Chaos theory and the current financial crisis

Years ago, in a letter to the editor of Physics Today (February, 1979) we noted that the “market may be considered an open system in which an adequate flow of money will effect a transition from disorder (random walk) to order (cooperative or crowd behavior).” Open systems in the physical sciences require a flow of energy to maintain an ordered state far from thermal equilibrium. For example, a laser requires energy to be pumped continuously to maintain a coherent state. In the financial markets, price stability requires a flow of money or credit. In the Great Depression, credit became scarce as the bubble in stock prices unwound after the "Roaring Twenties." The current credit crisis involves the unwinding of the housing bubble and associated derivative securities.

We define a market “attractor” as a conditional return map, i.e. the average return on the day after a prior day return, R(T-1), that falls into one of five intervals:


small price changes [-0.5% < R(T-1) < +0.5%]
moderate price increases [+0.5% < R(T-1) < +3.5%]
moderate price declines [-0.5% > R(T-1) > -3.5%]
large price increase [+3.5% < R(T-1)]
large price declines [-3.5%] > R(T-1)]

Figure 1 summarizes the average conditional return map for the Dow Jones Industrial Average over the 80 year period from 1929 to 2009. A nonlinear third order polynomial fit is shown and illustrates that the market has been trend persistent on average over this period. The slope of the return map is positive in the region of moderate returns.


Figure 1. Over the past eighty years the Dow Jones Industrial Average has been governed on average by a coherent, trend persistent dynamic.

The conditional return map in Figure 1 illustrates a bistable attractor for the market. Moderate positive returns are followed on average by further positive returns. Similarly, moderate negative returns are followed on average by further negative returns. These drifts are toward dynamic equilibrium points (where the return map crosses zero) far from the market’s long term average daily return.

Next, we identify state transitions from a mean regressive market attractor to a bistable state attractor. First we define a bifurcation parameter as the sum over 200 days of conditional returns following moderate price increases (as defined above), minus the sum over 200 days of conditional returns after moderate price declines. In a mean regressive market, where the return map has a negative slope, this metric is negative whereas in a trend persistent market it is positive. The market attractor bifurcates as this measure crosses zero.

The bifurcation parameter is plotted in Figure 2. The most significant mean regressive markets occurred in the Great Depression era of the 1930s, though there were some wild swings in this indicator. From the 1940s to about 1980, the market was primarily in a trend persistent state and fluctuations of the bifurcation parameter were primarily around a positive mean. The further the bifurcation parameter deviates from zero, the better the opportunities for short term trading: in the Great Depression era a mean reversion strategy would have offered the best chance for success; from 1940 to 1980, a trend following strategy had the odds in its favor. However, more recently with the advent of computerized trading and negotiated commissions, the market has become more efficient, with less opportunity for trading.


Figure 2. The bifurcation parameter is negative in mean reverting market states and positive in trend persistent coherent markets, reflecting the slope of the conditional return map for moderate returns.


Figure 3 illustrates the market return map or attractor for market periods between 1929 and 2009 when the bifurcation parameter is negative. In this situation, moderate positive returns are followed on average by negative returns on the following day; moderate negative returns are followed by positive returns on average.


Figure 3. The market is mean regressive when the slope of the conditional return map and the bifurcation parameter are negative.

Recently the bifurcation parameter has dropped deeply into negative territory. This is an unusual development since this indicator hasn’t fallen this far since the Great Depression era. A short term trading strategy designed to profit from the market’s regression to the mean after moderate returns is appropriate in this market state. A strategy of avoiding equity positions entirely when the bifurcation parameter drops below -10% is illustrated in Figure 4. This straategy would have outperformed a buy and hold both in the Crash of 1929 and also successfully avoided much of the recent market meltdown. However, there is no assurance as to how it will work in the future particularly as it is based on a lagging indicator of market dynamics.


Figure 4. Avoiding mean reverting markets (negative bifurcation parameter) has shown profitable back testing results, but may not work in future markets.