Theories and Models

There is nothing so terrible as activity without insight.
— Johann Wolfgang von Goethe (“Maxims and Reflections”, 1892)

The world is impossible to grasp in its entirety. The human mind can focus on only a small part of its vast confusion. Models project detailed and complex world onto a smaller subspace, where regularities appear and then, in that smaller subspace, allow us to extrapolate and interpolate from the observed to the unknown. At some point, of course, the extrapolation will break down. But this strategy of reduction works very well in the physical sciences. Models in finance, by extension, use the same strategy in the hope that some of its magic would rub off nicely.

The aim of finance, like that of physics, is to find not only the relationships between the abstractions themselves, e.g., markets, money, assets, securities, but also the relationships between the realities they represent. In both physics and finance the first major struggle is to gain some intuition about how to proceed; the second struggle is to transform that intuition into something more formulaic, a set of rules anyone can follow, rules that no longer require the original insight itself. One person’s breakthrough thus becomes everybody’s possession.

The Efficient Market Hypothesis imagines price movements to be a diffusion process, i.e., a random walk. One of its origins is in the description of the drift of pollen particles through a liquid as they collide with its molecules. Einstein used the diffusion model to successfully predict the square root of the average distance the pollen particles move through the liquid as a function of temperature and time, thus lending credence to the existence of hypothetical molecules and atoms too small to be seen.

For particles of pollen, the model is also a theory, and pretty close to a true one. For stock prices, however, it is only a model. It is how we choose to imagine the the way changes in stock prices occur, not what actually happens. Models are simplifications, and simplification can be dangerous. It is naïve to imagine that the risk of every stock in the market can be condensed into just one quantity, its volatility σ. Risk has too many aspects to be accurately captured by that one number. In short, the Efficient Market Model’s price movements are too constrained and elegant to reflect the market accurately. After all, the movements of stock prices are more like the movements of humans than of molecules.

Model parameters that are implied from market prices are often easier to have an intuition about than are the market prices themselves, especially if the model is itself intuitive. For example, being told that an option has a particular price means nearly nothing, but being told that an option has a particular implied volatility gives a sense of meaning to it, something that can be pondered, something on which an opinion could be formed and a trade proposed. This is the power of model parameters. The idea is not that the model is correct, or that the assumptions can never be violated, but simply that the model is useful in explaining the risks. The parameters help the intuition.

We need models to explain what we see and to predict what will occur. We use models for envisioning the future and influencing it. The world of people is unpredictable and begs for divination as well. At every moment we face choices with uncertain outcomes. Each decision, even one made on the spur of the moment involves some imagined model for how the future may evolve and how our choices will affect it. We are always weighing the odds, estimating the relative importance of causality and chance. As time passes, possibilities narrow. Because our lifetime is finite, time, choice, risk, and reward are of the essence. Unless one can live in the perpetual present, one needs theories and models to exert some control. Theories and models are thus a kind of magic that bridges the visible and invisible worlds.

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Models are analogies; they always describe one thing relative to something else. Models need a defense or an explanation. Theories, in contrast, are the real thing. The need confirmation rather than explanation. A theory describes an essence. The abstractions of mathematics are often more suitable than words for formulating theories. A successful theory can become a fact, i.e., by describing the object of its focus so accurately that the theory becomes virtually indistinguishable from the object itself. The creator of a theory is attempting to discover the invisible principles that hide behind the appearances. The role of theory is to make evident what is hidden. Unlike models, a theory doesn’t simplify. It observes the world and tries to describe the principles by which the world operates.

It takes hard work to master a model. But models splinter when you look at them closely. Theories are irreducible, the foundations on which new metaphors can be built. But a theory doesn’t have to be complete or unmodifiable. There are theories that are not exactly right, but they are not models. Theories are the thing itself; when you look closely, there isn’t anything more to see. The surface and the object, the outside and the inside, are one.

The similarity of physics and finance thus lies more in their syntax than their semantics. For example, financial modelers use a process similar to renormalization in physics to force their less than perfect, less than real models to fit the world they observe. They call this process calibration, the tuning of parameters in a model until it agrees with the observable prices of liquid securities whose values we know. But calibration in finance works much less well than renormalization in physics: in physics the normal and abnormal are governed by the same laws, whereas in markets the normal is normal only while people behave conventionally. In crises the behavior of people changes and normal models fail. While quantum electrodynamics is a genuine theory of all reality, financial models are only mediocre metaphors for a part of it. Financial models, because of their incompleteness, inevitably mask risk. When you use a model you are trying to shoehorn the real world into a container too small for it to fit perfectly.

In human affairs, history matters, and people are altered by every experience. But it’s not only the past that leaves its trace on humans. In physics, effects propagate only forward through time, and the future cannot affect the present. In the social sciences the imagined future can affect the present, and thereby the actual future, too. Despite this, the Efficient Market Model assumes that all uncertainties about the future are quantifiable. It claims that at any instant current prices reflect all current and past information, and that the best estimate of value is the current price. That’s why it is a model of a possible world rather than a theory about the one we live in.

In finance, a useful guiding principle is the Law of One Price: If you want to know the value of one financial security, your best bet is to use the known price of another security that’s as similar to it as possible. When we compare it with almost everything else in economics, the wonderful thing about this law of valuation by analogy is that it dispenses with utility functions, the undiscoverable hidden variables whose ghostly presence permeates economic theory. The Law of One Price, however, is not a consistent law of nature. It is a general reflection on the practices of fickle human beings, who, when they have enough time, resources, and information, would rather buy the cheaper of two similar securities and sell the richer, thereby bringing their prices into equilibrium. The law usually holds in the long run, in well-oiled markets with enough savvy participants. In crises, however, duress forces people to behave in what looks like irrational ways, and even in normal time there are persistent shorter- or longer-term exceptions to the law.

To use the Law of One Price that underpins financial modeling, one simply shows that a target security and its replicating portfolio have identical future payoffs under all circumstances. Most of the mathematical complexity of modeling in finance involves the description of the range of future behavior that composes all circumstances. One can easily invent more complicated models of risky stock prices that incorporate violent moves and ferocious outbursts of risk. But in using such models one gives up simplicity for a still imperfect but more complex model that doesn’t necessarily do better.

As with earthquakes, it may be wiser to ensure that one owns a portfolio that will not suffer too badly under disastrous scenarios than it is to try to estimate the probability of destruction. When models in physics fail, they fail precisely, and often expose a paradox that opens a door. When models in the social sciences fail, they fail bluntly, with no hint as to what went wrong and no clue as to what to do next. Financial models are always metaphors.

 How finance is fundamentally different from  physics.

How finance is fundamentally different from  physics.

Financial modeling is not the physics of markets. Physics models begin with the current state of the world and evolve it into the future. Financial models begin with the current perception about the future and use them to move back into the present to estimate current values. And it is humans doing the perceiving. In other words, financial models don’t forecast; they simply transform one’s forecasts of the future into present value. The point of a model in finance is not the same as the point of a model in physics. In physics one wants to predict or control the future. In finance one wants to determine present value and goes about it by forming opinions about the future, about the interest rates or defaults or volatilities or housing prices that will come to pass. One uses a model to turn those opinions about the future into an estimate of the appropriate price to pay today for a security that will be exposed to that imagined future.

Overall, models are useful in finance and here are some of their major benefits: (i) models facilitate interpolation; (ii) models transform intuition into a dollar value; (iii) models are used to rank securities by value. However, to confuse a model with a theory is to believe that humans obey mathematical rules, and so to invite future disaster. Therefore, financial modelers must compromise. They must decide what small part of the financial world is of greatest current interest to them, describe its key features, and then mock up those features only. A successful financial model must have limited scope and must work with simple analogies. In the end you are trying to rank complex objects by projecting them onto a scale with only a few dimensions.

A good model can advance fashion by ten years.
— Yves Saint Laurent

References:

  1. Derman, Emanuel (2011). Models. Behaving. Badly. Why Confusing Illusion with Reality can Lead to Disaster, on Wall Street and in Life. Free Press.
  2. Derman, Emanuel (2013). The Young Person’s Guide to Pricing and Hedging. Retrieved from: http://www.emanuelderman.com/media/The_Young_Persons_Guide.pdf
  3. Maymin, Philip Z. (2012). Financial Hacking: Evaluate risks, price derivatives, structure trades, and build your intuition quickly and easily. World Scientific.