and exciting. In many topics, like option pricing or portfolio optimization, it becomes a challenge! And given the known benefits, it is worth trying to overcome the technical difficulties. Nevertheless, to damp the quant excitement, let us enjoy normal time and try not being obsessed more than needed by extreme and, thus, rare events. Fascination is no guide to a responsible realistic professional.

References

1. Boulier, J.-F., Dalaud, R., Longin, F. Application de la théorie des valeurs extrêmes aux marchés financiers. Paris Banque et Marchés 1998;32:5–14.
2. Boulier, J.-F., Gaussel, N., Legras, J. Crashes to crashes. Quants Review 1999;33, CCF.