标记档案: 测不准原理

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. 毕竟, 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, Lee Smolin, described to us how the Earth could be considered a living organism. Using insightful arguments and precisely modulated articulation, Lee made a compelling case that the Earth, 事实上, satisfied all the conditions of being an organism. The point in Lee’s view was not so much whether or the Earth was literally alive, 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, it is not hard to imagine that we are indeed inflicting fetid atrocities to our host Earth. Can’t we think of our city sewers and the polluted air as the stinking, oozing ulcers on its body?

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 (该 “Wall Street got drunk”) literally? 但严重的是, Brownian motion has been a wildly successful model that we borrowed from physics. 同样, 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, 它是?

Chaos Everywhere

In physics, chaos is generally described as our inability to predict the outcome of experiments with arbitrarily close initial conditions. 例如, try balancing your pencil on its tip. 明确地, you won’t be able to, and the pencil will land on your desktop. 现在, 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 — 毕竟, 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. Take, 例如, the socio-economic phenomenon of globalization, which I can describe as follows, admittedly with an incredible amount of over-simplification. 很久以前, we used to barter agricultural and dairy products with our neighbours — 说, 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, 例如. 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 (rather, especially if) I could. 然而, 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. 例如, 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. Using this framework, I suspect one can predict the incidence and intensity of financial turmoils, though not their nature and causes. 然而, 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. 同样, 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. 换句话说, 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, 当然, is Einstein’s E = mc2.) 它说,一些看似简单的 — you can measure two complementary properties of a system only to a certain precision. 例如, 如果你试图找出其中一个电子是 (测量其位置, 就是说) 更多和更精确地, 它的速度逐渐变得更加不确定 (或, 动量测量变得不精确).

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 (哪, 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. 当然, if we can understand and model a product perfectly, we can price it right and expect to reap healthy rewards. 所以, 确定, modelling affects the risk-reward equation.

但, 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 (或缺乏) 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. (未来想起来, 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, 以及. 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. 更糟糕, 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. 毕竟, 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, 终极理论之梦, Nobel Prize winning physicist Steven Weinberg marvels at the uncanny ability of mathematics to anticipate physics needs. 同样, quants may marvel at the ability of physics to come up with phenomena and principles that can be directly applied to our field. 对我来说,, it looks like the repertoire of physics holds a few more gems that we can employ and exploit.

箱: 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, 我们要研究一下原理真正说. 这里有几个可能的解释:

  • 一个粒子的位置和动量本质上是相互关联的. 正如我们更精确地测量动量, 粒子种 “向外扩散,” 乔治·伽莫夫的性格, 先生. 汤普金斯, 把它. 换句话说, 这只是其中的一件事情; 世界的运作方式.
  • 当我们衡量的位置, 我们打​​扰势头. 我们的测量探头 “太胖,” 因为它是. 当我们提高位置精度 (由波长较短的光闪耀, 例如), 我们打​​扰的势头越来越多 (因为较短波长的光具有更高的能量/动量).
  • 与此密切相关的解释是认为测不准原理是一个感性的限制.
  • 我们也可以想到的不确定性原理的认知极限,如果我们考虑到未来的理论可能超越这些限制.

第一种观点是目前比较流行的,它与量子力学的所谓哥本哈根解释. 让我们忽略它,因为这是不讨论过于开放.

第二种解释通常被理解为一个实验困难. 但是,如果实验装置的概念被扩展到包括不可避免人类观察者, 我们在感性限制的第三种观点到货. 在此视图中, 它实际上是可能的 “派生” 测不准原理, based on how human perception works.

让我们假定,我们使用的是光的波长的光束 lambda 观察粒子. 在位置的精度,我们希望能做到的是顺序 lambda. 换句话说, Delta x approx lambda. 在量子力学, 在光束的每个光子的动量成反比的波长. 至少有一个光子被反射粒子,让我们可以看到它. 所以, 由经典守恒定律, the momentum of the particle has to change by at least this amount(approx constant/lambda) 从什么是测量前. 因此,, 通过感性论据, 我们得到类似海森堡测不准原理的东西

Delta x.Delta p approx constant

我们可以使这种说法更严格, 并获得恒定的值的估计. 显微镜的分辨率由经验式给出 0.61lambda/NA, 哪里 NA 是数值孔径, 其中有一个的最大值. 因此,, 最好的空间分辨率 0.61lambda. 在光束的每个光子具有动量 2pihbar/lambda, 这是在粒子动量的不确定性. 所以我们得到 Delta x.Delta p approx 4hbar, 震级大于量子力学极限大约订单.

通过更严格的统计参数, 相关的空间分辨率和预期的动量转移, 它可能可以通过这种推理得出的海森堡不确定原理.

如果我们考虑的哲学观点,即我们的现实是我们的知觉刺激认知模型 (这是有道理的我只能查看), 不确定性原理是一个认知的限制也是我第四次演绎持有一点水.

作者简介

作者是来自欧洲组织的科学家核子研究中心 (欧洲核子研究中心), 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. 表示此列中的观点只是他个人的看法, 它没有被影响了公司的业务或客户关系的考虑.

不确定性原理

不确定性原理是物理学中的第二件事情已经引起了公众的想象力. (第一个是 E=mc^2.) 它说,一些看似简单的 — 可以测量系统的两个互补的属性只能在一定的精度. 例如, 如果你试图找出其中一个电子是 (测量其位置, 就是说) 更多和更精确地, 它的速度逐渐变得更加不确定 (或, 动量测量变得不精确).

哪里这个原则来自? 之前,我们可以问这个问题, 我们要研究一下原理真正说. 这里有几个可能的解释:

  1. 一个粒子的位置和动量本质上是相互关联的. 正如我们更精确地测量动量, 粒子种 “向外扩散,” 乔治·伽莫夫的性格, 先生. 汤普金斯, 把它. 换句话说, 这只是其中的一件事情; 世界的运作方式.
  2. 当我们衡量的位置, 我们打​​扰势头. 我们的测量探头 “太胖,” 因为它是. 当我们提高位置精度 (由波长较短的光闪耀, 例如), 我们打​​扰的势头越来越多 (因为较短波长的光具有更高的能量/动量).
  3. 与此密切相关的解释是认为测不准原理是一个感性的限制.
  4. 我们也可以想到的不确定性原理的认知极限,如果我们考虑到未来的理论可能超越这些限制.

行, 最后两种解释都是我自己, 所以我们不会详细讨论这里.

第一种观点是目前比较流行的,它与量子力学的所谓哥本哈根解释. 这是一种像印度教的封闭声明 — “这样是绝对的性质,” 例如. 准确, 可能是. 但没有什么实际用途. 让我们忽略它,因为这是不讨论过于开放.

第二种解释通常被理解为一个实验困难. 但是,如果实验装置的概念被扩展到包括不可避免人类观察者, 我们在感性限制的第三种观点到货. 在此视图中, 它实际上是可能的 “派生” 测不准原理.

让我们假定,我们使用的是光的波长的光束 \lambda 观察粒子. 在位置的精度,我们希望能做到的是顺序 \lambda. 换句话说, \Delta x \approx \lambda. 在量子力学, 在光束的每个光子的动量成反比的波长. 至少有一个光子被反射粒子,让我们可以看到它. 所以, 由经典守恒定律, 粒子的动量具有至少改变由 \Delta p \approx 不变\lambda 从什么是测量前. 因此,, 通过感性论据, 我们得到类似海森堡测不准原理的东西 \Delta x \Delta p = 不变.

我们可以使这种说法更严格, 并获得恒定的值的估计. 显微镜的分辨率由经验式给出 0.61\lambda/NA, 哪里 NA 是数值孔径, 其中有一个的最大值. 因此,, 最好的空间分辨率 0.61\lambda. 在光束的每个光子具有动量 2\pi\hbar/\lambda, 这是在粒子动量的不确定性. 所以我们得到 \Delta x \Delta p = (0.61\lambda)(2\pi\hbar) \approx 4\hbar, 震级大于量子力学极限大约订单. 通过更严格的统计参数, 相关的空间分辨率和预期的动量转移, 它可能可以通过这种推理得出的海森堡不确定原理.

如果我们考虑的哲学观点,即我们的现实是我们的知觉刺激认知模型 (这是有道理的我只能查看), 不确定性原理是一个认知的限制也是我第四次演绎持有一点水.

参考

这篇文章的后半部分是从我的书的摘录, 虚幻宇宙.