Finance and Modeling

Most of the concepts introduced thus far are foundational to a number of disciplines like economics, accounting and so forth. In a very real sense, these are core concepts of Computational Finance. Yet, as you no doubt noticed, there is no indisputable clarity and rigour to these notions as there is to their analogues in established fields such as physics¹. You may, of course, argue that lose definitions sourced from Wikipedia aren't suitable building material to those seeking rigour — and you'd certainly have a point; but it's worthwhile bearing in mind that this approach was chosen after a great deal of deliberation and practical experience.

In order to understand why Computational Finance so greatly differs from physics, one must first make a distinction between a theory and a model. Availing ourselves with Derman's Metaphors, Models & Theories (Derman, Emanuel, 2011a), we start with his clear demarcation (emphasis ours):

Theories deal with the world on its own terms, absolutely. Models are metaphors, relative descriptions of the object of their attention that compare it to something similar already better understood via theories. Models are reductions in dimensionality that always simplify and sweep dirt under the rug. Theories tell you what something is. Models tell you merely what something is partially like.

Further on, he deals the knock-out blow:

There are no genuine theories in finance. Financial models are always models of comparison, of relative value. They are metaphors. The efficient market model assumes stock prices behave like smoke diffusing through a room. These are comparisons that have some reasonableness, but they’re not even remotely fact. Newton’s laws and Maxwell’s equations are. There is almost no gap between the object and their description. You can say that stock prices behave like smoke. You cannot say that light behaves like Maxwell’s equations. Light is Maxwell’s equations. All concepts, perhaps all things, are mental. But there are no genuine theories in finance because finance is concerned with value, an even more subjective concept than heat or pressure.

Furthermore, it is very difficult to find the scientific laws or even regularities governing the behavior of economies: there are very few isolated economic machines, and so one cannot carry out the repeated experiments that science requires. History is important in economics. Credit markets tomorrow won't behave like credit markets last year because we have learned what happened last year, and cannot get back to the initial conditions of a year ago. Human beings and societies learn; physical systems by and large don't.

Stern stuff. To be sure, the kernel of these ideas had already been touched upon earlier in Soros' work, when he introduced — or perhaps recovered — the notion of relflexity. For me, personally, relflexity marks the first time I encountered philosophical discussions around trading, and it made a lasting impression; to this day, I still find it to be one of the most crucial properties of the field. Soros defines reflexivity and fallibility as follows (Soros, George, 2013) (emphasis his):

My conceptual framework is built on two relatively simple propositions. The first is that in situations that have thinking participants, the participants' views of the world never perfectly correspond to the actual state of affairs. People can gain knowledge of individual facts, but when it comes to formulating theories or forming an overall view, their perspective is bound to be either biased or inconsistent or both. That is the principle of fallibility.

The second proposition is that these imperfect views can influence the situation to which they relate through the actions of the participants. For example, if investors believe that markets are efficient then that belief will change the way they invest, which in turn will change the nature of the markets in which they are participating (though not necessarily making them more efficient). That is the principle of reflexivity.

In other words, all models are and will always be imperfect; and, when they do work, it is not entirely clear whether it is due to their predictive power and ability to model the real world, or whether adoption itself is causing the model to be good at prediction, or whether it is due to something else entirely. Perversely, in many cases, as the model increases in popularity so do its predictive powers, reinforcing the belief that the model works. All of these forces are probably always at play, in varying degrees, but what makes the situation difficult is that it is not possible to unpick them within any reasonable confidence interval because of the inoperativeness of the scientific method in this realm.

One may be tempted to read the words of Derman and Soros and visualise complex mathematical equations describing highly specialised financial products. This, in our view, does not get to the truth of the matter, for the issue lies much deeper within. It starts with value, as Derman says, and continues, to the point that almost all concepts within Computational Finance are models — and those that aren't, are so deeply intermingled with all the rest that they might as well be models too.

When we use terms like "price", "money" or "currency", we are inadvertently referring to models, for these things are not of the world in the same way an atom or a molecule is, and the best one can do is to create models as metaphors to represent them. These models are hazy neo-cortical constructs, programs running in people's brains that have a faint connection to the real world²; and the field is made up of models within models which point to other models, ad infinitum³. To get a flavour of what is meant by this, we shall dip our toes on the famous debate on the origins of money, as reported by Svizzero and Tisdell (Svizzero, Serge and Tisdell, Clement, 2019) (emphasis ours):

Metallists believe that money developed spontaneously as a medium of exchange in order to eliminate the obvious limitations of barter. In other words, the origins and the early evolution of money are viewed as an unintended consequence of spontaneous individual actions in the context of barter. Thus, money emerged via a natural process of transaction cost minimization. Metallism was an economic principle stating that the value of money derived from the purchasing power of the commodity upon which it is based. […]

On the other hand, there is the Platonic vision of currency, also called the credit theory of money. This contends that money is a social construction rather than a physical commodity. The abstract entity in question is a credit relationship; that is, a promise from someone to repay a favor (product or service) to the holder of the token.

In other words, even at this late stage, we still have no consensus as to what exactly money is; and whatever it is, it appears to be deeply intermingled with how it came to be. As is with money, so it is with most, if not all, of the notions put forward in this work. Therefore, there is hardly a need for elaborate definitions, since little or nothing is gained with regards to subjectiveness and much — when not all — is lost in terms of comprehension. One should search instead for sufficient explanations, for intuitive descriptions, for the simplest of things that allow us to progress in constructing our tower of models. This is the guiding principle of the present work⁴.

At this juncture, the alert reader will probably question the purpose of the entire enterprise, given its wanton disregard for rigour. Here, the words of the experienced practitioner are of great assistance. Box informed us that "all models are wrong, but some are useful" (Box, George EP, 1979) and Wilmott went one step further by claiming, not without justification, that "every financial axiom I've ever seen is demonstrably wrong — the real question is how wrong is the theory and how useful is it regardless of its validity." (Wilmott, Paul, 1998) In this vein, one must keep in mind that none of these challenges make Computational Finance any less useful; after all, for better or worse, the world's economy runs on constructs such as these, proving there is no shortage of uses. Instead, what we must not lose sight of are the properties of models, and how they relate to the real world. Wilmott and Derman stated on this regard (Derman, Emanuel and Wilmott, Paul, 2009) (emphasis ours):

We do need models and mathematics — you cannot think about finance and economics without them — but one must never forget that models are not the world. Whenever we make a model of something involving human beings, we are trying to force the ugly stepsister’s foot into Cinderella’s pretty glass slipper. It doesn't fit without cutting off some essential parts. And in cutting off parts for the sake of beauty and precision, models inevitably mask the true risk rather than exposing it. The most important question about any financial model is how wrong it is likely to be, and how useful it is despite its assumptions. You must start with models and then overlay them with common sense and experience.

These are all wise words, but the word "assumption" is one of disproportionate importance. This is because underlying all models lie assumptions, and its not necessarily just because you might misuse models deliberately; much more dangerous are those assumptions which are hidden. Henney puts it quite poignantly:

Epistemologically speaking, assumptions are the barefoot-trodden Lego bricks in the dark of knowledge. You don't know they're there until you know that they're there. And even if you know there are some there, you don't know exactly where and you'll still end up stepping on some.

With all of this said, it should by now be clear that we must understand the modeling activity better, given its all we'll be doing. For models to be truly of use one must first know their limits, and that effort must surely begin by developing a clear understanding what is meant by the term. Regrettably, we cannot investigate Model Theory at the level of detail it demands, but we are at least able to provide some basic intuition via Stachowiak's work, and his classic Allgemeine Modelltheorie (General Model Theory) in particular, wherein he proposes a model-based concept of cognition. He identifies three principal features of model⁵:

• Mapping: Models map individuals, original or artificial, to a category of all such individuals sharing similar properties. The object of the mapping could itself be a model, thus allowing for complex composition. - Reduction: Models focus only on a subset of the individual's properties, ignoring aspects that are deemed irrelevant. - Pragmatism: Models have a purpose as defined by its creators, which guides the modeling process. Stachowiak states (emphasis ours): "[models] are not only models of something. They are also models for somebody, a human or an artificial model user. They perform therefore their functions in time, within a time interval. And they are finally models for a definite purpose." (Podnieks, Karlis, 2017)

Much of the expressive power of models arises from these three fundamental properties; this is what models are good for. They must therefore be used with great care, and for their intended purposes. However, once you find yourself outside the model, anything can happen. And, given sufficient time, you will always find yourself outside the model. Lewis, in his usually prosaic manner, put it best (Lewis, Michael M, 2009):

Everything, in retrospect, is obvious. But if everything were obvious, authors of histories of financial folly would be rich.

With these parting words, we have reached the summit of our brief philosophical ascent. In the context of the broader domain, this discussion is roughly placed within the porous boundaries of Model Risk, which Wikipedia defines like so (emphasis ours):

Definition 2.23: In finance, model risk is the risk of loss resulting from using insufficiently accurate models to make decisions, originally and frequently in the context of valuing financial securities.

We won't delve further into model risk for now⁶, but do keep in mind that we have barely scratched the surface on this topic; in particular, we've ignored significant voices such as Mandelbrot (Mandelbrot, Benoit and Hudson, Richard L, 2007), Taleb (Taleb, Nassim, 2005) and many others — Derman, for one, had much more to say on the subject. Nonetheless, the purpose of this section is merely to sketch out the why for our approach and to give sufficient context for the decisions taken, rather than to perform a comprehensive literature review, which, sadly, must be left as an exercise to the diligent reader.