Mga Archive ng Tag: kawalan ng katiyakan prinsipyo

Ganap na kaguluhan at Kawalang-kasiguraduhin

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. Pagkatapos ng lahat, 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 ginawa ng nakakahimok na kaso na ang Earth, sa katunayan, nasiyahan sa lahat ng mga kondisyon ng pagiging isang organismo. Ang punto sa view Lee ng ay hindi kaya magkano man o ang Earth ay literal na buhay, 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, ito ay hindi mahirap na isipin na sa katunayan kami ay inflicting mabaho mga kalupitan sa aming host ng Earth. Can’t we think of our city sewers and the polluted air as the stinking, oozing ulcers sa katawan nito?

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 (that “Wall Street got drunk”) literally? Ngunit sineseryoso, Brownian motion has been a wildly successful model that we borrowed from physics. Muli, 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, ay ito?

Chaos Everywhere

Sa pisika, chaos is generally described as our inability to predict the outcome of experiments with arbitrarily close initial conditions. Halimbawa, try balancing your pencil on its tip. Malinaw, you won’t be able to, and the pencil will land on your desktop. Ngayon, 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 — pagkatapos ng lahat, 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. Kumuha, halimbawa, the socio-economic phenomenon of globalization, which I can describe as follows, admittedly with an incredible amount of over-simplification. Mahabang oras ang nakalipas, we used to barter agricultural and dairy products with our neighbours — sabihin, 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, halimbawa. 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. Gayunpaman, 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. Halimbawa, 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. Gamit ang balangkas, I suspect one can predict the incidence and intensity of financial turmoils, though not their nature and causes. Gayunpaman, 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. Katulad nito, 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. Sa ibang salita, 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, oo naman, is Einstein’s E = mc2.) Sinasabi nito ang isang bagay na tila tapat — you can measure two complementary properties of a system only to a certain precision. Halimbawa, kung susubukan mo upang malaman kung saan ang isang elektron ay (sukatin ang posisyon nito, na) mas at mas tiyak, bilis nito ay nagiging patuloy na mas hindi sigurado (o, ang pagsukat ng momentum nagiging imprecise).

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 (kung saan, 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. Oo naman, if we can understand and model a product perfectly, we can price it right and expect to reap healthy rewards. Kaya, sigurado, modelling affects the risk-reward equation.

Pero, 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 (o kulang nito) 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. (Paparating na sa tingin ng mga ito, 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, pati na rin. 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. Mas masama, 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. Pagkatapos ng lahat, 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, Dreams ng isang Final Teorya, Nobel Prize winning physicist Steven Weinberg marvels at the uncanny ability of mathematics to anticipate physics needs. Katulad nito, quants may marvel at the ability of physics to come up with phenomena and principles that can be directly applied to our field. Akin, 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, mayroon kaming upang suriin kung ano talaga ang sinasabi ng mga prinsipyo. Narito ang ilang mga posibleng pagpapakahulugan:

  • Posisyon at momentum ng isang maliit na butil ay intrinsically interconnected. Bilang namin masukat ang momentum mas tumpak, ang maliit na butil ng uri ng “kumakalat out,” bilang karakter George Gamow ni, Mr. Tompkins, inilalagay ito. Sa ibang salita, ito ay isa sa mga bagay lamang; ang paraan gumagana ang mundo.
  • Kapag namin masukat ang posisyon, mang-istorbo namin ang momentum. Ang aming mga probes pagsukat ay “masyadong mataba,” parang. Bilang namin dagdagan ang katumpakan sa posisyon (sa pamamagitan ng nagniningning na liwanag ng mas maikling wavelength, halimbawa), mang-istorbo namin ang momentum at mas higit pa (dahil mas maikling wavelength ilaw ay may mas mataas na enerhiya / momentum).
  • Malapit na nauugnay sa interpretasyon na ito ay isang view na ang kawalan ng katiyakan prinsipyo ay isang perceptual limit.
  • Maaari din naming mag-isip ng uncertainly prinsipyo bilang isang nagbibigay-malay na limit kung isaalang-alang namin na ang isang hinaharap na teorya ay maaaring malampasan tulad limitasyon.

Ang unang view ay kasalukuyang popular at may kaugnayan sa mga tinatawag Copenhagen interpretasyon ng kabuuan mekanika. Hayaan balewalain ni ito para sa mga ito ay hindi masyadong bukas sa mga talakayan.

Ang ikalawang pagpapakahulugan ay karaniwang nauunawaan bilang isang pang-eksperimentong kahirapan. Ngunit kung ang mga kuru-kuro ng mga pang-eksperimentong setup ay pinalawak upang isama ang mga tiyak na mangyayari tagamasid ng tao, dumating kami sa ikatlong view ng perceptual limitasyon. Sa view na ito, ito ay tunay na posible upang “nakukuha” kawalan ng katiyakan prinsipyo, based on how human perception works.

Ipagpalagay natin na kami ay gumagamit ng isang sinag ng liwanag ng wavelength Ipaalam lambda upang obserbahan ang maliit na butil. Ang katumpakan sa posisyon maaari naming pag-asa upang makamit ang mga pagkakasunud-sunod ng lambda. Sa ibang salita, Delta x approx lambda. Sa kabuuan mekanika, ang momentum ng bawat poton sa liwanag beam ay inversely proporsyonal sa wavelength. Hindi bababa sa isang photon ay makikita sa pamamagitan ng mga butil upang maaari naming makita ang mga ito. Kaya, sa pamamagitan ng batas ang mga klasikal na conservation, the momentum of the particle has to change by at least this amount(approx constant/lambda) mula sa kung ano ito ay bago ang pagsukat. Kaya, sa pamamagitan ng perceptual arguments, makakakuha tayo ng isang bagay na katulad ng Heisenberg kawalan ng katiyakan prinsipyo

Delta x.Delta p approx constant

Maaari naming gumawa ng mas mahigpit na ito argument, at makakuha ng isang pagtatantya ng halaga ng pare-pareho. Ang resolution ng isang mikroskopyo ay ibinigay sa pamamagitan ng mga empirical formula 0.61lambda/NA, where NA ay ang numerical siwang, na kung saan ay may isang maximum na halaga ng isang. Kaya, ang pinakamahusay na spatial resolution ay 0.61lambda. Ang bawat poton sa liwanag beam may momentum 2pihbar/lambda, kung saan ay ang kawalan ng katiyakan sa butil momentum. Kaya makuha namin Delta x.Delta p approx 4hbar, humigit-kumulang sa isang order ng magnitude na mas malaki kaysa sa kabuuan ng makina limit.

Sa pamamagitan ng mas mahigpit na statistical arguments, kaugnayan sa spatial resolution at ililipat ang inaasahang momentum, ito ay maaaring posible upang kunin ang Heisenberg kawalan ng katiyakan prinsipyo sa pamamagitan na ito linya ng pangangatwiran.

Kung isaalang-alang namin ang mga pilosopiko view na ang aming mga katotohanan ay isang nagbibigay-malay na mga modelo sa aming perceptual stimuli (na kung saan ay ang tanging view na akma sa akin), aking ika-apat na interpretasyon ng kawalan ng katiyakan prinsipyo sa pagiging isang cognitive limitasyon din ay mayroong isang piraso ng tubig.

Tungkol sa May-akda

Ang may-akda ay isang scientist mula sa European Organization para sa 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. Ang tanawin ng ipinahayag sa hanay na ito ay lamang ang kanyang personal na tanawin, na kung saan ay hindi nai-naiimpluwensyahan sa pamamagitan ng pagsasaalang-alang ng mga relasyon ng negosyo o client ng kumpanya.

Uncertainly Prinsipyo

Ang kawalan ng katiyakan prinsipyo ay ang ikalawang bagay sa physics na may nakuha ng pampublikong imahinasyon. (Ang unang isa ay E=mc^2.) Sinasabi nito ang isang bagay na tila tapat — Maaari mong sukatin ang dalawang komplimentaryong katangian ng isang sistema lamang sa isang tiyak na katumpakan. Halimbawa, kung susubukan mo upang malaman kung saan ang isang elektron ay (sukatin ang posisyon nito, na) mas at mas tiyak, bilis nito ay nagiging patuloy na mas hindi sigurado (o, ang pagsukat ng momentum nagiging imprecise).

Saan nagmumula ang mga prinsipyo na ito mula sa? Bago namin hinihiling na tanong, mayroon kaming upang suriin kung ano talaga ang sinasabi ng mga prinsipyo. Narito ang ilang mga posibleng pagpapakahulugan:

  1. Posisyon at momentum ng isang maliit na butil ay intrinsically interconnected. Bilang namin masukat ang momentum mas tumpak, ang maliit na butil ng uri ng “kumakalat out,” bilang karakter George Gamow ni, Mr. Tompkins, inilalagay ito. Sa ibang salita, ito ay isa sa mga bagay lamang; ang paraan gumagana ang mundo.
  2. Kapag namin masukat ang posisyon, mang-istorbo namin ang momentum. Ang aming mga probes pagsukat ay “masyadong mataba,” parang. Bilang namin dagdagan ang katumpakan sa posisyon (sa pamamagitan ng nagniningning na liwanag ng mas maikling wavelength, halimbawa), mang-istorbo namin ang momentum at mas higit pa (dahil mas maikling wavelength ilaw ay may mas mataas na enerhiya / momentum).
  3. Malapit na nauugnay sa interpretasyon na ito ay isang view na ang kawalan ng katiyakan prinsipyo ay isang perceptual limit.
  4. Maaari din naming mag-isip ng uncertainly prinsipyo bilang isang nagbibigay-malay na limit kung isaalang-alang namin na ang isang hinaharap na teorya ay maaaring malampasan tulad limitasyon.

Ayos lang, sa huling dalawang pagpapakahulugan ang aking sariling, kaya hindi namin ay talakayin ang mga ito sa mga detalye dito.

Ang unang view ay kasalukuyang popular at may kaugnayan sa mga tinatawag Copenhagen interpretasyon ng kabuuan mekanika. Ito ay uri ng tulad ng closed mga pahayag ng Hinduism — “Tulad ay ang likas na katangian ng Absolute,” halimbawa. Tumpak, Baka. Ngunit ng maliit na mga praktikal na paggamit. Hayaan balewalain ni ito para sa mga ito ay hindi masyadong bukas sa mga talakayan.

Ang ikalawang pagpapakahulugan ay karaniwang nauunawaan bilang isang pang-eksperimentong kahirapan. Ngunit kung ang mga kuru-kuro ng mga pang-eksperimentong setup ay pinalawak upang isama ang mga tiyak na mangyayari tagamasid ng tao, dumating kami sa ikatlong view ng perceptual limitasyon. Sa view na ito, ito ay tunay na posible upang “nakukuha” kawalan ng katiyakan prinsipyo.

Ipagpalagay natin na kami ay gumagamit ng isang sinag ng liwanag ng wavelength Ipaalam \lambda upang obserbahan ang maliit na butil. Ang katumpakan sa posisyon maaari naming pag-asa upang makamit ang mga pagkakasunud-sunod ng \lambda. Sa ibang salita, \Delta x \approx \lambda. Sa kabuuan mekanika, ang momentum ng bawat poton sa liwanag beam ay inversely proporsyonal sa wavelength. Hindi bababa sa isang photon ay makikita sa pamamagitan ng mga butil upang maaari naming makita ang mga ito. Kaya, sa pamamagitan ng batas ang mga klasikal na conservation, ang momentum ng maliit na butil ay may upang baguhin sa pamamagitan ng hindi bababa sa \Delta p \approx palagian\lambda mula sa kung ano ito ay bago ang pagsukat. Kaya, sa pamamagitan ng perceptual arguments, makakakuha tayo ng isang bagay na katulad ng Heisenberg kawalan ng katiyakan prinsipyo \Delta x \Delta p = palagian.

Maaari naming gumawa ng mas mahigpit na ito argument, at makakuha ng isang pagtatantya ng halaga ng pare-pareho. Ang resolution ng isang mikroskopyo ay ibinigay sa pamamagitan ng mga empirical formula 0.61\lambda/NA, where NA ay ang numerical siwang, na kung saan ay may isang maximum na halaga ng isang. Kaya, ang pinakamahusay na spatial resolution ay 0.61\lambda. Ang bawat poton sa liwanag beam may momentum 2\pi\hbar/\lambda, kung saan ay ang kawalan ng katiyakan sa butil momentum. Kaya makuha namin \Delta x \Delta p = (0.61\lambda)(2\pi\hbar) \approx 4\hbar, humigit-kumulang sa isang order ng magnitude na mas malaki kaysa sa kabuuan ng makina limit. Sa pamamagitan ng mas mahigpit na statistical arguments, kaugnayan sa spatial resolution at ililipat ang inaasahang momentum, ito ay maaaring posible upang kunin ang Heisenberg kawalan ng katiyakan prinsipyo sa pamamagitan na ito linya ng pangangatwiran.

Kung isaalang-alang namin ang mga pilosopiko view na ang aming mga katotohanan ay isang nagbibigay-malay na mga modelo sa aming perceptual stimuli (na kung saan ay ang tanging view na akma sa akin), aking ika-apat na interpretasyon ng kawalan ng katiyakan prinsipyo sa pagiging isang cognitive limitasyon din ay mayroong isang piraso ng tubig.

Reference

Ang huling bahagi ng post na ito ay isang sipi mula sa aking mga libro, Ang imitasyon Universe.