Usually financial crises go along with bubbles in asset prices, such as the housing bubble in the US in 2007. This paper attempts to build a mathematical model of financial bubbles from an econophysics, and thus a new perspective. I find that agents identify bubbles only with a time delay. Furthermore, I demonstrate that the detection of bubbles is different on either the individual or collective point of view. Second, I utilize the findings for a new definition of asset bubbles in finance. Finally, I extend the model to the study of asset price dynamics with news. In conclusion, the model provides unique insights into the properties and developments of financial bubbles.
The phenomenon of a financial bubble is debated in economics for centuries. There are several important questions: What is a bubble? How can we identify a bubble? What is the root cause of a bubble and what triggers a bubble burst? So far, different economic and mathematical models study different issues of bubbles. However to my knowledge, there is no consistent model that addresses all issues at once. In addition, the complexity in financial markets due to social interaction and behavioural elements makes the study of those bubbles difficult.
This paper attempts to develop a new model of financial bubbles. My model utilizes the idea of particle dynamics in physics. I translate this idea into the dynamics of financial assets without relying on stochastic and martingale theory [
The remainder of the paper is structured as follows. Section 2 discusses the different forms of financial bubbles and presents a literature review. In Section 3, I derive the mathematical model and study the main implications. Section 4 extends the model and studies the impact of news. Finally, Section 5 provides concluding remarks.
Let me start with a quote by Charles MacKay about the forces of financial bubbles in general: “Men, it has been well said, think in herds; it will be seen that they go mad in herds, while they only recover their senses slowly, and one by one.” No doubt, herd behaviour turns out to be important during almost all financial bubbles. Thus it was studied in the economic literature for decades. In general, there are different types of herd behaviour and they are characterized by imperfect information, a systematic bias in expectations, and regulatory arbitrage due to perverse incentives.
The need for better models in this field has to do with the tremendous social costs of a bubble burst [
First, economists have analyzed “information-based” or “rational” herd behaviour. That research is pioneered by DeLong et al. [
A second line of literature in economics argues for “reputation-based herding”. This was first explored by Scharfstein and Stein [
This paper develops a unique link between the different types of herd behaviour and a general “Model of Financial Bubbles”. To analyze the complex financial dynamics, I build a mathematical model based on particle physics. In fact, there exist an obvious parallel of both particle and financial dynamics. The dynamics of both systems is based on the interaction of single elements/agents that lead to a collective outcome. From an aggregate perspective, the frequent events of traffic jams are often the result of uncoordinated interaction of individual’s at overcrowded highways and not always a car accident. Interestingly, this is similar to the root cause of herd behaviour in financial markets. Consequently, a financial bubble, i.e. a massive price increase (decrease), can be caused by a large number of buying (selling) individuals, especially if they build a herd. Thus, bubbles can come into existence without any change of the fundamental value. The ups and downs of asset prices attract similar agents and thus form a herd. Soon later, I obtain a bubble or crash. Unfortunately, the collective behaviour creates even more volatility, higher risks, and massive welfare losses. A second parallel between physics and financial bubbles is the impact of news as a root cause or trigger of a bubble. Ad-hoc news either positive or negative can be compared to an unforeseeable red light or traffic flows via an exit or entrance to a highway. Hence, the main objective of my model is twofold: 1) build a coherent model and 2) find new implications for empirical finance in future.
There is also a large descriptive literature about collective behaviour in speculative bubbles [
The financial asset price dynamic is denoted by
where k captures the agents’ expectation or reputation level. I assume
Definition 1. Trading volume (flow) of asset i is defined as the amount of buy and sell orders (number of trades) times the buy and sell price per day. It is denoted by the function
Definition 2. Trading density of asset i is defined as the number of trades within a certain price range
Next, based on both definitions, I define a fundamental relationships:
Definition 3. There exists a relationship between trading volume, trading density and the benefit-loss-field, such as
The last equation is a fundamental law in applied physics and in my econophysics model on financial assets. In the following subsections, I derive the model and study the identification of a financial bubble.
Next, I consider the two fundamental variables
Solving this equation would determine the asset price for which an agent is willing to buy or sell at a later time under a uniform stock market. However, finding the function
In general, the number of trades change over time. It increases with the inflow of buying agents at the lower bound
Combining Equations (4) and (5), per definition yields
The last equation can be labelled a “conversation law”. Hence, the equation denotes that the number of trades in a certain price range is equal to the difference of the trading volume in this price range. Starting with that equation, I derive the model. Consider the integral conversation law over a small interval from
Now, divide by
The right-hand side of that equation is the first derivative of
The average number of trades in the defined price interval
In limit for
The left-hand side of Equation (8) equals now
which is a wave equation in form of a partial differential equation (PDE). This equation can be rewritten with the help of the fundamental law from Equation (2), where q is defined as
Solving this PDE requires further assumptions. Suppose that the BLF is defined as a function of the density, such as
the function value,
The solution of the non-linear PDE Equation (13) determines the trading density and thus the price at all future times. Hence, I solve the following initial value problem
This problem requires a linear approximation. Suppose the density is uniform, then it can be approximated as
where
where
Taylor-series approximation of the second term, yields
where
I solve this PDE via variable transformation. Use both
Substitute both findings in Equation (19), yields finally
Proposition 1. The general solution of the PDE in Equation (19) and (21), is given as
Proof. First, compute
and then substitute both expressions in Equation (19):
The solution of the model enables me to define a financial bubble in general.
Definition 4. A bubble is defined as a trading density greater than
Proposition 2. The existence of a bubble implies that the benefit-loss-field (BLF) is different to the specific BLR of an asset.
Proof. Due to
A negative slope of the BLF is a necessary and sufficient condition for a bubble in my model. In this constellation all traders buy (sell) the respective asset and that lowers the risk of the individual’s BLF. However, an increasing trading density creates herd behaviour and finally a financial bubble in the overall market (
Proposition 3. In general, a bubble is defined by an upper limit for
side is less than the BLF
Proof. If
This proposition demonstrates that a bubble can be characterized as a social interaction problem. Hence, a financial bubble is not just determined by individual rationality or irrationality as assumed in economic models. On the contrary, more buyers automatically indicate a higher trading density and higher risk for an exuberance. But from the individual point of view it is the other way round. A high density imply a small price movement and thus a lower risk and higher willingness to buy (sell) assets. This interaction and finally the imbalance of both mechanisms trigger a financial bubble. This new insight is in line with findings in sociology and psychology, for instance in the social system theory [
Finally, I model the impact of “news” in financial markets with bubbles. I utilize the idea of a “jump-disconti- nuity”. In this case, I obtain
This integral is well defined if even
Let’s assume that the trading density changes on both ends of the interval at the same amount,
where
Proposition 4. A financial bubble is characterized by a negative BLF in respect of time:
Proof. Use the properties of the function
Consequently, the proposition confirms that good news imply a jump with a decline in the change of asset prices (BLR) and thus a lower probability of a bubble burst. Even if the result seems to be counterintuitive at first, it has a rational. News reduce asymmetric information and thus lowers the BLF. This reduces the market reaction time. In other words, the bubble may burst in case of small price jumps. Consequently, efficient markets with many news may trigger only small financial bubbles, but inefficient markets (=high asymmetric information) with little news may trigger huge financial bubbles. The solution of the ODE (Equation (25)), demonstrates the evolution of the asset price with news and it yields
This equation explains the fact that with increasing time, t, the asset price declines automatically due to the lag of news. Therefore, more agents enter the market and buy/sell assets. In the end, this leads to herd behaviour and a financial bubble.
This paper attempts to explain the major properties and developments of financial bubbles from an econophysics point of view. I utilize a new theoretical model to detect ex ante financial bubbles. This model makes the complexity tractable without stochastic and martingale theory. In general, this approach has several advantages. First, it is a simple model and uniquely defines a financial bubble. Second, the model is general and thus explains the origins and developments of bubbles, too. Third, the model contains novel implications for empirical studies on financial bubbles. Finally, and most importantly, this model enables you to study the effect of news on financial bubbles for the first time.
I thank the editor and two anonymous referees for their helpful comments and suggestions. Research is funded by the RRI―Reutlingen Research Institute. This support is greatly appreciated.