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Risk Professional, Dec/Jan 1999
Richard B. Hoppe
In my previous essay in this magazine (Risk Professional, October 1999, page 18), I argued that current mathematical models of markets are inappropriate for the tasks they are set in finance. I gave three related reasons. First I argued that real markets do not meet the assumptions of the mathematics that are typically used to model them, and therefore the maths is not a veridical representation of market behavior.
Second, I argued that markets are historical systems. The route by which a market reaches its current state is critical to understanding that state and anticipating likely future states. However, the maths typically used to model market behavior stochastic statistics and differential equations is ahistorical. It either discards history altogether by casting returns into timeless frequency distributions or trivialises it by using time-reversible equations. Even the various species of GARCH models are typically sensitive to only one aspect of a markets past, its recent volatility -- they differ only in how volatility is taken into account in otherwise ahistorical distributional models.
I asserted that in order to model markets, one must have a substantive theory of markets. To draw a map of a territory one must first know what the territory looks like. Once one knows that, once one has a working theory of the terrain to be mapped and knows the purpose of the map, then one can decide on the mapping conventions the model -- that one will use to represent it.
A substantive theory plays several roles. It guides ones choice or invention of appropriate modeling technologies. It provides a basis for determining which properties of markets must be modeled and which may be safely disregarded. It justifies the mapping of market entities and processes into the medium provided by the modeling technology. It provides interpretations of the results of manipulations of the model, allowing one to translate the models behaviour into meaningful statements about the behaviour of the real market system. In short, a substantive theory gives meaning to the model and justifies its use to support action in the real world.
This essay addresses the concerns I raised by sketching a set of substantive theoretical assumptions that in my opinion are fundamental to creating plausible and useful models of financial markets. It provides the skeleton of a theory of market behaviour that can support the creation of models to guide effective actions in real markets. It does not describe modeling technologies as such. It is intended to stimulate thinking and research, not provide a cookbook.
This is only a sketch for two reasons. First, I have neither the time nor the inclination to write a book. Second, I am under legal and ethical constraints. My firm is using the general approach described here to create novel methods of modeling market systems to inform risk estimation and to drive trading, and I cannot disclose proprietary material.
My conception of market systems grows out my background in cognitive psychology and the literature of complex adaptive systems. Taken as a group, books by Ruelle (1991), Simon (1996), Kauffman (1993), and Holland (1995) describe the problems associated with theorizing about and modeling complex adaptive systems very well.
Defining a Market
In this essay I refer to markets as though they were entities. That is shorthand. I regard a market as a collection of people who interact by regularly trading with each other and occasionally with outsiders in a common geographic and/or abstract communication space. Market participants attend to information, process that information, make decisions in the light of their goals, knowledge, and expectations, and take actions to buy or sell a particular instrument or small set of closely related instruments.
A market is the aggregation of those participants. Its properties are generated by the modal behaviours of the participants. A market is in some ways analogous to a biological species an inter-breeding (trading) population. Like a species, a market evolves through time in response to environmental pressures and as its participants learn new information processing, decision-making, and behavioural rules.
There are some obvious measures of the aggregate behaviour of markets, most notably price and volume time series data. Those time series are manifestations of the collective behaviour of the markets participants.
The behaviour of a market depends on the emotional, cognitive and behavioural characteristics of its participants. Those characteristics put boundaries on the aggregate behaviour that a market can display. The microstructure of a composite system imposes constraints on properties of the composite system. Knowledge about the components of a system informs theories of the behavior of the system. It can also inform the model building enterprise.
There are three levels of analysis to take into account in analyzing market behavior participants, individual markets, and the market system as a whole. Constraints and properties propagate from participants through individual markets to the market system. Each level depends on, and is constrained by, properties of the levels below it in the hierarchy.
Fundamental Assumptions
The theory sketch consists of a set of fundamental assumptions. I will not give justifications for them here. Some seem to me to be obvious. Others are not so obvious but seem necessary given the relevant properties of market participants. They can be regarded as hypotheses or working assumptions.
The central assumption is that the major financial markets form a completely coupled (but not closed) interacting system, with a pair of connections linking each market with every other market in the system, forming a web of doubly linked nodes. The two links connecting any pair of markets, one link pointing in each direction, are not assumed to be symmetrical in transmission properties. Indeed, they are almost certainly asymmetrical. The links between markets carry information, or influence, such that changes in the behavior of one market can affect, to a greater or lesser degree, the behavior of all the other markets. The market system as a whole is open. It is subject to exogenous influences: the system has permeable boundaries.
I assume that the connections between markets are nonlinear transmission links in the sense that the transfer function of any connection may be characterized by variable time delays and nonlinear transformations of the information carried on the link. This is equivalent to assuming that the response of the receiving market is complexly nonlinear in the input from another market. As a pragmatic matter I regard the transmission links as embodying the complexity because that is consonant with the particular modeling technologies my firm has developed.
I assume that symbolic equations to describe the transfer functions of the links are presently unknown, change through time, and even if the equations were known they would not converge to a unique solution.
I assume that market participants learn they change their modal information processing methods and decision-making rules in the light of experience. As participants collectively learn, both the transmission properties of the links among markets and each markets response to its inputs change. As a consequence, the market systems gross behaviour changes through time. The interactions within the market system and each markets responses to its inputs change over time.
I assume that the evolution of the market system through time is historically contingent. Its temporal course or trajectory is what it is by virtue of its previous history, and that given a different history, the course of its evolution from any particular state to possible future states would be different. In other words, an adequate description of the systems current state must include reference to how it got to that state.
The next two assumptions seem necessary to proceed with the modeling enterprise in an applied context. They are not purely ad hoc -- we have some empirical evidence for them.
I assume that while the abstract topography of the systems linking dynamics changes through time, the rate of change is usually fairly slow and there are periods of relative stability with a given dynamical regime persisting for single- or double-digit months. For a given subset of the system, a dynamical regime might even last for several years.
Finally, I assume that there are boundaries on the permitted dynamical regimes of the market system. While a particular regime will not last forever, there is a limited range of potential ways in which the markets can interact. A given segment of market system history may not precisely repeat itself, but the range of possibilities is not unlimited and dynamical regimes may be revisited or nearly revisited.
Implications for Market Models
A number of implications for modeling a market system flow directly from the assumptions. I will suggest just a few here. First, one cannot adequately model the behaviour of a single market in isolation from the larger market system in which it is embedded. In order to understand the behaviour of a target market of interest it is necessary to know (at least) the pattern of influences it receives from other markets. Therefore, an adequate market modeling technology must be capable of representing the system as a whole, or at least significant subsets of it.
Second, the number of links increases approximately as the square of the number of markets in the system. Modeling could get expensive in computational terms and unwieldy to interpret if one cannot somehow limit either the number of markets under analysis or the complexity of the linking system. Pruning is a practical necessity, and how the pruning is accomplished will affect the veridicality and utility of the model that results. If the market system is nearly decomposable in Simons (1996) sense, as I believe it is, judicious pruning is less damaging to a models utility than if it is not.
Third, one must reject static models. Since market dynamics change through time as the external environment changes and as participants learn from experience, the specific form of the model that is appropriate at one time will be inappropriate at some future time. Therefore the modeling technology must be able to change, adapting its properties as the real market system evolves. There must be provision for continuous interplay between the model and the real world. Because market systems evolve, market models must learn.
Fourth, because the links among markets form complex nonlinear feedback loops with variable time delays, the behaviour of the system as a whole in all likelihood is chaotic in the technical sense of the word. This means that a models predictions will deteriorate as time horizons lengthen. Given that the market system is open and subject to exogenous shocks, it also means that any theory or model that assumes or implies that there is an equilibrium state is doomed to academic irrelevance.
Fifth, because history is crucial, a models representation of the current state of the system must include reference to its past, to how it got to its current state. To know a vector composed of todays returns across the market system is not sufficient to characterize the current state of the market system. One must know how that vector follows from the history of behaviour that led to today.
Finally, because I assume that the market system can revisit dynamical regimes, a model that is able to represent and remember past patterns of dynamics can query its memory and compare todays regime with past regimes and make predictive inferences from the systems behaviour during past regimes that are similar.
Some Candidate Modeling Technologies
Two closely related approaches to modeling a market system will immediately come to the minds of economists and financial physicists given the assumption that a market system consists of a network of dynamically linked nodes. One is the orthodox econometric approach, which can involve creating massive systems of partial differential equations. The other is a physics-based approach that looks at the web of markets as analogous to an array of coupled oscillators or similar sorts of physical systems. Riccardo Rebonato, in his response (RP 1/6) to my essay Its Time We Buried Value at Risk, described that sort of physical system when he wrote of deriving a model of molecular interactions from quantum mechanics. (Incidentally, Rebonatos anecdote is irrelevant to market modeling because there is nothing analogous to quantum mechanics from which to borrow.)
I reject those sorts of candidate models because they do not adequately represent the evolving transfer functions that characterize the links among markets. They offer no obvious or natural adaptive properties. Adaptation, if it is present in such models at all, must be pasted on as an ad hoc adjunct; it is not an intrinsic property of the modeling technology.
I also reject statistical models that depend on time-independent distributions. Collapsing a time series into a frequency distribution (typically interpreted as a probability density function) discards history and therefore throws away critically important information about the system. Trying to understand a market systems behaviour based on a frequency distribution of returns is like trying to understand the rules of basketball by plotting a frequency distribution of the running speed, sampled at one-minute intervals, of one player on the court.
Simulation models can provide a more fruitful route to understanding market systems. Rather than trying to write a set of symbolic equations to describe the behaviour of ant colonies in the wild, I think we need something analogous to an artificial ant farm. That is, we need models that are analogue representations of the important components, processes, and relationships defined by our theory rather than abstract digital symbolic representations. Representation is a critical ingredient of creative problem solving. The way one represents a domain is a strong determinant of the kinds of solutions that one can invent.
Given the theoretical assumptions we can specify some properties that a simulation model of the market system must have. It must be structured as an interactive multi-component system. One way to conceptualize it is as a web of intelligent (market) nodes, the nodes connected by the pairs of links I described above. Each node must know its current mode(s) of responding to the array of inputs it receives. Each node must have a memory in which to store its history of interactions with other nodes. Each node must be capable of acquiring new modes of behaviour based on feedback from its tracking of the real market system it must be able to learn.
The absolutely crucial aspect of the simulation model is how it represents the connecting links among markets. It must have a way of representing the links that can encode changing nonlinear transfer functions and that can adapt as the dynamical topography of the real market system changes.
There are various technologies that may be able to provide models with the properties we need. Genetic algorithms, neural nets, adaptive agents, and adaptive fuzzy logic expert systems may have potential utility. Mapping the market system as I have conceived it into some combination of them is not easy or straightforward, but it is not in principle impossible. It is also possible to invent novel models and (especially) novel representation technologies given the guidance provided by the theoretical assumptions.
References
Holland, John H. (1995) Hidden Order. Cambridge, MA: Addison-Wesley.
Kauffman, Stuart A. (1993) The Origins of Order. New York: Oxford University Press.
Ruelle, David (1991) Chance and Chaos. Princeton, NJ: Princeton University Press.
Simon, Herbert A. (1996) The Sciences of the Artificial, Third Edition. Cambridge, MA: The MIT Press.
Richard Hoppe spent the 1960s in the aerospace and defence industry before becoming professor of psychology at Kenyon College in Ohio in the 1970s and 1980s. He has been associated with the trading industry since 1990 as a consultant in artificial intelligence and as a trader. Richard is currently a principal at IntelliTrade, a market risk decision support firm, and can be contacted at itrac@itrac.com.
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