Friday, 24 October 2014

Mapping Ebola Event Risk by Postulating a Financial Markets Ontology

Mapping Ebola Event Risk by Postulating a Financial Markets Ontology.










1.  As we said last time, correlation is not causation.  This is an important distinction in science.  If we wish to do science in law and finance, then using a Category Theory approach, we should translate our statements made of legal and financial terms into SCIENTIFIC PROPOSITIONS that adhere to at least a FIRST ORDER LOGIC.  Translated into this first order logic, we can check our premises and inspect our deductions and inferences for their weaknesses and soundness.  If you would like to know what scientific language would look like, you can turn to Spivak's A Category Theory for Scientists (Old Version) 2013-14 which is freely available on the Web.  In brief, theories and models within theories are written in propositional form using "arrow-language", thus, f:A-->B, where f designates the name of the arrow between A and B.  If you trace Spivak's work who is at MIT, you'll note that he wrote a paper together with Robert E. Kent on "Ologs", which are basically "ontological statements" of real processes.  Robert E. Kent (who's affiliation to any academic institution appears non-existent) has written on the INSTITUTIONAL APPROACH which is a version of Category Theory that is translated for use among people who work and play in the area of INSTITUTIONAL THEORY.  As far as I can tell with my almost total ignorance of the field, the first successful transplantation of a fundamental device from Category Theory into Institutional Theory was by Dimaggio & Powell in their extremely well-cited paper (over 27,000 citations so far) on "The Iron Cage Revisited: Institutional Isomorphism and Collective Rationality in Organizational Fields" (1983) see Jistor.  They used the concept of isomorphism which in Category Theory means f:A->B and g:B->A, so that the morphisms f and g which represent "processes" link the two objects A and B in a manner such that the objects are isomorphic, that is, that A and B are unique up to isomorphism.  Even if you don't quite understand what this "definition" means, you should understand that isomorphism is the way Category Theorists think about "equivalence."  It is all process driven.  

2.  Now Robert E. Kent's work on the Institutional Approach is part of a blossoming field called ONTOLOGIES.  An ontology is simply a model of real processes involving people and machines, especially computational devices that can be linked.  The link between Category Theory and Ontology goes back to a prescient genius named Goguen who was at the University of California San Diego.  His paper on "A Categorical Manifesto" 1991 is a must read.  And the paper co-authored with Burstal on:  INSTITUTIONS:  ABSTRACT MODEL THEORY FOR SPECIFICATION AND PROGRAMMING (1992).  This paper lays out a fantastic theory of institutions that is rigged according to category theory formalities.  Now how do these theories relate to the Ebola versus Financial Charts above?

3.  Well, the point is that it is genuinely difficult to understand what the above charts MEAN unless we have some kind of FINANCIAL MARKET ONTOLOGY buzzing away in the back of our minds.  That ontology would need to be made explicit in order for us to have an explicit understanding of the EVENT RISK that Ebola (as an event) poses onto the minds of financial traders.

4.  To do that which is required in 3 may seem very complicated indeed!  But I don't think so.  We will examine that space--the space of financial market ontology--in further blogs.      

Wednesday, 22 October 2014


Let's Get Rid of Causation in Finance.

This chart comes from Zerohedge today.  It makes me laugh.  Is the world one?  That is, it is so interconnected in terms of web-based and email connections, that messages from the market manipulators instantaneously infects our own wet-ware so that we unconsciously push buttons that amplify particular messages in the noosphere?  "Correlation is not causation" so says Zerohedge repeating the mantra that distinguishes the knee-jerk Palovian-Skinnerian stimulus-response reductionism of our poor little brain from the rationality and decision-making processes of the game-theoretic monsters that can crunch big data until a particular solution fits the circumstances just inside the horizon of noncomputability (NP-Hard problems).  The best papers still in this area where SPIRITUALITY meets PHYSICS are by Emilie Nother -- Einstein's tutor in infinite-dimensional Hilbert space and lovely well-loved teacher who only wanted to teach but the idiotic German Universities could not even give her a proper teaching post.  In the long run of a future history, Germany in the late 19th and 20th centuries will be castigated not only for Nazism but also because its chauvinistic attitude against women--and Noether will be remembered far into the future, farther than when people forget about Hitler.  When Noether died, Einstein wrote something to the effect that she had discovered the first spiritual law of physics.  There are a couple Noether Theorems, but the one FINANCE THEORETICIANS SHOULD PAY ATTENTION TO but do not is the one that allows one to move from an energy conservation law to dimensional space.  How she moves from correlation-covariance is very similar to what the finance theorists from Bachelier to Miller tried with almost brute force.  Basically, Noether moves from correlation through covariance using symmetries. Symmetries as we know in the early to first half of the 20th century is the algebra of group theory, and allegedly, our standard model is based in the ideal on our methods developed in group theory.  Group theory to me looks like it depends on identity, associativity and invertibility (reversal).  I was studying this area for a while back in 2006-07, and then imagine my delight and surprise to find a theory that could encompass and ground group theory using only identity and associativity!  That is, why study Group Theory when Category Theory could do all of group theory and go even much farther?

So now we can come back to the two graphs above.  How do we compare them?  You might say, "Do a correlation analysis and determine the variance."  OK. Then you can do the "co-variance analysis" and come up with a number you can compare against other co-variances.  But this assumes a standard deviation metric underneath, in other words, a normal distribution.  As everyone knows in finance except the crazy financial regulators, there is no such animal in the de facto.  "Volatility" is not a good measure.  What would be a much measure would be a fractal dimension a la Mandelbrot.  If we got use to a Mandelbrotian measure, we'd have a much better feel for the "jitteriness" and "emotionality" of prices, and very importantly, we'd have much better metrics and therefore, a better language to gauge and communicate our intimations and observations of what's actually going on.

To realize these intuitions, I believe we have to build from the ground up, and link the above charts to particular geometries, and to talk about them properly, we would need a new vocabulary that would allow us to make comparisons between the charts and the contexts that surround them in an absolutely precise way.

While we would never say Chart 1 causes Chart 2 or vice versa, we could say without any problem that Chart 1 is isomorphic to Chart 2, and then we would be forced to make explicit exactly WHAT THE PROCESS that makes the isomorphism up.  If we could do that, we would not ever again bother with the concept of "causation" in finance.       
A Category Theory of Financial Instruments and Financial Institutions -  A Fundamental Ontology of Law and Finance


-Draft Abstract-

In an early paper examining the regulation of private placement memorandum (PPM) regulation, we found that there are in general two approaches to financial regulations: (1) the regulation of the behaviours of the financial institution; and (2) regulation of the financial instruments.  PPMs are interesting financial instruments since they sit in-between two extremes of financial regulatory types: (1) purely private form of financial intermediation (that is, at the extreme, we have bilateral financial contracts) and (2) highly ritualized guidance on what can be communicated in the raising of capital (that is, prospectus-type regulations) stemming from the Securities Act of 1933 and Securities Exchange At of 1934.  The unnatural divide between the regulation of financial institutions and the regulation of financial instruments has played itself out in terms regulations which aim at adjusting the incentives relating to particular types of businesses within a financial institutional framework.  In general, the play out has been a border conflict between banking regulations and capital market regulations.  This is not to say that any particular jurisdiction uses one type of financial regulation exclusively to the exclusion of the other, but rather there is a combination or admixture of financial regulations, that in total, ascribe to one general tendency or the other.  Recall the legal theorists who attempted to justify financial regulations on the basis that “Law Matters”, stating in particular that law matters in that it set outs the initial permit or license to practice a certain form of business.  As we shall see, if we apply certain fundamental risk and return models, which the financial industry itself uses to measure its own financial instruments, we can distinguish different sorts of businesses according to these risk and return financial instrument components.  Thus, from an extreme financial institutional perspective, which applies financial theory to its own institutional design, behaviours and assessment of behaviours, financial instrument theory applies laws to accomplish the institution’s particular objectives and the discipline and guidance therefore are simple financial theory models of risk and return, which captures the market definition of money.  From the institutional perspective apply financial theory, an institution is simply a financial instrument with certain risk and return characteristics.  Thus, financial institutions regarding themselves as financial instruments are constrained by the “rules of the game” of finance, which are basically arbitrage (“the law of one price”) which forever use laws such as contracts, and every other sort of law and regulation, as merely instrumental.  There are researchers who assert that there should be a legal theory of finance and use legal ideas to promote the logical priority of law to finance. There is no argument with this thesis if we add the distinction that law is the context in which finance exists.  However, it is not entirely plausible to say that finance exists because of law, nor is it plausible to assert that law is absolutely necessary for finance to exist in the world.  In any case, the purpose of this paper is not to adjust or determine where the horse and carriage can become one or the other—we believe that is actually an obvious distinction--but rather to understand if possible the system, model or ecology in which law and finance co-exist.  As a theoretical approach, we shall combine the financial instrument view and the financial institutional view into one totality, and call that totality, a law and finance ontology.  Our view, which is slightly complicated because of the distinctness of disciplines and variety of sub-disciplines involved in law and finance, is to start with the simplest ideal financial instrument and then to ask ourselves what would happen to that financial instrument in the real world of law and finance.

This model of moving from a well-defined financial instrument (which is the particular in the Aristotelian sense of a genus-specie category or more modernly, the event) to how this financial instrument operates in the real world (the generalized reality of abstract continuity) is in effect a study of the reality of finance given a legal context in the form of a mapping or under a mapping technology.   


Thursday, 21 August 2014

“If-then’s” generate “choices” or “how to free yourself of other’s causal claims”.

I read a bit of news—mainstream, twitter & FB--and I have this desperate urge to review a bit of logic.

For example, when someone declares "Do it my way or the highway," you don't have to worry.  You can reconstruct that statement into:

If it is x, then it must be y.
It is either not x or it is y.

For example:
If he wears brown shorts, then he must be a terrorist.
Either he does not wear brown shorts or he is a terrorist.

Translating the above threat, “Do it my way or the highway,” means “If you don’t do it my way then you must take the highway.”  The phrase “do it my way” is negated in the antecedent.

Let’s go further.  How about the statement:

“If you believe in me, then you will go to heaven”?

Translates to:

“Either you don’t believe in me or you will go to heaven.”

That’s proper.

But suppose that statement gets warped into:

“If you don’t believe in me then you will go to hell.”

Which translated into “or” form becomes:

“Either you believe in me or you will go to hell.”

Bottom-line:  when you get into arguments about choices, watch out for causation type statements as blame that turn out to be dichotomies.  Once you see this rule operating, you don't have to take sides.

You are free to mosey along.

Friday, 19 July 2013

Default Invariance:  The Three Approximations of Legal and Financial Reality

I took a break from blogging to work on a theory paper entitled, "Default Invariance, A Naive Category Theory of Law and Finance."  You can see its abstract on www.ssrn.org.   I think the idea of taking a simple logical structure implied by the simplest form of a legal-financial phenomenon, namely, a financial contract with a one-period payment, and looking to the topological space implied by its terminal object will forever change the way we do law and finance.  In the paper there are three approximations of law and finance that correspond to the structure implied by the terminal objects, Pay, Not-Pay and Pay & Not-Pay.  In the simplest rough and ready terms, these terminal objects imply a point, a line ("risk homological chain complex") and a cyclic matrix ("a ring structure").  Each approximation defines the context-environment of the legal financial structures.  And I'm happy to say that we can may make explicit conceptual calculations which improve on ("correct") the works of three Nobel laureates in economics, Arrow, Debreu and Sharpe.  Arrow-Debreu-Sharpe basically set out the contingent claim model (Debreu, by the way applied abstract algebraic topological methods in his seminal work, A Theory of Value, which got rid of probability for one-period claims) which underlies everything we know and do in risk management, corporate governance, portfolio theory and practically, everything else we call "modern finance theory."  So, one way to read my Default Invariance paper is that it puts Arrow-Debreu-Sharpe into the perspective of a naive category theory, and shows our syntactical structures pre-dispose our conceptual calculations.  Anyway, you can read the paper to find out for yourself.  It's got 35 original diagrams that are meant to help "re-wire" one's own internal mapping of how the law and finance world works.

Fourth Approximation:  Taking Parts and Partitions Seriously

If we wanted to study law and finance as a physical process, we might find that there is a Darwinian-light version to the selection of laws and financial products that appears to apply.  Recall Darwin used the principle random selection for the adaptation of certain macro-features appearing to differentiate species according to external environments.

Consider Lancelot Law Whyte (1965) Internal Factors in Evolution, cited by John Bonner (20 July 2013) "Evolution, by chance?" New Scientist, 26-27, 26.  As Bonner states, "His [Whyte's] thesis was straightforward. Not only is there selection of organisms in the environment--Darwinian natural selection, which is eternal--but there is also continuous internal selection during development.  Maybe the idea was too simple and straightforward to have taken root." Bonner then goes on to state his own thesis, "This fits in neatly with my contention that the shape of microorganisms is more affected by randomness than for large, complex organisms.  Being small means very few development steps, with little or internal selection.  The effect of a mutation is likely to be immediately evident in the eternal morphology, so adult variants are produced with large numbers of different shapes and there is an increased chance that some of these will be untouched by natural selection. Compare this with what happens in a big, complex organism--a mammal, say. Only those mutations that occur at a late stage of development are likely to be viable--eye or hair colour in humans are obvious examples.  Any unfavourable mutation will likely be eliminated by internal selection."  He points out the evidence that the shapes of microorganisms are "less likely to be culled by natural selection" by citing Radiolaria (50,000 species) and diatoms (100,000 species) and Foraminifera (270,000 species).   Then he states, "If you are a strict adaptionist, you have to find a separate explanation for each shape. If you favour my suggestion that their shapes arose through random mutation and there is little or no selection, the problem vanishes." [p. 27]

What structure is implied by Bonner's internal versus external environment selection thesis?  I find his terminology a bit confusing.  For what is the external environment of a micro-organism?  Isn't everything outside it in a sense a micro-structure and therefore, could be in it, as well?  Perhaps we can clarify the thesis by translating the situation into a morphism f: A-->2.  Imagine the object A population with lots and lots of elements but having one partition such that you can maximise or minimise either part.  The f-morphism are injections to either of the two elements in 2.  So long as the partition exists, the 2 separate values will exist in 2.  So, we don't need the internal versus external division.  In category theory, there is a theorem which just gets rid of all "internal diagrams" so that anything and everything that can be possibly expressed can be done with external diagrams only.  I think the same can be said about Whyte's and Bonner's thesis above.  In other words, Darwin's random selection to adaption is preserved in the structure of parts and partition via a morphism f: A-->2.

The Fourth Approximation is taking the Third Approximation of Pay and Not-Pay as parts with a partition.  The structured implied from this terminal object is "Continuous Contingencies" (CC) to "Infinitely Discrete Randomness" (IDR).  This sounds extraordinarily vague, but what it means is that which is undifferentiable can be made into discrete unit choice.  In syntactic form:  g: CC-->IDR.  This is similar to the conceptual step of moving from "God as ubiquitous being" to "eating a properly cooked vegetarian meal is a morally correct choice of being."

Tuesday, 25 September 2012

Lecture 1 post hoc notes - Legal Aspects of Corporate Finance

Lecture 1: Legal Aspects of Corporate Finance Guest instructors:  Professor Edmond Curtin and PhD Candidate Rezarte Vukatana I walked in a few minutes late with a bundle of papers and just started talking about THEORY as if it were the most natural thing in the world.  I told them about a Russian table tennis star whose training regime included 6 hours of chalk and blackboard theory everyday. But the main point came from Hohfeld's definition of theory: "A theory is not even a theory unless it can be used by practitioners in their practice."  I don't think I introduced myself but I did introduce Edmond and later Rezi.  I mentioned a few themes: (1) WEAK EQUIVALENCE as the subtle equivalence of thoughts; (2) the UNITY OF SCIENCE CRITERION as the main ground for adjudicating theories-- a theory should be judged on how it helps us understand the unity of all knowledge of being; (3) sign, symbol (Edmond mentioned "signifier" pointing to the picture of the green man in the exit sign--everyone turned to look); (4) HOHFELD the undergrad chemistry student turned professor of Yale Law School who in early 20th century wrote only 6 articles and invented a periodic table for the law - 4 JURAL OPPOSITES and 4 JURAL CORRELATIVES with enormous theoretical effects; (5) CORBIN and WILLISTON who wrote encyclopediac tomes on contracts law, and how Corbin (a Hohfeldian student) took just one jural correlative, rights versus duties, and turned that tiny almost trivial legal distinction into 7 (or was it 9?) volumes of contract law; (6) And Where are Contracts anyway? shock horror to the civil law students ["on paper", "after the signature" they say] but no, says the common law jurisprudentem--CONTRACTS EXIST IN THE MIND [Edmond]; horror of horrors, is this the pure subjectivism, relativism and thus, total discretionary totalitarianism of the law?; (7) Why some questions within professional discourse make no sense ("What's north of the north pole, eh?"] and is there a way of understanding that transcends the bounds of discourse?  Later, the astute Russian student answering a question about "material information" asked a rhetorical question about the distinguishment of various risks.  Then I told a long story about Yuanjia, the Great Wun Chin master, who when cajoled by a Japanese martial artist that there are levels in the artistry of tea, replied, "The tea makes no such distinctions and is thoroughly enjoyed."   Thankfully, Edmond gave us a brief rendition on some of the essential legal principles of DERIVATIVES--how they actually create MORE RISK and MORE ANXIETY, and never less. Rezi described part of her PhD dissertation research--theory of self-fulfilling prophecy a la Merton (?) and how this can be used to help explain the strange behaviours of very complex nodes of financial system called intermediated securities accounts.   I passed around 3 LLM dissertations for the students' inspection, and gave them a homework assignment.   I filed some prospectuses at LLMCFL2012@gmail.com [if you want the password, you need to contact me] with my notes, and asked the students to write 2,000 words on (1) the risks of the prospectus transaction (either Salvatore Ferragamo or Prada); and (2) determine whether and what parts of the selected prospectus would need to be changed under the Directive 2010/73 Nov 2010.  They'll need to review about 600 to 800 pages and email me their work by 12noon Monday.  Nice shock therapy. 

Tuesday, 11 September 2012

Extreme Philosophy: On the Limits of Self-Referential Truth: Why Paradox Has Been Binned By Naive Category Theory

1. Here are two papers of EXTREME PHILOSOPHICAL SIGNIFICANCE: [1] Lawvere, F. William, "Diagonal arguments and cartesian closed categories with Author Commentary,"  Lecture Notes in Mathematics, 92 (1969), 134-145, available at:   http://www.tac.mta.ca/tac/reprints/articles/15/tr15.pdf [2] Yanosky, Noson (2003) "A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points,"  available at: http://arxiv.org/pdf/math/0305282v1 2. Unless you've studied a bit of category theory, i.e., read Lawvere and Schanuel (2008, 2nd edition) and Lawvere and Roseburgh (2003), Lawvere [1] will be very obscure even with Lawvere's commentary. But take a look and get a feel. Then, look at Yanosky [2] which explains in a more breezy (but precise) way what the genius Lawvere was up to, and even more cleverly in order to reach a "wider audience", dropped category theory altogether and explains Lawvere's discoveries in easy enough "set and function" language. 3. I realize that category theory is not for everybody (yet) and recently, in the literature, there is a push-back accusing category theory of making "foundational claims" that are unjustified. For example, that the entirety of mathematics can be put on a category theory footing and replace set theory as the fundamental theory which all other theories must bow down to. But I don't think category theory as it is practiced sets out to make any really big claims like these--that would be the job of propogandists. Rather it "solves" some rather apparent fundamental problems by "resolving" the problems into a diagrammatic logic. If you buy the diagrams as BEING DENOTIVE then you might also see how category theory IS linked to Aristotle's great work On Categories. Mac Lane in a footnote joked about how the title "category theory" came from "purloining words from the philosophoers, Aristotle and Kant" [pp. 29-30 of Categories for the Working Mathematician]. He doesn't say anything more about this jokey link. But if you read and understand Aristotle's motive in his Categories, you can see immediately that Aristotle set up foundational problems so they can be resolved. He analysed knowledged into what might be called "said-of" and "thing-in" and asked what are those abstractions that are primary, that is, what are those properties that are extended and therefore, must be. He listed 10 categories [what they are appears arbitrary] and he showed how you can use these primary categories to categorize everything else, that is, that which is not so extended and universal. Now, this mental-conceptual move to abstraction in order to solve a particular problem is a natural function. Lawvere & Schanuel in Conceptual Mathematics explain this movement in terms of isomorphisms: e.g. think of how you can understand what's happening in a film even after walking into the cinema late. In media res, you know Humphrey Bogart is playing a particular character and Audrey Hepburn is playing another character, and when you sort out who's who in the film, suddenly, you can follow the plot in the film with the actors as playing their roles. Similarly, being born in the middle of things, we open our eyes, stretch our arms and legs, and explore the universe, fully confident that we will be able to sort EVERYTHING out. This confidence comes from something pretty powerful within ourselves that enables us to gain knowledge. And the point here is that knowledge isn't at its rock bottom paradoxical. It is in all likelihood isomorphic. 4. Lawvere [1] takes a swipe at the propogandists who have been using some of the great theoretical work of theorists (such as Russell, Cantor, Godel, Tarski) and turned them into very general claims about the nature of paradox at the heart of knowledge. To put this into a general philosophical context, Aristotle's optimism was founded on his discovery of a general scientific method which if simply re-iterated, would eventually uncover all the mysteries of the universe. It was based on observing that which is and translating those into propositions which could be understood. If at the heart of heart of "proposition making" we have paradox, then this whole enterprise is doomed to failure. So, burdened with the prospect of failure, why start the programme of knowledge? 5. The answer by Lawvere [1] and Yanofsky [2] shows why the propogandists of paradox are simply wrong. In Yanofsky's terms, Lawvere's great little paper [1] has been largely ignored by category theorists and philosophers alike because it is written in a forbidding unpopular formalism. Yanofsky translates the results of Lawvere's paper by saying the classical paradoxes of self-referential truth (e.g. Liar's paradox, Russell paradox, Godel's incompleteness and so on) are just instances of overstepping the limitations of a discourse ("discourse" is my term). There must be a way of limiting what a discourse can say about itself. This "problem" comes up in law and finance whenever they try to talk about themselves. I call it the problem of structure. That is, there is no such question in law and finance that says, "What is the structure of law? What is the structure of statements about finance?" There is no call for self-consciousness within laws or financial practice. Rather, the call for such professional consciousness comes from without. But there is a way of understanding such questions about professional discourses from a category theory perspective. And not only do the questions about the structure of law and finance make sense, they actually direct in some fashion a resolution to answers about the structure of law and finance. For example, one of the things I have been harping on in this blog is that there is a fundamental structure to law and finance in the forms of an individual unit which I have dubbed the "financial contract" and the "great cycle of default invariance." From these structures, we can explain a lot of current practice at the individual-to-individual level of financial transactions on up to historical and contemporary nausea of continuously impending financial catastrophes. It's all a matter of "mapping" and translating apparent limitations within the discourse of law and finance into a notation which allows for mental journeys and conceptual calculations. By the way, one of the virtues of seeing how paradoxes are slain in [1] and [2] is that we can recover a sense of optimism that Aristotle once had in the unity of science. Again, I say, judge the value of a theory by its contribution to the unity of science.