How do you understand/appreciate math on a more philosophical level?

[Stefan Zota]

No, it's not futile. What is futile is the attempt to mathematise the field -- as if it's subject to general laws like celestial mechanics, rather than being subject to ad hoc and expedient political decisions. If it's to be studied, it should be empirically, without theoretical preconceptions, and using rules of thumb one is prepared to jettison the moment they look as if they are not working. Finance is an artificial man-made system, where the ground rules keep getting changed by the people at the pinnacle. The math is there to dress it up as something subject to laws and regularities. And to provide gainful employment to dishonest nincompoop professors who drone on about sigma-algebras.

Your approach starts from a hidden assumption: "search for absolute modeling truth".
In other words, you see quants as searchers for a perfect model in a market that doesn't follow strict laws. False. Markets are not perfect and most people are not claiming that.
The goal is to estimate different factors or extract some behavior from available data. Everything is based on a set of assumptions. If data source is changed, assumptions may be invalid. No problem. No model works forever in any market.
We are not looking for absolute laws. So the trend won't stop, markets participants cannot go back to paper and pencil estimations.

 

[bigbadwolf]

Your approach starts from a hidden assumption: "search for absolute modeling truth".
In other words, you see quants as searchers for a perfect model in a market that doesn't follow strict laws. False. Markets are not perfect and most people are not claiming that.
The goal is to estimate different factors or extract some behavior from available data. Everything is based on a set of assumptions. If data source is changed, assumptions may be invalid. No problem. No model works forever in any market.
We are not looking for absolute laws. So the trend won't stop, markets participants cannot go back to paper and pencil estimations.

I think perhaps you don't understand my position and it could be I'm not explaining things in as limpid and trenchant a style as I might wish. The problem is "the set of assumptions." The models don't "work," in that financial products are mispriced and risks are miscalculated -- as commentators like Taleb have been pointing out for some time. Yet these models continue to be taught. How come? Partly sheer force of inertia. Something has to be "taught" in MFE programs. And the lay public has to continue to be bamboozled into thinking that there is some science, some ratonal approach, being used by practitioners. In reality I think paper and pencil estimations, backed by a feel for the market and its underlying forces, are going to be at least as good as anything coming from more complex models. The issue is trying to model something inherently relcacitrant to being modeled. This is not the case for physical or engineering phenomena. Yet this "feel" for the markets cannot be taught, at least not by nincompoop professors who have no understanding themselves. Sigma-algebras can be taught -- no matter how inapplicable and irrelevant they may really be.

 

[IlyaKEightSix]

The models don't work?

O RLY?

RenTec...DESCo...Ed Thorp...

The models don't work. Yep. You're absolutely correct.

 

[AnoopRN]

The issue is trying to model something inherently relcacitrant to being modeled.

I'm loath to admit that perhaps bbw has a point here. However, that math modelling in finance is not an exact science should not be a terminal limitation or reason for us to abandon this field of study altogether. Instead, we need to promote a temperament of disquisitive enquiry amongst practitioners and students alike, where we constantly question and evaluate existing and forthcoming works in this field. Admittedly, there is a need for a cohesive framework governing its usage. But in all fairness, mathematical finance is in its nascency. It's too early to judge if math is salubrious or detrimental to the understanding of financial markets.

 

[IlyaKEightSix]

No, BBW has very little point. In math, which seems to be his niche, all you need to do is find one counterexample to prove a stream of logic incorrect. Jim Simons, David Shaw, and Ed Thorp are three counterexamples.

Just because it's difficult to do doesn't mean it's impossible.

 

[alain]

The models don't work?

O RLY?

RenTec...DESCo...Ed Thorp...

The models don't work. Yep. You're absolutely correct.

Again, another example of you running your mouth. Do you know what exactly is RenTec, DESCo and Ed Thorp modeling or modeled?

I think the BBW is talking math applied to model the Q world (the so called risk neutral probabilities world). Those are the models used to price complex or exotic derivatives. The names you mentioned are famous for trading simpler products.

 

[bigbadwolf]

I think the BBW is talking math applied to model the Q world (the so called risk neutral probabilities world). Those are the models used to price complex or exotic derivatives. The names you mentioned are famous for trading simpler products.

Yes. And in general I'm sceptical of the applications of stochastic processes, probability, time series, and PDEs to finance. But by the same token, I'm equally sceptical of the applications of mathematics to economics, and I much prefer it when economics was known as "political economy," for the term suggests the social and political aspect of economic phenomenon, whereas the term "economics" suggests an area where independent laws and patterns exist. Economics and finance are not physics and the mathematical attempt to treat them as such is leading to mistaken notions of what insight is available into such phenomena.

---------- Post added 11-16-2009 at 12:02 AM ---------- Previous post was 11-15-2009 at 11:44 PM ----------

However, that math modelling in finance is not an exact science should not be a terminal limitation or reason for us to abandon this field of study altogether.

Your phrasing is unfortunate for it suggests math modeling in finance might be approximate -- if not exact -- and thus still afford us some qualitative insight. My contention is it cannot do even this much. There are foundational issues at the root of math modeling in economics and finance that have not yet been carefully examined. We cannot test our models by repeatable experiments. We cannot usually test the predictive validity of our models because we can't empirically measure the phenomenon we're supposed to be measuring. When discrepancies arise between theory and experiment (i.e., to the extent they arise), there is a tendency to introduce fudge factors or explain away the differences by citing the difficulty in measuring. So I am sceptical we our models even afford us qualitative insight. In sharp contrast, in physics, predictive accuracy is often of the decimal place variety -- if memory serves, QED accurately predicts the mass of an electron to 39 decimal places. The predictions of general relativity and quantum mechanics can be quantitatively checked.

Instead, we need to promote a temperament of disquisitive enquiry amongst practitioners and students alike, where we constantly question and evaluate existing and forthcoming works in this field. Admittedly, there is a need for a cohesive framework governing its usage. But in all fairness, mathematical finance is in its nascency. It's too early to judge if math is salubrious or detrimental to the understanding of financial markets.

The point is that with academic departments and an army of practitioners, there is a strong vested interest to not critically attack the foundations of the field. It becomes ideology, dogma. Not to be questioned. Except by the occasional maverick.

 

[Aj Kappor]

Please elaborate a bit more on your insight

Yes. And in general I'm sceptical of the applications of stochastic processes, probability, time series, and PDEs to finance.

BBW, kindly elaborate upon your thoughts:

Are you skeptical about "how" stochastic processes, probability, time series, and PDEs are applied without understanding their boundaries and limits in modeling the specific phenomena in finance?

Or, are you skeptical about "why" they are applied and think that their application – _regardless of how they are applied_ – in all shapes and forms by everyone is "causing" mis-measurement and inaccurate modeling of the specific phenomena?

_Do you advise that such tools _not_ be used at all for modeling risk in finance_? Or do you advise that they be used differently than they have been used by their dogmatic and unquestioning adherents.

Apparently, Derman, Wilmott, and Taleb all seem to be cautioning against the “blind and uncaring” application of tools and models of natural science world to the social domain of finance. They seem to be critical of "how" the above tools are applied. None of them seems to be questioning "why" they are applied and asking that the tools be banished from their applications in modeling finance and risk. Are they?

In sum, are the tools at fault or are those 'blindly' using the tools? Given your extensive knowledge about those tools, your response can help both experts and novices hoping to understand 'correct' application of the tools for solving the problems that matter to most on this forum to recognize the tools for what they are.

Of course, the above question applies to somewhat steadier state of the world a few years ahead as given the current socio-political environment [that you observe] most bets about any kind of deterministic modeling seem to be off for most players in the market.

No, it's not futile. What is futile is the attempt to mathematise the field -- as if it's subject to general laws like celestial mechanics, rather than being subject to ad hoc and expedient political decisions. If it's to be studied, it should be empirically, without theoretical preconceptions, and using rules of thumb one is prepared to jettison the moment they look as if they are not working. Finance is an artificial man-made system, where the ground rules keep getting changed by the people at the pinnacle. The math is there to dress it up as something subject to laws and regularities.

When you refer to 'theoretical preconceptions', do you imply theoretical economic assumptions such as those about 'efficient markets' and 'rational behaviors' etc?

 

[IlyaKEightSix]

Again, another example of you running your mouth. Do you know what exactly is RenTec, DESCo and Ed Thorp modeling or modeled?

I think the BBW is talking math applied to model the Q world (the so called risk neutral probabilities world). Those are the models used to price complex or exotic derivatives. The names you mentioned are famous for trading simpler products.

If I knew what they did, I'd be far richer than I am right now. As for what BBW was talking about, he was painting in broad swathes. If you're talking about pricing securities so exotic that the top funds are repulsed from them, that's another thing altogether.

 

[bigbadwolf]

Are you skeptical about "how" stochastic processes, probability, time series, and PDEs are applied without understanding their boundaries and limits in modeling the specific phenomena in finance?

Or, are you skeptical about "why" they are applied and think that their application – _regardless of how they are applied_ – in all shapes and forms by everyone is "causing" mis-measurement and inaccurate modeling of the specific phenomena?

_Do you advise that such tools _not_ be used at all for modeling risk in finance_? Or do you advise that they be used differently than they have been used by their dogmatic and unquestioning adherents.

Apparently, Derman, Wilmott, and Taleb all seem to be cautioning against the “blind and uncaring” application of tools and models of natural science world to the social domain of finance. They seem to be critical of "how" the above tools are applied. None of them seems to be questioning "why" they are applied and asking that the tools be banished from their applications in modeling finance and risk. Are they?

In sum, are the tools at fault or are those 'blindly' using the tools? Given your extensive knowledge about those tools, your response can help both experts and novices hoping to understand 'correct' application of the tools for solving the problems that matter to most on this forum to recognize the tools for what they are.

When you refer to 'theoretical preconceptions', do you imply theoretical economic assumptions such as those about 'efficient markets' and 'rational behaviors' etc?

This is just a friendly and informal discussion. I don't have the answers to all your questions nor am I necessarily correct in my outlook.

If you look at the philosophy and history of science, you see that it is (almost?) never the case that some towering genius invents a complete theoretical structure for some aspect of physical experience. What seems to occur, rather, is a slow and uncertain convergence between theory and experiment, with one honing the other. Theory tells us what to see, what to look for (often erroneously, with hindsight), and experiment informs us whether our theoretical outlook makes sense. The connection between theory and experiment is often tenuous -- for instance in cosmology or particle physics -- but that's the modus operandi. And often the theoretical tools develop with reference to a particular area. ODEs and PDEs developed with reference to specific problems, specific areas, in physics and engineering. If you look at quant finance, however, this slow natural process of evolution, of interaction between theory and experiment, has not been the one followed. Instead, theoretical tools developed in and for other areas have been used wholesale and unthinkingly in an area where their applicability can be questioned, where experimental feedback is difficult (if not impossible), and hence one where predictive power cannot be gauged. This is no longer science; it is something disingenuously masquerading as science by being dressed up in the language of science (i.e., mathematics).

My contention -- perhaps erroneous -- is that the math we use in quant finance (stochastic, PDEs. etc.) does not give us insight into finance. These are tools designed for other areas. Finance, and economic phenomena generally, are social and political endeavors, where those at the apex can arbitrarily change the ground rules -- e.g., by "quantitative easing", or by bailing out losers. What use are general models when the rules keep changing? For example, the models used to assess "risk" become useless because, well, there is no downside risk for the largest players -- they will get bailed out. In contrast, phenomena in mechanics cannot be tempered by human agency. The orbit of Jupiter cannot be affected by a decision from the Secretary of the Treasury. The equation for its orbit will remain valid.

 

[Aj Kappor]

Is it 'tools' or the 'men'? Or are all 'blind men' around an elephant?

Your point may be probably stretched to suggest that the mathematical tools and methods of natural sciences such as physics are probably not most suitable for analyzing social science phenomena. One can observe a different set of tools such as those based on traditional statistical methods in contrast to statistical physics being applied to study of empirical social science phenomena. Even in applications of those tools, fundamental assumptions along with those about normal and outliers seem to have similar consequences about defective measurement and modeling. Probably, there is some hope for better understanding and explaining complex and messy social political phenomena such as finance by using a combination of above tools. Or, by maintaining professional skepticism about tools and methods and keeping a close eye on whether assumptions and findings bear semblance to the empirical reality of the phenomena.

Even the powers at the pinnacle have to try to manage the sheer complexity and enormity of data and variables to make sense of the messy reality, however tenuous such sense may be. Given different sets of assumptions, one may come to diverging findings as the ongoing debate about regulation of derivatives and exotic instruments seems to suggest. Regardless, just the sheer enormity of data and variables requires some means of processing it for decision-making by regular folks as well as those who may appear to have semblance of power to control messy phenomena such as finance. Hence, despite their limitations, some tools will probably need to be used. They may not be the best tools at the moment as you suggest. Probably real expertise may lie more in knowing how to adroitly apply them while maintaining healthy empirical skepticism about them rather than in blind adherence to tools such as Gaussian copula.

 

[Ergodic]

The orbit of Jupiter cannot be affected by a decision from the Secretary of the Treasury.

$$$



(Give it time.)

 

[Aj Kappor]

Some Natural Science Phenomena are As Fickle

In contrast, phenomena in mechanics cannot be tempered by human agency. The orbit of Jupiter cannot be affected by a decision from the Secretary of the Treasury. The equation for its orbit will remain valid.

Perhaps some [or more] natural science phenomena are also as much subject to mis-modeling and mis-measurement [and probably to interpretivism] just like the social science phenomena.

One example that you noted earlier is the Ptolemaic theory of planetary motions.

A more recent example is that of Pluto, the "ninth planet" that "existed" for as long as most of us studied the structure of "nine planets" in school. Now there "exist" only "eight planets."

Hence, from the philosophy and history of science perspective, Kuhn's structure of scientific revolutions seems applicable to both natural and social sciences.

 

[bigbadwolf]

Perhaps some [or more] natural science phenomena are also as much subject to mis-modeling and mis-measurement [and probably to interpretivism] just like the social science phenomena.

This quickly gets into deep philosophical waters. Is mathematics a structure that is superimposed on physical phenomena? In other words, are we putting on rose-tinted glasses and exclaiming in astonishment that the world looks red? Or are our mathemaftical models an approximation to an underlying mathematical structure that our increasingly sophisticated models are converging towards? A couple of essays attempt to shed some light on this. The first is by Wigner:

http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

and the second one, with the same title, by Hamming:

http://www.dartmouth.edu/~matc/MathDrama/reading/Hamming.html

An extract from Hamming:

I will arrange my explanations of the unreasonable effectiveness of mathematics under four headings.

1. We see what we look for. No one is surprised if after putting on blue tinted glasses the world appears bluish. I propose to show some examples of how much this is true in current science.​

...​

As another example of what has often been thought to be a physical discovery but which turns out to have been put in there by ourselves, I turn to the well-known fact that the distribution of physical constants is not uniform; rather the probability of a random physical constant having a leading digit of 1. 2, or 3 is approximately 60%, and of course the leading digits of 5, 6, 7, 8, and 9 occur in total only about 40% of the time. This distribution applies to many types of numbers, including the distribution of the coefficients of a power series having only one singularity on the circle of convergence. A close examination of this phenomenon shows that it is mainly an artifact of the way we use numbers.​

Having given four widely different examples of nontrivial situations where it turns out that the original phenomenon arises from the mathematical tools we use and not from the real world, I am ready to strongly suggest that a lot of what we see comes from the glasses we put on. Of course this goes against much of what you have been taught, but consider the arguments carefully. You can say that it was the experiment that forced the model on us, but I suggest that the more you think about the four examples the more uncomfortable you are apt to become. They are not arbitrary theories that I have selected, but ones which are central to physics,

Thus my first answer to the implied question about the unreasonable effectiveness of mathematics is that we approach the situations with an intellectual apparatus so that we can only find what we do in many cases. It is both that simple, and that awful. What we were taught about the basis of science being experiments in the real world is only partially true. Eddington went further than this; he claimed that a sufficiently wise mind could deduce all of physics. I am only suggesting that a surprising amount can be so deduced. Eddington gave a lovely parable to illustrate this point. He said, "Some men went fishing in the sea with a net, and upon examining what they caught they concluded that there was a minimum size to the fish in the sea."

With regard to quant finance, I can recommend the book, "An Engine, not a Camera: How Financial Models Shape Markets," by Donald MacKenzie. If memory serves, the author contends that the raison d'etre for the development and the use of quant models was not so much to understand risk and market behavior as to provide legitimacy to trading practices and a veneer of intellectual respectability.

 

[Aj Kappor]

In Defense of Math and Quant Models: By Ripping Them Apart

BBW, please allow me to re-frame the philosophical debate for a moment from a purely utilitarian pragmatic perspective.

Assertion 1. Regardless of why math or quant models are used in finance and regardless of the fact if and to what extent they represent the realism of the ‘real world’ [assuming its homogeneous version exists ‘out there’ – here I seem to be consistent with the philosophers you quote, however my skepticism is not only limited to relying upon math as a ‘lens’ for understanding reality but to any other _singular_ field of human knowledge too], complex math and quant models are an innate part of modern finance.

Assertion 2. As far as one can see, there doesn’t seem to be a highly probable future wherein modern finance [with its derivatives: options, swaps, futures, swaptions, and, other yet unimagined innovations to come] will not remain dependent or perhaps not become even more reliant upon high level math and complex quant models.

Assertion 3. Hence, given the asking price for understanding, practicing, refuting, de-constructing, re-constructing, or even rejecting modern finance and all the complexity (whether right or wrong; whether necessary or necessary) it entails, one would have _no choice_ but to become “literate” in its language.

Assertion 4: Conclusion. Ergo, if one sees any viable future of immersion in the world of modern finance, one _must_ become facile in finance related stochastic, PDEs, etc.

The above statement is offered as a strong statement in favor of math and quant models as applied to finance so that it can generate the strong dialectic necessary for unraveling the truth for whatever it is even if it requires [or, better yet, preferably by] ripping apart the above assertions to shreds.

BBW, please feel free to rip apart the above assertions, so that any fault in the rationale or the above logic can help those who have “pragmatic” interest in quant finance [for whatever reason] assess any fallacy in justifying its pursuit.

[BBW, thank you for sharing various related books and articles. I look forward to savor each of these gems one day given time and glean whatever insights are possible even though I would be skeptical of any one singular specific viewpoint - including sociology or MacKenzie - except one’s own reason based on first hand empirical experience.]

 

[chaoticrambler]

Damn thats a lot of hard GRE words you used there, AJ.

 

[atreides]

Aj Kappor and BigbadWolf are in a league of their own.....phew

 

[Stefan Zota]

I think perhaps you don't understand my position and it could be I'm not explaining things in as limpid and trenchant a style as I might wish. The problem is "the set of assumptions." The models don't "work," in that financial products are mispriced and risks are miscalculated -- as commentators like Taleb have been pointing out for some time. Yet these models continue to be taught. How come? Partly sheer force of inertia. Something has to be "taught" in MFE programs. And the lay public has to continue to be bamboozled into thinking that there is some science, some ratonal approach, being used by practitioners. In reality I think paper and pencil estimations, backed by a feel for the market and its underlying forces, are going to be at least as good as anything coming from more complex models. The issue is trying to model something inherently relcacitrant to being modeled. This is not the case for physical or engineering phenomena. Yet this "feel" for the markets cannot be taught, at least not by nincompoop professors who have no understanding themselves. Sigma-algebras can be taught -- no matter how inapplicable and irrelevant they may really be.

We are on opposite positions on this essential topic, all the more reason to discuss
Even though you don't admit it, you are still looking for the "truth", for the model that works as a law of nature. No such thing. A model can work for 2 years, in certain market conditions, then it can be changed or thrown away completely. I don't see any problem.

Financial markets are not "eternal" or ideal, it is a "living organism", morphs continuously.
If 5 years ago, Black Scholes would be the main stochastic foundation, now it may move to Poisson processes or t-distribution increments. 5 years from now, who knows?
The impression that entire modeling is based on lognormal stock movements with deterministic volatility or interest rates is wrong.

 

[Aj Kappor]

"How do you understand/appreciate math on a more philosophical level?"

Aj Kappor and BigbadWolf are in a league of their own.....phew

atreides: As I recall your original question was:

"How do you understand/appreciate math on a _more philosophical level_?"

Welcome to the world of philosophy,
i.e., "love of knowledge" (Greek: philein + sophos)!

Weary so soon? Hope you are not aspiring for a PhD ;-)
(In case you are aspiring for a PhD, wish you luck as you will need plenty of it ;-) )

Now perhaps you realize why most folks continue living with 'plug and chugg' or 'formulaic' math/calculus: something that is attributed for the GFS (global financial crisis).

We are on opposite positions on this essential topic, all the more reason to discuss

From a "philosophical" perspective, setting up an argument from two mutually [strong] contradictory viewpoints helps unravel underlying assumptions to know the real truth. In philosophy, the process is often described as Hegelian dialectic.
[Georg Wilhelm Friedrich Hegel was considered as one of the most influential philosophers of the Age of Enlightenment. The word "dialectic" originates in Ancient Greece, and was made popular by Plato's Socratic dialogues.]

Damn thats a lot of hard GRE words you used there, AJ.

It is informal chit-chat in philosophy, typical intro to PhD for some.
Not sure if GRE started covering Philosophy.

 

[bigbadwolf]

Assertion 1. Regardless of why math or quant models are used in finance and regardless of the fact if and to what extent they represent the realism of the real world [assuming its homogeneous version exists out there here I seem to be consistent with the philosophers you quote, however my skepticism is not only limited to relying upon math as a lens for understanding reality but to any other _singular_ field of human knowledge too], complex math and quant models are an innate part of modern finance.

Nothing innate about it. It's a political decision made by those in power that the math should serve as a fig leaf to disguise the nakedness of how prices are determined and to camouflage a casino-like activity (which MacKenzie discusses).

Assertion 2. As far as one can see, there doesnt seem to be a highly probable future wherein modern finance [with its derivatives: options, swaps, futures, swaptions, and, other yet unimagined innovations to come] will not remain dependent or perhaps not become even more reliant upon high level math and complex quant models.

If the models aren't working, why continue to use them -- except to deceive the lay public into how abstruse the whole area is? It's a case of the emperor without any clothes.

Assertion 3. Hence, given the asking price for understanding, practicing, refuting, de-constructing, re-constructing, or even rejecting modern finance and all the complexity (whether right or wrong; whether necessary or necessary) it entails, one would have _no choice_ but to become literate in its language.

Use "Ockham's razor": search for the simplest possible explanations that make phenomena intelligible. In general, complex theories and explanations die a quick death while simple ideas -- or at least simple foundations -- endure. Be suspicious of complexity: it is usually there to hide intellectual poverty, to camouflage base and mercenary motives, to confuse and deceive people.

Genius consists in finding simple ideas and explanations for seemingly complex phenomena while mediocrity consists in devising complex and confusing theories.

Assertion 4: Conclusion. Ergo, if one sees any viable future of immersion in the world of modern finance, one _must_ become facile in finance related stochastic, PDEs, etc.

If you want a job in the field, you have to demonstrate some mastery of these things to those interviewing you. If that's what you mean, I agree.

The above statement is offered as a strong statement in favor of math and quant models as applied to finance so that it can generate the strong dialectic necessary for unraveling the truth for whatever it is even if it requires [or, better yet, preferably by] ripping apart the above assertions to shreds.

If you mean you have to master complex models in order to demonstrate their inapplicability or falsehood, you are wasting your time. Life is too short for this kind of pointless demonstration. You don't have to master the Ptolemaic calculations in order to refute the Ptolemaic outlook on planetary motions.