Linking agent-based models and stochastic models of financial markets

0 · Context

Two ways of modelling prices, each missing what the other has

Two empirical facts about stock returns are known to hold across markets and decades. Large price changes are far more common than a Gaussian bell curve allows: the probability that a daily return exceeds $x$ falls off as a power law, $P(|r|>x)\sim x^{-\xi}$, with a tail exponent $\xi$ of about 3 to 4 (a fat tail). And the size of price changes has a long memory: a turbulent day is followed by turbulent days, and the correlation of absolute returns decays slowly, as a power of the time lag, over months.

Two traditions model these facts. Stochastic models, starting with Engle’s ARCH and Bollerslev’s GARCH processes, let the variance of tomorrow’s return depend on past returns. They reproduce the statistics accurately, but their parameters are fitted to the data and say nothing about why traders produce them. Agent-based models, from statistical physics and economics, simulate individual traders and give a qualitative picture of where the regularities come from. But they rarely match the data quantitatively, and in many of them the fat tails fade away when the number of agents grows unless the interaction strength is re-tuned.

In our paper we combine the two: an agent-based model built only from documented trading behaviour, a stochastic model derived from it, and parameters that are measured in markets and surveys instead of fitted.

1 · Evidence

Technical traders, not fundamentalists, drive daily price changes

Fundamentalists try to work out what a company is worth. They hold most of the shares, institutional owners held 83 % of the outstanding shares on average in our data, but they trade rarely. Technical traders instead make decisions from price patterns, and they trade their smaller holdings much more often, so they account for most of the daily trading. We therefore model only technical traders and treat fundamentalists as background noise. The data end in 2006, before high-frequency trading became significant.

The model rests on four documented features of their behaviour:

1. Daily decisions that look random. Each trader uses a different strategy, so from outside, the decision to buy, sell or hold appears random. Decisions are made once a day, because intraday trading shows little persistence.
2. Price change follows the order imbalance. The price moves with the difference between buy and sell orders on a day; the largest move happens when everyone buys or everyone sells.
3. Interaction through the price. Technical traders all read the same input, the past price movement, so the price itself couples them. The coupling then grows with the number of traders and needs no network of who talks to whom.
4. Opinions converge after large moves. Traders copy the most profitable strategy. After a small price change, differences in their strategies (timing, risk tolerance, portfolio) dominate and they act independently. After a large one, such as a crash, they act in concert regardless of those differences, as in the spreading panic of a sell-off. This is the mechanism that distinguishes our model from others.

2 · Model

Three steps, and a large price move today merges traders into fewer groups tomorrow

The agent-based model has $n_0$ traders, each trading one unit. Every day it runs three steps.

The trading probability $p$ is measured, not fitted. For the 309 companies that stayed in the S&P 500 throughout 1997–2006, a share changed hands $V=1.64$ times a year on average (the trading velocity, between 1.3 and 1.9 over the decade). Taking institutional owners, who held 83 % of the shares, as fundamentalists who trade less than once a year (velocity $V_f<1$), the technical traders’ velocity is $V_c=(V-0.83\,V_f)/(1-0.83)$, and with about 250 trading days a year, $p=V_c/(250\cdot 2)$. For $V_f=0.2$ to $0.8$ this gives $p=0.0174$ to $0.0115$; with $V_f=0.4$, $p=0.0154$ and technical traders make about 80 % of the volume.

3 · Fat tails

The model matches the S&P 500 in returns and in the number of trades, for any number of agents

We compared simulated returns with about 800,000 daily returns of the 309 S&P 500 stocks (open to close, so overnight news is excluded, each stock normalized to zero mean and unit variance). Both the distribution of absolute returns and the distribution of the daily number of trades agree with the data over the whole range, centre and tail. The Hill estimator gives tail exponents $\xi_r=3.69\pm0.07$ (S&P 500) and $4.08\pm0.08$ (simulation) for returns, and $\xi_n=4.02\pm0.07$ and $4.02\pm0.08$ for the number of trades. Capturing both of these highly correlated quantities at once makes the mechanism plausible.

The exponent $\omega$ in Eq. 3 sets how strongly a price move merges opinions, and $\omega=1$ is the value closest to the data:

The tail exponent also does not depend on the number of agents or on the fundamentalists’ trading velocity: it stays between 3.66 and 3.89 for $n_0$ from $2^8$ to $2^{14}$ and $V_f$ from 0.2 to 0.8 (Table 2 of the paper). Most agent-based models lose their fat tails as the population grows; here the interaction runs through the price, so its strength grows with the population by construction.

Fat tails need no tuning: they come from traders who act together after a large move.

4 · Stochastic model

Averaging over traders turns the behaviour into an equation for the volatility

Step 3 makes tomorrow’s price change random, with a variance set by today’s move. On average each of the $c_t=n_0/|r_{t-1}|$ groups has $|r_{t-1}|$ members and moves the price by $\pm|r_{t-1}|$ with probability $p$ each, so

$$\sigma_t^2 \equiv E\!\left(r_t^2 \mid r_{t-1}\right) = 2p\,n_0\,|r_{t-1}| .$$

(8)

Writing $r_t=\sigma_t\eta_t$ with $\eta_t$ a random number of zero mean and unit variance (we use a normal distribution; a $t$-distribution gives similar results) gives a time-dependent variance of the kind modelled by ARCH processes. Here it has a behavioural reading: a large price change brings traders’ opinions closer, and that produces large price changes afterwards.

Equation 8 remembers only yesterday. Two more pieces of evidence extend it. (v) Technical strategies are applied at different horizons, from one day to more than a year, and the survey of US fund managers in Fig. 2 shows that the share $\alpha_i$ of traders with horizon $i$ days decays as $\alpha_i\propto i^{-d}$ with $d\approx1.12$. (vi) Traders trade more after large moves, because their strategies set price thresholds, so the trading probability rises with past price changes: $p_{t+1}\approx p_0+\alpha'\Sigma_t$, with $p_0$ the background activity, mostly of fundamentalists.

A trader with horizon $i$ reacts to the price change over the last $i$ days, $|s_t-s_{t-i}|$, where $s_t$ is the log price. Opinion convergence then depends on a weighted sum over horizons up to a maximum $M$,

and with both extensions, keeping the dominant term, the volatility becomes

Here $\alpha=\big[\sum_{i=1}^{M}i^{-d}\big]^{-1}$ normalizes the horizon weights. $A$ sets the typical size of returns and $B$ the share of trading triggered by technical thresholds relative to background trading. The form resembles fractionally integrated volatility models, with one difference: $\sigma_{t+1}$ depends on absolute aggregate returns over past days, weeks and months rather than on past daily returns, and each term has a behavioural meaning.

5 · Long memory

One survey exponent fixes how slowly the market forgets its volatility

Long memory shows up in the autocorrelation function (ACF) of absolute returns, $\rho_\ell\propto\ell^{-\gamma_1}$, and of squared returns, $\rho'_\ell\propto\ell^{-\gamma_2}$, at lag $\ell$ days. From Eq. 11 we derive scaling relations between these exponents and the survey exponent $d$, much like the relations between critical exponents at a phase transition:

$$\left\{\begin{aligned} &\gamma_1=2d-2,\\ &2d-2\le\gamma_2\le4d-4,\\ &\gamma_1\le\gamma_2\le2\gamma_1 . \end{aligned}\right.$$

(12)

Ding, Granger and Engle had observed that absolute returns stay correlated longer than squared returns. Eq. 12 makes this quantitative and ties it to the behaviour of traders through $d$.

To simulate, every parameter comes from data: $M=500$ trading days (about two years, the longest horizon in the survey), $d=1.2$ in line with the survey, and $A=0.002$, $B=0.05$ from trading-volume fluctuations and the unconditional variance of price changes. So $\sigma_{t+1}\approx0.002+0.05\sum_{i=1}^{500}i^{-1.2}|s_t-s_{t-i}|$. Simulation and the S&P 500 index (daily closes 1971–2010) give similar ACFs and return distributions (Fig. 4), with $\gamma_1=0.40$, $\gamma_2=0.53$ for the simulation and $\gamma_1=0.44$, $\gamma_2=0.7$ for the index. Eq. 11 is still a stationary process, unlike many other power-law memory processes.

Long memory in volatility is the trace of traders who watch different horizons, mostly short ones.

6 · Summary

Behaviour in, stylized facts out, parameters measured

The agent-based model singles out the dominant mechanism; the stochastic model allows a detailed analysis. Both share the same mechanism and both take their parameters from market data with a clear behavioural meaning. The study covers data before 2006, ignores high-frequency trading, and treats fundamentalists only as background noise.