Understanding Market Crashes Through the Lens of Physics
Written on
Chapter 1: Historical Context of Market Crashes
Throughout history, numerous market crashes have occurred, such as the infamous 1929 crash, often referred to as the "Great Crash." The following figure illustrates the Dow Jones Industrial Average's decline during this tumultuous period:
Another significant event was the "Black Monday" of 1987, which marked a pivotal moment in financial history. This chart displays the Dow's fluctuations from June 1987 to January 1988:
More recently, fears surrounding the coronavirus outbreak resulted in a significant market downturn, reminiscent of the 2008 financial crisis:
Market crashes are rare occurrences. As demonstrated through the work of Paul and Baschnagel, these events happen far more frequently than traditional Gaussian or Lévy distributions would suggest. The graphic below compares these two statistical models:
When analyzing market behavior, it can be categorized into distinct phases:
- A standard trading phase with minimal price fluctuations.
- A build-up phase leading to a crash, marked by pronounced correlations in price movements.
This transformation in market behavior can be likened to a phase transition in physics, where specific points witness significant changes in physical properties.
Section 1.1: Analyzing the Properties of Market Crashes
As previously discussed, the distribution of stock returns aligns more closely with a Lévy distribution than a Gaussian one, providing better predictive capabilities for rare events such as crashes. It is reasonable to assume that a crash arises from extreme fluctuations at the tails of this distribution:
Following the findings of Paul and Baschnagel, we simulate stock prices using a truncated drift-free Lévy flight. This process can be described mathematically:
Equation 1: Truncated Lévy distribution.
In this equation, N signifies a normalization constant. The figure below depicts both the truncated Lévy distribution and the standard Lévy distribution for comparison:
According to Stanley and Mantegna, the main body of this distribution can be approximated by a Lévy distribution with an exponent of α = 1.4 for fluctuations less than 6σ, where σ is the standard deviation. The increments can be produced using the algorithm outlined in this paper:
Equation 2: Drift-free Lévy flight.
If the distribution maintains symmetry with an exponent other than 1, it can be expressed as follows:
Equation 3: The Lévy flight for a symmetric distribution.
The definition of a maximum excursion in a stochastic process is crucial for understanding extreme events:
Equation 4: The maximum value of a sample path x(t).
To evaluate the distribution of maximum excursions, we must establish a stopping time T. If we assume that the process ceases when x(t) first reaches some value a, we can define:
Equation 5: The stopping time for the process x(t).
The probability density function of the maximum, or maximal excursion, for standard Brownian motion is:
Equation 6: Probability density of the maximum M(t) for standard Brownian motion.
This insight is vital for assessing risks in financial markets.
Chapter 2: Drawdowns and Market Dynamics
Extreme events can also be analyzed through the concept of drawdowns, defined as the percentage decline from a relative peak to the subsequent trough in price data. This method offers a solution to the fixed time horizon issue encountered when analyzing extreme excursions:
In the following figure, we observe two distinct regimes in the DJI. For drawdowns less than 15%, Paul and Baschnagel's model fits well to an exponential distribution. However, when examining drawdowns around the 30% mark, the frequency in the DJI is approximately ten times higher compared to simulation data, which suggests a greater likelihood of significant drawdowns:
Section 2.1: The Concept of Regime Change
The data leads to a hypothesis that these rare events do not merely represent outliers but signify a transition between two distinct regimes: one characterized by "normal" dynamics and another marked by extreme fluctuations.
Consider the daily closing prices of the DJIA leading up to the crashes of 1929 and 1987:
As observed, the price series steadily rise in increments before both crashes, indicating a pattern where the intervals between increments decrease as the crash approaches.
The impact of these price drops suggests that they are not merely rare fluctuations within 'normal' price series.
Section 2.2: The Role of Cooperativity and Criticality
As a crash nears, a substantial number of traders may decide to sell, leading to increased market liquidity issues. This collective behavior fosters correlations among traders, which can precipitate a crash.
In thermodynamics, similar phenomena occur when systems approach critical points. This article connects market behavior with thermodynamics through simple models that illustrate this relationship.
The Cont-Bouchaud Model
This model considers a scenario with N(t) traders, each with an average trading volume. A trader can either buy, sell, or remain inactive. The resulting supply-demand difference is expressed as follows:
Equation 7: Supply-demand difference.
Two assumptions underpin this model:
- Price changes are proportional to the supply-demand imbalance.
- The proportionality constant indicates the market's susceptibility to supply-demand variations.
Interactions among traders occur directly and indirectly through price observations. This interconnectedness fosters a bond between traders, which can be quantified:
Equation 8: Bond creation probability.
Traders form clusters based on their trading strategies. The probability distribution of these clusters can be expressed as:
Equation 9: Sum rewritten in terms of clusters.
Utilizing principles from percolation theory, we derive the cluster-size probability distribution, which is valid under specific conditions:
Equation 10: Cluster-size probability distribution.
When the cluster size exceeds a critical threshold, a percolation effect occurs, leading to a shift in market dynamics. This critical behavior can result in significant price drops if a majority of traders decide to sell.
Physics often involves systems that gravitate toward dynamic attractors, exhibiting critical behavior as they approach these points. The market's intrinsic dynamics appear to guide it toward this critical state, especially during periods of stability.
The concept of self-organized criticality emerges, suggesting that the stock market functions as a self-organizing critical system.
The first video, "What can physics tell us about stock market crashes?" by Dragan Mihailović at TEDxLjubljana, explores the link between physics and market behavior.
The second video, "From Earthquakes to Financial Bubbles: The Science of Predicting Crashes," delves into the methods used to forecast financial downturns.
For more insights into finance, mathematics, data science, and physics, visit my personal website at www.marcotavora.me.