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Chaos and Uncertainty

The last couple of months in finance industry can be summarized in two words — chaos and uncertainty. The aptness of this laconic description is all too evident. The sub-prime crisis where everybody lost, the dizzying commodity price movements, the pink slip syndrome, the spectacular bank busts and the gargantuan bail-outs all vouch for it.

The financial meltdown is such a rich topic with reasons and ramifications so overarching that all self-respecting columnists will be remiss to let it slide. Ведь, a columnist who keeps his opinions to himself is a columnist only in his imagination. I too will share my views on causes and effects of this turmoil that is sure to affect our lives more directly than anybody else’s, but perhaps in a future column.

The chaos and uncertainty I want to talk about are of different kind — the physics kind. The terms chaos and uncertainty have a different and specific meanings in physics. How those meanings apply to the world of finance is what this column is about.

Symmetries and Patterns

Physicists are a strange bunch. They seek and find symmetries and patterns where none exists. I remember once when our brilliant professor, Ли Смолин, described to us how the Earth could be considered a living organism. Using insightful arguments and precisely modulated articulation, Ли сделал убедительные аргументы, что Земля, на самом деле, довольные все состояния бытия организм. Дело в силу Ли не столько в том или Земля была буквально живы, but that thinking of it as an organism was a viable intellectual pattern. Once we represent the Earth in that model, we can use the patterns pertaining to organism to draw further predictions or conclusions.

Expanding on this pattern, I recently published a column presenting the global warming as a bout of fever caused by a virus (us humans) on this host organism. Don’t we plunder the raw material of our planet with the same abandon with which a virus usurps the genetic material of its host? In addition to fever, typical viral symptoms include sores and blisters as well. Looking at the cities and other eye sores that have replaced pristine forests and other natural landscapes, это не трудно представить, что мы действительно нанося зловонные зверств нашего хозяина Земли. Can’t we think of our city sewers and the polluted air as the stinking, сочилась язвы на его теле?

While these analogies may sound farfetched, we have imported equally distant ideas from physics to mathematical finance. Why would stock prices behave anything like a random walk, unless we want to take Bush’s words (что “Wall Street got drunk”) literally? А если серьезно, Brownian motion has been a wildly successful model that we borrowed from physics. Снова, once we accept that the pattern is similar between molecules getting bumped around and the equity price movements, the formidable mathematical machinery and physical intuitions available in one phenomenon can be brought to bear on the other.

Looking at the chaotic financial landscape now, I wonder if physics has other insights to offer so that we can duck and dodge as needed in the future. Of the many principles from physics, chaos seems such a natural concept to apply to the current situation. Are there lessons to be learned from chaos and nonlinear dynamics that we can make use of? May be it is Heisenberg’s uncertainty principle that holds new insights.

Perhaps I chose these concepts as a linguistic or emotional response to the baffling problems confronting us now, but let’s look at them any way. It is not like the powers that be have anything better to offer, это?

Chaos Everywhere

В физике, chaos is generally described as our inability to predict the outcome of experiments with arbitrarily close initial conditions. Например, try balancing your pencil on its tip. Очевидно, you won’t be able to, and the pencil will land on your desktop. Сейчас, note this line along which it falls, and repeat the experiment. Regardless of how closely you match the initial conditions (of how you hold and balance the pencil), the outcome (the line along which it falls) is pretty much random. Although this randomness may look natural to us — ведь, we have been trying to balance pencils on their tips ever since we were four, if my son’s endeavours are anything to go by — it is indeed strange that we cannot bring the initial conditions close enough to be confident of the outcome.

Even stranger is the fact that similar randomness shows up in systems that are not quite as physical as pencils or experiments. Возьмите, например, the socio-economic phenomenon of globalization, which I can describe as follows, admittedly with an incredible amount of over-simplification. Давным-давно, we used to barter agricultural and dairy products with our neighbours — сказать, a few eggs for a litre (or was it pint?) of milk. Our self-interest ensured a certain level of honesty. We didn’t want to get beaten up for adding white paint to milk, например. These days, thanks to globalization, people don’t see their customers. A company buys milk from a farmer, adds god knows what, makes powder and other assorted chemicals in automated factories and ships them to New Zealand and Peru. The absence of a human face in the supply chain and in the flow of money results in increasingly unscrupulous behaviour.

Increasing chaos can be seen in the form of violently fluctuating concentrations of wealth and fortunes, increasing amplitudes and frequency of boom and bust cycles, exponential explosion in technological innovation and adaptation cycles, and the accelerated pace of paradigm shifts across all aspects of our lives.

It is one thing to say that things are getting chaotic, quite another matter to exploit that insight and do anything useful with it. I won’t pretend that I can predict the future even if (довольно, especially if) I could. Однако, let me show you a possible approach using chaos.

One of the classic examples of chaos is the transition from a regular, laminar flow of a fluid to a chaotic, turbulent flow. Например, when you open a faucet slowly, if you do it carefully, you can have a pretty nice continuous column of water, thicker near the top and stretched thinner near the bottom. The stretching force is gravity, and the cohesive forces are surface tension and inter-molecular forces. As you open the faucet still further, ripples begin to appear on the surface of the column which, at higher rates of flow, rip apart the column into complete chaos.

In a laminar flow, macroscopic forces tend to smooth out microscopic irregularities. Like gravity and surface tension in our faucet example, we have analogues of macroscopic forces in finance. The stretching force is probably greed, and the cohesive ones are efficient markets.

There is a rich mathematical framework available to describe chaos. Используя эту структуру, I suspect one can predict the incidence and intensity of financial turmoils, though not their nature and causes. Однако, I am not sure such a prediction is useful. Imagine if I wrote two years ago that in 2008, there would be a financial crisis resulting in about one trillion dollar of losses. Even if people believed me, would it have helped?

Usefulness is one thing, but physicists and mathematicians derive pleasure also from useless titbits of knowledge. What is interesting about the faucet-flow example is this: if you follow the progress two water molecules starting off their careers pretty close to each other, in the laminar case, you will find that they end up pretty much next to each other. But once the flow turns turbulent, there is not telling where the molecules will end up. Аналогично, in finance, suppose two banks start off roughly from the same position — say Bear Stearns and Lehman. Under normal, laminar conditions, their stock prices would track similar patterns. But during a financial turbulence, they end up in totally different recycle bins of history, as we have seen.

If whole financial institutions are tossed around into uncertain paths during chaotic times, imagine where two roughly similar employees might end up. Другими словами, don’t feel bad if you get a pink slip. There are forces well beyond your control at play here.

Uncertainty Principle in Quantitative Finance

The Heisenberg uncertainty principle is perhaps the second most popular theme from physics that has captured the public imagination. (The first one, конечно, is Einstein’s E = mc2.) Это говорит что-то, казалось бы, проста — you can measure two complementary properties of a system only to a certain precision. Например, если вы пытаетесь выяснить, где электрон (измерить свою позицию, то есть) более и более точно, его скорость становится все более неопределенной (или, измерение импульса становится неточным).

Quantitative finance has a natural counterpart to the uncertainty principle — risks and rewards. When you try to minimize the risks, the rewards themselves go down. If you hedge out all risks, you get only risk-free returns. Since risk is the same as the uncertainty in rewards, the risk-reward relation is not quite the same as the uncertainty principle (которые, as described in the box, deals with complementary variables), but it is close enough to draw some parallels.

To link the quantum uncertainty principle to quantitative finance, let’s look at its interpretation as observation altering results. Does modelling affect how much money we can make out of a product? This is a trick question. The answer might look obvious at first glance. Конечно, if we can understand and model a product perfectly, we can price it right and expect to reap healthy rewards. Так, уверен, modelling affects the risk-reward equation.

Но, a model is only as good as its assumptions. And the most basic assumption in any model is that the market is efficient and liquid. The validity of this assumption (или ее отсутствие) is precisely what precipitated the current financial crisis. If our modelling effort actually changes the underlying assumptions (usually in terms of liquidity or market efficiency), we have to pay close attention to the quant equivalent of the uncertainty principle.

Look at it this way — a pyramid scheme is a perfectly valid money making model, but based on one unfortunate assumption on the infinite number of idiots at the bottom of the pyramid. (Coming думать об этом, the underlying assumption in the sub-prime crisis, though more sophisticated, may not have been that different.) Similar pyramid assumptions can be seen in social security schemes, а также. We know that pyramid assumptions are incorrect. But at what point do they become incorrect enough for us to change the model?

There is an even more insidious assumption in using models — that we are the only ones who use them. In order to make a killing in a market, we always have to know a bit more than the rest of them. Once everybody starts using the same model, I think the returns will plummet to risk-free levels. Why else do you think we keep inventing more and more complex exotics?

Summing up…

The current financial crisis has been blamed on many things. One favourite theory has been that it was brought about by the greed in Wall Street — the so-called privatization of profits and socialization of losses. Incentive schemes skewed in such a way as to encourage risk taking and limit risk management must take at least part of the blame. A more tempered view regards the turmoil as a result of a risk management failure or a regulatory failure.

This column presents my personal view that the turmoil is the inevitable consequence of the interplay between opposing forces in financial markets — risk and rewards, speculation and regulation, risk taking and risk management and so on. To the extent that the risk appetite of a financial institute is implemented through a conflict between such opposing forces, these crises cannot be avoided. Хуже, the intensity and frequency of similar meltdowns are going to increase as the volume of transactions increases. This is the inescapable conclusion from non-linear dynamics. Ведь, such turbulence has always existed in the real economy in the form cyclical booms and busts. In free market economies, selfishness and the inherent conflicts between selfish interests provide the stretching and cohesive forces, setting the stage for chaotic turbulence.

Physics has always been a source of talent and ideas for quantitative finance, much like mathematics provides a rich toolkit to physics. In his book, Сны окончательной теории, Nobel Prize winning physicist Steven Weinberg marvels at the uncanny ability of mathematics to anticipate physics needs. Аналогично, quants may marvel at the ability of physics to come up with phenomena and principles that can be directly applied to our field. Мне, it looks like the repertoire of physics holds a few more gems that we can employ and exploit.

Box: Heisenberg’s Uncertainty Principle

Where does this famous principle come from? It is considered a question beyond the realms of physics. Before we can ask the question, мы должны изучить, что принцип действительно говорит. Вот несколько возможных интерпретаций:

  • Положение и импульс частицы неразрывно взаимосвязаны. Как мы измерить импульс более точно, вид частиц “распространяется,” как характер Джорджа Гамова, Г-н. Томпкинс, кладет это. Другими словами, это просто одна из тех вещей; то, как устроен мир.
  • Когда мы измеряем положение, мы нарушить импульс. Наши измерительные зонды являются “слишком жирный,” так сказать. Как мы увеличиваем точность позиционирования (по сияющий свет более коротких длин волн, например), мы нарушить импульс более (так как сокращение длины волны света имеет более высокую энергию / импульс).
  • Тесно связана с этой интерпретацией является мнение, что принцип неопределенности является перцептивно предел.
  • Мы также можем думать о принцип неопределенности как когнитивный предела, если учесть, что будущая теория может превзойти таких ограничений.

Первая точка зрения в настоящее время популярны и связана с так называемой копенгагенской интерпретации квантовой механики. Давайте игнорировать его для это не слишком открыты для дискуссий.

Вторая интерпретация обычно понимается в качестве экспериментальной сложности. Но если понятие экспериментальной установки расширена за счет включения неизбежный человека-наблюдателя, мы приходим к третьему зрения восприятия ограничения. С этой точки зрения, это на самом деле возможно “получать” Принцип неопределенности, based on how human perception works.

Давайте предположим, что мы используем луч света с длиной волны lambda наблюдать частицу. Точность в положении мы можем надеяться достичь, это порядка lambda. Другими словами, Delta x approx lambda. В квантовой механике, импульс каждого фотона в световом пучке обратно пропорционально длине волны. По крайней мере один фотон отражается частицы, так что мы можем видеть его. Так, классическим законом сохранения, the momentum of the particle has to change by at least this amount(approx constant/lambda) от того, что было перед измерением. Таким образом, через восприятия аргументов, мы получаем нечто похожее на принципе неопределенности Гейзенберга

Delta x.Delta p approx constant

Мы можем сделать этот аргумент более строгий, и получить оценку стоимости постоянная. Разрешение микроскопа определяется по эмпирической формуле 0.61lambda/NA, где NA это числовая апертура, который имеет максимальную величину одного. Таким образом, Наилучшее пространственное разрешение составляет 0.61lambda. Каждый фотон в световом пучке имеет импульс 2pihbar/lambda, который является неопределенность импульса частиц. Итак, мы получаем Delta x.Delta p approx 4hbar, примерно на порядок больше, чем квантово-механического предела.

Благодаря более строгих статистических аргументов, связаны с пространственным разрешением и ожидаемый импульс, передаваемый, это может можно вывести принцип неопределенности Гейзенберга через эту линию рассуждений.

Если мы рассмотрим философскую точку зрения, что наша реальность является когнитивная модель нашего чувственного раздражителей (который является единственным мнение, что имеет смысл для меня), мой четвертый толкование принципа неопределенности составляет когнитивный ограничение также имеет немного воды.

Об авторе

The author is a scientist from the European Organization for Nuclear Research (CERN), who currently works as a senior quantitative professional at Standard Chartered in Singapore. More information about the author can be found at his blog: http//www.Thulasidas.com. The views expressed in this column are only his personal views, which have not been influenced by considerations of the firm’s business or client relationships.

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(Эта шутка была найдена на E-mail Flotsam страница в мире Майка)