Mortgage dynamics

My latest Bloomberg column should appear sometime Sunday night 7 April. I've written about some fascinating work that explores the origins of the housing bubble and the financial crisis by using lots of data on the buying/selling behaviour of more than 2 million people over the period in question. It essentially reconstructs the crisis in silico and tests which factors had the most influence as causes of the bubble, i.e. leverage, interest rates and so on.

I think this is a hugely promising way of trying to answer such questions, and I wanted to point to one interesting angle in the history of this work: it came out of efforts on Wall St. to build better models of mortgage prepayments, using any technique that would work practically. The answer was detailed modelling of the actual actions of millions of individuals, backed up by lots of good data.

First, take a look at the figure below:

This figure shows the actual (solid line) rate of repayment of a pool of mortgages that were originally issued in 1986. It also shows the predictions (dashed line) for this rate made by an agent-based model of mortgage repayments developed by John Geanakoplos working for two different Wall St. firms. There are two things to notice. First, obviously, the model works very well over the entire period up to 1999. The second, not obvious, is that the model works well even over a period for which it was not designed, by the data, to fit. The sample of data used to build the model went from 1986 through early 1996. The model continues to work well even out of sample over the final three years of this period, roughly 30% beyond the period of fitting. (The model did not work in subsequent years and had to be adjusted due to a major changes in the market itself, after 2000, especially new possibilities to refinance and take cash out of mortgages that were not there before.).

How was this model built? Almost all mortgages give the borrower the right in any month to repay the mortgage in its entirely. Traditionally, models aiming to predict how many would do so worked by trying to guess or develop some function to describe the aggregate behavior of all the mortgage owners, reflecting ideas about individual behavior in some crude way in the aggregate. As Geanakoplos et al. put it:

The conventional model essentially reduced to estimating an equation with an assumed functional form for prepayment rate... Prepay(t) = F(age(t), seasonality(t), old rate – new rate(t), burnout(t), parameters), where old rate – new rate is meant to capture the benefit to refinancing at a given time t, and burnout is the summation of this incentive over past periods. Mortgage pools with large burnout tended to prepay more slowly, presumably because the most alert homeowners prepay first. ...

Note that the conventional prepayment model uses exogenously specified functional forms to describe aggregate behavior directly, even when the motivation for the functional forms, like burnout, is explicitly based on heterogeneous individuals.

There is of course nothing wrong with this. It's an attempt to do something practically useful with the data then available (which wasn't generally detailed at the level of individual loans). The contrasting approach, seeks instead to start from the characteristics of individual homeowners and to model their behavior, as a population, as it evolves through time:

the new prepayment model... starts from the individual homeowner and in principle follows every single individual mortgage. It produces aggregate prepayment forecasts by simply adding up over all the individual agents. Each homeowner is assumed to be subject to a cost c of prepaying, which include some quantifiable costs such as closing costs, as well as less tangible costs like time, inconvenience, and psychological costs. Each homeowner is also subject to an alertness parameter a, which represents the probability the agent is paying attention each month. The agent is assumed aware of his cost and alertness, and subject to those limitations chooses his prepayment optimally to minimize the expected present value of his mortgage payments, given the expectations that are implied by the derivatives market about future interest rates.

Agent heterogeneity is a fact of nature. It shows up in the model as a distribution of costs and alertness, and turnover rates. Each agent is characterized by an ordered pair (c,a) of cost and alertness, and also a turnover rate t denoting the probability of selling the house. The distribution of these characteristics throughout the population is inferred by fitting the model to past prepayments. The effects of observable borrower characteristics can be incorporated in the model (when they become available) by allowing them to modify the cost, alertness, and turnover.

By way of analogy, this is essentially modelling the prepayment behavior of a population of homeowners as an ecologist might model, say, the biomass consumption of some population of insects. The idea would be to  follow the density of insects as a function of their size, age and other features that influence how and when and how much they tend to consume. The more you model such features explicitly as a distribution of influential factors, the more likely your model will take on aspects of the real population, and the more likely it will be to make predictions about the future, because it has captured real aspects of the causal factors in the past.

Models of this kind also capture in a more natural way, with no extra work, things that have to be put in by hand when working only at the aggregate level. In this mortgage example, this is true of the "burnout" -- the gradual lessening of prepayment rates over time (other things being equal):

... burnout is a natural consequence of the agent-based approach; there is no need to add it in afterwards. The agents with low costs and high alertness prepay faster, leaving the remaining pool with slower homeowners, automatically causing burnout. The same heterogeneity that explains why only part of the pool prepays in any month also explains why the rate of prepayment burns out over time.

One other thing worth noting is that those developing this model found that to fit the data well they had to include an effect of "contagion", i.e. the spread of behavior directly from one person to another. When prepayment rates go up, it appears they do so not solely because people have independently made optimal decisions to prepay. Fitting the data well demands an assumption that some people become aware of the benefit of prepaying because they have seen or heard about others who have done so.

This is how it was possible, going back up to the figure above, to make accurate predictions of prepayment rates three years out of sample. In a sense, the lesson is that you do better if you really try to make contact with reality, modelling as many realistic details as you have access to. Mathematics alone won't perform miracles, but mathematics based on realistic dynamical factors, however crudely captured, can do some impressive things.

I suggest reading the original, fairly short paper, which was eventually published in the American Economic Review. That alone speaks to at least grudging respect on the part of the larger economics community to the promise of agent based modelling. The paper takes this work on mortgage prepayments as a starting point and an inspiration, and tries to model the housing market in the Washington DC area in a similar way through the period of the housing bubble.

Approaching the singularity -- in global finance

In a new paper on trends in high-frequency trading, Neil Johnson  and colleagues note that:

... a new dedicated transatlantic cable is being built just to shave 5 milliseconds off transatlantic communication times between US and UK traders, while a new purpose-built chip iX-eCute is being launched which prepares trades in 740 nanoseconds ...

This just illustrates the technological arms race underway as firms try to out-compete each other to gain an edge through speed. None of the players in this market worries too much about what this arms race might mean for the longer term systemic stability of market; it's just race ahead and hope for the best. I've written before (here and here) about some analyses (notably from Andrew Haldane of the Bank of England) suggesting that this race is generally increasing market volatility and will likely lead to disaster in one form or another. 

We may be getting close. If Johnson and his colleagues are correct, the markets are already showing signs of having already made a transition into a machine-dominated phase in which humans have little control.

Most readers here will probably know about futurist Ray Kurzweil's prediction of the approaching "singularity" -- the idea that as our technology becomes increasingly intelligent it will at some point create self-sustaining positive feedback loops that drive explosively faster science and development leading to a kind of super-intelligence residing in machines. Humans will be out of the loop and left behind. Given that so much of the future vision of computing now centers on bio-inspired computing -- computers that operate more along the lines of living organisms, being able do things like self-repair and adaptation, true reproduction, etc, -- it's easy to imagine this super-intelligence again ultimately being strongly biological in form, while also exploiting technologies that earlier evolution was unable to harness (superconductivity, quantum computing, etc.). In that case -- again, if you believe this conjecture has some merit -- it could turn out ironically that all of our computing technology will act as a kind of mid-wife aiding a transition from Homo sapiens to some future non-human but super-intelligent species.

But forget that. Think singularity, but in the smaller world of the markets. Johnson and his colleagues ask the question of whether today's high-frequency markets are moving toward a boundary of speed where human intervention and control is effectively impossible:

The downside of society’s continuing drive toward larger, faster, and more interconnected socio-technical systems such as global financial markets, is that future catastrophes may be less easy to forsee and manage -- as witnessed by the recent emergence of financial flash-crashes. In traditional human-machine systems, real-time human intervention may be possible if the undesired changes occur within typical human reaction times. However,... in many areas of human activity, the quickest that someone can notice such a cue and physically react, is approximately 1000 milliseconds (1 second)

Obviously, most trading now happens much faster than this. Is this worrying? With the authors, let's look at the data.

In the period from 2006-2011, they found (looking at many stocks on multiple exchanges) that there were about 18,500 specific episodes in which markets, in less than 1.5 seconds, either 1. ticked down at least 10 times in a row, dropping by more than 0.8% or 2. ticked up at least 10 times in a row, rising by more than 0.8%. The figure below shows two typical events, a crash and a spike (upward), both lasting only 25 ms.



Apparently, these very brief and momentary downward crashes or upward spikes -- the authors refer to them as "fractures" or "Black Swan events" -- are about equally likely. And they become more likely as one goes to shorter time intervals:

... our data set shows a far greater tendency for these financial fractures to occur, within a given duration time-window, as we move to smaller timescales, e.g. 100-200ms has approximately ten times more than 900-1000ms.

But they also find something much more significant. They studied the distribution of these events by size, and considered if this distribution changes when looking at events taking place on different timescales. The data suggests that it does. For times above about 0.8 seconds or so, the distribution closely fits a power law, in agreement with countless other studies of market returns on times of one second or longer. For times shorter than about 0.8 seconds, the distribution begins to depart from the power law form. (It's NOT that it becomes more Gaussian, but it does become something else that is not a power law.) The conclusion is that something significant happens in the market when we reach times going below 1 second -- roughly the timescale of human action.

Ok. Now for the punchline -- an effort to understand how this transition might happen. In my last blog post I wrote about the Minority Game -- a simple model of a market in which adaptive agents attempt to profit by using a variety of different strategies. It reproduces the realistic statistics of real markets, despite its simplicity. I expect that some people may wonder if this model can really be useful in exploring real markets. If so, this new work by Johnson and colleagues offers a powerful example of how valuable the minority game can be in action.

Their hypothesis is that the observed transition in market dynamics below one second reflects "a new fundamental transition from a mixed phase of humans and machines, in which humans have time to assess information and act, to an ultrafast all-machine phase in which machines dictate price changes."  They explore this in a model that...

...considers an ecology of N heterogenous agents (machines and/or humans) who repeatedly compete to win in a competition for limited resources. Each agent possesses s > 1 strategies. An agent only participates if it has a strategy that has performed sufficiently well in the recent past. It uses its best strategy at a given timestep. The agents sit watching a common source of information, e.g. recent price movements encoded as a bit-string of length M, and act on potentially profitable patterns they observe.

This is just the minority game as I described it a few days ago. One of the truly significant lessons emerging from its study is that we should expect markets to have two fundamentally distinct phases of dynamics depending on the parameter α=P/N, where P is the number of different past histories the agents can perceive, and N is the number of agents in the game. [P=2^M if the agents use bit strings of length M in forming their strategies]. If α is small, then there are lots of players relative to the number of different market histories they can perceive. If α is big, then there are many different possible histories relative to only a few people. These two extremes lead to very different market behaviour.

Johnson and colleagues suggests that the transition between these regimes is just what shows up in the statistics around the one second threshold. They first argue that the regime for large α (many strategies per agent) should be associated with the trading regime above one second, where both people and machines take part. Why? As they suggest,

We associate this regime (see Fig. 3) with a market in which both humans and machines are dictating prices, and hence timescales above the transition (>1s), for these reasons: The presence of humans actively trading -- and hence their ‘free will’ together with the myriad ways in which they can manually override algorithms -- means that the effective number (i.e. α > 1). Moreover α > 1 implies m is large, hence there are more pieces of information available which suggests longer timescales...  in this α > 1 regime, the average number of agents per strategy is less than 1, hence any crowding effects due to agents coincidentally using the same strategy will be small. This lack of crowding leads our model to predict that any large price movements arising for α > 1 will be rare and take place over a longer duration – exactly as observed in our data for timescales above 1000ms. Indeed, our model’s price output (e.g. Fig. 3, right-hand panel) reproduces the stylized facts associated with financial markets over longer timescales, including a power-law distribution.

What they're getting at here is that crowding in the space of strategies, by creating strong correlations in the strategies of different agents, should tend to make large market movements more likely. After all, if lots of agents come to use the very same strategy, they will all trade the same way at the same time. In this regime above one second, with humans and machine, they suggests there shouldn't be much crowding; the dynamics here do give a power law distribution of movements, but it is what is found in all markets in this regime.

In contrast, they suggest that the sub one second regime should be associated the the α < 1 phase of the minority game:

Our association of the α < 1 regime with an all-machine phase is consistent with the fact that trading algorithms in the sub-second regime need to be executable extremely quickly and hence be relatively simple, without calling on much memory concerning past information: α < 1 regime with an all-machine phase is consistent with the fact that trading algorithms in the sub-second regime need to be executable extremely quickly and hence be relatively simple, without calling on much memory concerning past information: Hence M will be small, so the total number of strategies will be small and therefore... α < 1. Our model also predicts that the size distribution for the black swans in this ultrafast regime (α < 1) should not have a power law since changes of all sizes do not appear – this is again consistent with the results in Fig. 2.

And...

Our model undergoes a transition around α = 1 to a regime characterized by significant strategy crowding and hence large fluctuations. The price output for α < 1 (Fig. 3, left-hand panel) shows frequent abrupt changes due to agents moving as unintentional groups into particular strategies. Our model therefore predicts a rapidly increasing number of ultrafast black swan events as we move to smaller α and hence smaller subsecond timescales – as observed in our data.

 The authors go on to quantify this transition in a little more detail. In particular, they calculate in the simple minority game model the standard deviation of the price fluctuations. In the regime α < 1 this turns out to be roughly proportional to the number N of agents in the market. In contrast, it goes in proportion only to the square root of N in the α > 1 regime. Hence, the model predicts a sharp increase in the size of market fluctuations when entering the machine dominated phase below one second.

The paper as a whole takes a bit of time to get your head around, but it is, I think, a beautiful example of how a simple model that explores some of the rich dynamics of how strategies interact in a market can give rise to some deep insights. The analysis suggests, first, that the high frequency markets have moved past "the singularity," their dynamics having become fundamentally different -- uncoupled from the control, or at least strong influence, of human trading. It also suggests, second, that the change in dynamics derives directly from the crowding of strategies that operate on very short timescales, this crowding caused by the need for relative simplicity in these strategies.

This kind of analysis really should have some bearing on the consideration of potential new regulations on HFT. But that's another big topic. Quite aside from practical matters, the paper shows how valuable perspectives and toy models like the minority game might be.

Natural models of markets

I've written quite a bit about the shortcomings of traditional economic models of markets. Most notably, such models -- typically characterized by rational agents the actions of whom lead (by assumption) to a market equilibrium of some sort -- generally fail to explain basic dynamical features of real markets. These include, most prominently, 1) a pronounced tendency in all markets to large price fluctuations reflected in "fat tailed" distributions of returns, and 2) vigorous and persistent volatility with delicate long-memory features, which is closely linked to the way that episodes of volatility "cluster" in the market.

These phenomena show up in specific mathematical signatures in the statistics of price movements, and these signatures present some of the most obvious details that any decent model of markets ought to reproduce quite naturally. Models capable of doing this have only emerged in the past two decades, and only succeed by making a clear break with the neo-classical tradition of equilibrium. This short review from 2006 by economist Blake LeBaron gives a nice introduction to the motivation for this kind of work, which very much follows in the tradition of natural science (as opposed to much of modern economics) in pursuing explanations in models with make plausible assumptions and explore the conclusions which follow from them.

This is a topic I'll be writing about more in the next few months (in connection with a longer term project I'm working on). Le Baron's review is a little old (5 years) and needs updating, but he describes some crucial points very effectively. A few highlights are worth mentioning:

Le Baron first outlines some of what he calls the "major puzzles of financial markets," which are directly linked to the things I just mentioned. First is simply the existence of pronounced volatility:

Volatility is the most obvious and probably the most important puzzle in finance. Why do financial prices and foreign exchange rates move around so much relative to other macro series both on a short term and long term basis? The difficulty of overall financial volatility was first demonstrated in Shiller (1981), and an update is in Shiller (2003). The issue has been that it is difficult to find financial or macro economic fundamentals that move around enough to justify the large swings observed in financial markets. As a potential policy problem, and an issue for long range investors, this might be the most important puzzle faced by financial modelers.

As Le Baron notes, lots of models (ARCH, GARCH, etc) have been produced which reproduce the mathematical character of volatility but without even attempting to understanding its origin. Moreover, most models in traditional economics, by sticking to the view that individuals are more or less identical and have rational expectations, simply shy away from the very phenomenon needing to be modeled:

The persistence of volatility in many financial markets has led to an entire industry of models, and is an area of intense interest both in academic and commercial areas. However, although there is a lot of empirical activity, the underlying microeconomic motives for volatility persistence are still not well understood. There are very few models which have even tackled this problem. This is probably due to the fact that in a homogeneous agent framework this is simply a very difficult problem.

The second puzzle of markets that Le Baron lists is what I mentioned first above: the fat tails of market returns or  "excess kurtosis" in the statistical lingo:

Financial returns at relatively high frequencies (less than one month) are not normally distributed. There is not much of a strong theoretical reason that they need to be, but the hope has often been that some form of the central limit theorem should drive returns close to normality when aggregated over time. Recently, a new field, Econophysics, has appeared which stresses that returns also have additional structure that can be described using power laws. The determination and testing of power laws remains a somewhat open area, and the set of processes that generate acceptable power law pictures is also not well understood. 

Aside from the long memory of volatility and fat tailed returns, he also mentions the rich dynamics of trading volume, which is equally as interesting as those of prices. Unfortunately, when it comes to the dynamics of volume, Le Baron notes, "Most traditional financial models remain completely silent."

Le Baron rightly points out that these rich dynamics in time are almost certainly linked directly to two things: 1) the fact that different market participants have different expectations at any moment (indeed, it is such differences which drive trading) and 2) that these differences undergo perpetual evolution through time. This suggests that a decent explanation of the origin of these so-called "stylised facts" (and others like them) will likely emerge from models which attempt to follow and capture something about the dynamics of beliefs and expectations in a population of diverse interacting agents:

Many of the most puzzling results from finance deal with problems of behavioral heterogeneity, and the dynamics of heterogeneity. The study of market heterogeneity as a kind of complicated dynamic state variable that needs to be modeled is probably one of the defining features of agent-based models. Empirical features such as trading volume are directly related to the amount of heterogeneity in the market, and demand models that can speak to this issue. Other empirical features are probably indirectly related. Large moves, excess kurtosis, and market crashes all probably stem from some type of strategy correlation that keeps the law of large numbers from functioning well across the market. These changing patterns can only be explored in a framework that allows agent strategies to adapt and adjust over time...

I think this point about the failure of the law of large numbers is, while obvious, still worth emphasizing. Something in the market ruins the simple picture of normal statistics, which would emerge from independent factors driving prices changes. Somehow the actions of different market participants must come to be correlated, thereby leading to non-normal fat tailed returns and long memory effects. In principle, of course, there may be many mechanisms contributing to this lack of independence (some participants following trends could be enough, for example).

Now, I doubt anyone actually working in finance would find this observation anything but banal. Yet many economists seem determined to continue modelling markets as if this were not the case. I don't know enough economic history to know why the homogeneous rational expectations view has such a following, but it seems very weird to me indeed.

The rest of Le Baron's review explores an example of the kind of model -- an agent based model for a market very much out of equilibrium -- which reproduces the above features quite naturally, at least qualitatively. The idea is simply to respect the fact that agents in a market are different and change their behaviour over time, adapting to what happens in the market. There is no presupposition that market prices must settle down to an equilibrium; the market does what it does as people interact and trade and try to profit as well as they can. The simple model explored here is one developed by Le Baron in 2002, but shares basic features with many other models developed by others. Agents can choose between a risky asset (a stock) and a risk free bond, and they use a variety of information to make their trading strategies:

Agents chose over a set of portfolio strategies that map current asset market information into a recommended portfolio fraction of wealth in the risky asset. This fraction can vary from zero to one since short selling and borrowing are not allowed. Information includes lagged returns, dividend price ratios, and several trend indicators. Agents must evaluate rules using past performance, and it is in this dimension where they are assumed to be heterogeneous. Agents use differing amounts of past information to evaluate rules. In other words, they have different memory lengths when it comes to evaluating strategies. Some agents use 30 years worth of data, while others might use only 6 months. In this way this model implements to behavioral features. First, agents are clearly boundedly rational in that they do not attempt to determine the entire
state space of the economy, which would be unwieldy if they attempted this. Also, they are assumed to have “small sample bias” since they don’t all choose to use as much data as possible.

The paper is an easy read so I won't get into much detail, but what emerges from the interaction of learning and adapting agents in a setting like this is immediately much more realistic and interesting than anything coming from traditional models. For example, the figure below shows the price of the risky asset as a function of time. In the model, there is a true equilibrium price which is made to fluctuate in a normal, Gaussian way (this price being linked to dividends). The actual price in the market is rarely at this equilibrium, but instead has large fluctuations around it, being some times far too high and at others too low, and often moving very rapidly from one point to another. 



Now there are some features of this time series that don't look realistic. There seems to be a periodicity of sorts, for example. But this is a very simple model and the exciting thing is what it gets right -- easily giving a model of a market which never settles down to a prices, has persisting volatility, fat tails in returns, rich dynamics for trading volume and so on. Details can be found in the paper.
 
As Le Baron concludes the review,

Agent-based models make more progress than other frameworks in explaining these features due to that fact that at their core is a world of people who process information differently, and try hard to continually adjust and adapt their behavior over time. This market may never reach anything that looks like an equilibrium efficient market, but it is in a continual struggle toward this. The range of facts these models explain, and the robustness of their explanations to different structures and parameters, is impressive. At the moment, no other models can capture this many facts with this kind of simplicity and style.

As I said, that was 5 years ago. The final statement remains very much true, however. I think it is becoming increasingly clear, even to most economists, that agent-based models present currently the best technique to go beyond the restricting and unrealistic assumption of equilibrium economics and to build models of markets which get their basic behaviour right.

I intend this post just to be a beginning of an exploration of what is currently happening in the development of more realistic non-equilibrium market models. The field is currently exploding and there are many interesting questions to tackle. For example, models of this kind often involve quite a large number of parameters describing agents strategies and so forth. Yet many models work very similarly despite large differences in such parameters. An open question is whether it may be possible to develop a kind of map of the space of possible models, showing a set of classes into which different models fall. Physicist Luciano Pietronero has done some interesting work in this direction.

In any event, I find it hard to understand why anyone today would go on studying homogeneous rational equilibrium, at least in application to the financial markets.