A brief overview of how it is structured and broken.
The specific model as originally developed, modified in the 80s and applied to options pricing is explained fairly well in its Wikipedia article. The way it has been adapted and extended to apply to the larger, growing world of derivatives is broad. You may want to browse through the Journal of Derivatives to get an idea of how broad the spectrum is.
Here, we worry only about the basic structure of the technique.
Continuity
The various methods all build on the basic structure of a differential equation. Differential equations are the basic stuff of Newtonian physics, and indeed, Newton developed the differential calculus for that very purpose. When applied to physics, It assumes that the world is continuous, that there are no abrupt tears in the fabric of reality. Further, this implies that if things are connected, they are connected smoothly. There can be abrupt swings in effects, but if you zoom in on the transitions, they have no breaks or corners.
This suited Newton well, because no observation of the human-scale world ever presented such a tear or corner. Everything is well-behaved; and it has to be because of the way mass and forces work in the model. A way of thinking about this is that the stuff of the world constrains itself to be well-behaved.
But financial instruments are inventions: human ideas, not real stuff. There is nothing intrinsic in the reality of these things (other than the model we create) to constrain it to be well-behaved and continuous. The behaviors that are captured include impressions by humans. Impressions and certain concepts tend to be non-continuous in the human mind. We have tostruggle, for instance, with shades of grey in our justice system: someone is guilty or not. Something is true or false.
A clear example can be seen in how numbers are handled. In everyday life we manage numbers naturally, but in two fundamentally different ways.
One way is continuously: for instance in how we measure time. We think of time as smooth, and if we wanted, we could measure every second, or tenth or millionth. There are no jumps in time, nor the numbers we use in measuring it. This in fact is the notion of time and number that Newton uses.
But at the same time we have a more fundamental notion of numbers as integers. You can have a certain number of children, or cars. You buy cans of food at the grocery store in discrete units. This notion of discreteness is also a foundation of Newtonian physics: the presumption that the matter in the world consists of objects. Planets are discrete things, as are leaves, atoms, even non-corporal things like breaths and concepts. They are all discrete, with no meaningful notion of in-between.
In the real world, we balance these twofundamentally different notions of numbers without noticing, even though we use the same notation and much of the same arithmetic.
Black-Scholes deals well with the one notion of number that is continuous and ungracefully with those that are discontinuous.
This matters not just because of numbers themselves; they were just the example. The numbers stand for something, often complex dynamics. When those dynamics are based on human reasoning, we find that they do have jumps, tears, discontinuous breaks.
First Class Independence
Another problem would be characterized by a formalist as the flattening of the type system. All concepts that are factored into the model are considered to be equal in their existence and independent except as described in the model. The base case first addressed was option trading where the four factors were duration of the option, prices, interest rates, and market volatility. These four factors are related by the model, but are assumed to exist in a world outside the model as real things. The existence of each of these things is equal. This is a very big deal.
The analogy in physics would be to say, for instance, that quarks, magnetic fields, light and color can be considered as qualities of the universe that are equal in their being. Also that so far as any consideration of how they interact, they are independent except for the equations we would state that capture the model of their interaction.
But the world as we deal with it has some things that are more fundamental than others. There is a hierarchy of types, in other words. Though different models and theories propose different hierarchies, no one would dispute that such a hierarchy exists. That is what science is all about.
How does the financial community deal with this? By fudging. Human judgement is supposed to be used in coloring these inputs. Every application of the model, across all the derivative applications hides the fact that the supposedly impressive formal model depends on providing the ‘correctly colored’ inputs. (This is the problem that Merton claimed was behind the failure of the very simple application of the approach to options in the late 70s.)
One can say that the system collapsed because the model is good but the inputs scurrilously or ignorantly miscolored. But it should be impossible to disentangle the relationships of the inputs outside the model from inside the model in any application.
Ordinary Causal Logic
The final fatal problem with the model is in the concept of causality used. The reader should be careful to sort this out from a complaint that the model gets what causes what wrong. No, it gets the fundamental notion of cause wrong. This is not so much a problem with what Merton and company bring to the table with the model, but the larger issue of how science is applied in the domain of economics.
In physics, something happens, it affects something else in the neighborhood and some change occurs. The change is said to have been caused by the initial events through the interaction. This is intuitive. Even in physics, the notion has someissues, but let’s say here that it just works.
Economics does involve arithmetic. But regardless of how apparently complex that arithmetic, the underlying phenomena are human actions. And human actions are not based on logic, not always or even usually. This is not a novel observation: the laws associated with physics have always sat uncomfortably in the domain of the ‘soft sciences.’ The conversation about this is long, but traditionally not very deep after the observation that no ‘other science’ is applicable.
Greater insight comes from the biology community who has a similar problem. Physics-centric science likes to have a notion of cause associated with objects and effects as we just noted. But microbiology in the medical domain likes to think of complex systems and some degree of selforganization. Insights from people looking at this problem inform some new notions about cause coming simultaneously from all the individual elements as physics would have it, but also from some sort of system awareness, perhaps at different levels of system.
Some tools associated with expanded logics are emerging, and should be considered where they serve the model well. Black-Scholes has no room for this, though some researchers in these new geometric logics propose some useful approaches.
That means we allow for a number of ‘quantum effects’ that are normally associated with the areas where Newton fails, even in the domain of physics.
One is that because economic systems are human systems, they are by their nature introspective: they see themselves. Whenever we measure ourselves, the observation process perturbs the measurement. This is well understood even in the simplest of public polls.
A second, intriguing phenomenon is that not all the causal linkages are knowable at any given time. So inexplicable linkages will be affecting the system, analogous to spooky action at a distance.
Even more fascinating is because the governing frameworks are narrative-based. We seem to be hardwired to organize our understanding of systems in narrative structures. These structures build tentative, dynamic causal structures. They are dynamic in the sense that things that happen late in the narrative reactively change the meaning of salient relationships early in the narrative. An example in fiction is when you have a murder mystery and at the end you are given some causal connections that force you to go back through your stored memory of the story and reinvent some causal connections.
We do this in real life as well. For instance, the science shows that eye-witnesses are notoriously unreliable when they have gone over the events several times and made sense of them by changing key facts and causal connections to create an order in our memory. We see this in action in political campaigns where the winner is the fellow who can reinvent the past of the opponent to make a sensible but unattractive narrative. Once this narrative settles, it is all but unshakable as truth.
We’ll give the example of the financial crisis that started this note. When the events were happening, there were no indications that things were wrong beyond the normal level of analysts that predict disaster. The events certainly unfolded as a result of bad models and metrics. Yet after the fact, certain narratives have gone back and ‘changed history’ so that we have a simple explanation based on bad guys and unsophisticated dynamics.
This final effect is much like the effect in quantum physics where some phenomenon can be understood as changing the past.
The way Black-Scholes deals with these logical softnesses is by introducing probability at the level of the logic. This confounds the causal model.
We don’t propose — as many do — that we use quantum physics as a basis for economic modeling. We think a deeper, more comprehensive and basic foundation is needed, one that addresses the three big problems with Black-Scholes inspired models:
- A model that federates ontologies at least of sequence, discreteness and ordinary continuity.
- A structured type system that is sensitive to introspective selforganizing systems.
- A notion of causal logic that takes advantage of new mathematical foundations in this area.
With this, we may do better in understanding and reasoning about how we coexist.
The Bottom Line
The bottom line of this lengthy note:
- We can reasonably pin the recent financial disasters — and coming ones — on the inadequacy of the way we measure and assign cause.
- These same inadequacies face us in engineering advanced collaborative enterprises, and must be addressed if we want a more fulfilling world.
- In our efforts, we roll these into a project to define a science base for advanced future enterprise engineering. This may be associated with the European cluster of projects called Future Internet Enterprise Systems. A note with some details on our approach will appear soon.