Thought Viruses and Asset Prices

TL;DR

This paper models how investment ideas spread like viruses using epidemiological concepts, revealing patterns in market booms and busts, including early warning signs and the impact of rational agents on market dynamics.

Contribution

It introduces a novel epidemiological model of investment idea spread to analyze asset price bubbles and market top characteristics.

Findings

Market prices peak after the number of infected buyers continues to rise.

Fully rational agents accelerate booms and produce broader market tops.

The model identifies early symptoms indicating different stages of market bubbles.

Abstract

We use insights from epidemiology, namely the SIR model, to study how agents infect each other with "investment ideas." Once an investment idea "goes viral," equilibrium prices exhibit the typical "fever peak," which is characteristic for speculative excesses. Using our model, we identify a time line of symptoms that indicate whether a boom is in its early or later stages. Regarding the market's top, we find that prices start to decline while the number of infected agents, who buy the asset, is still rising. Moreover, the presence of fully rational agents (i) accelerates booms (ii) lowers peak prices and (iii) produces broad, drawn-out, market tops.

Equations

$$\dot{S}$$

$$\dot{I}$$

$$\dot{R}$$

$$N$$

$$I$$

$$R$$

$$I=-S+\frac{\gamma }{\beta }ln(S)+C_I,C_I=N_1+N_2-\frac{\gamma }{\beta }\ln (N_1).$$

$$\frac{dI}{dS}=-1+\frac{\gamma }{\beta }\frac{1}{S}=0,$$

$$\frac{d^{2}I}{d{S}^2}=-\frac{\gamma }{\beta }\frac{1}{S^2}<0.$$

$$c_v=\beta I_v{S}_v.$$

$$I=\beta {\int }_0^t{I}_v{S}_v{e}^{-\gamma (t-v)}dv.$$

$$\frac{dI}{dt}=\beta I(t)S(t)-\beta \gamma {\int }_0^t{I}_v{S}_v{e}^{-\gamma (t-v)}dv=0.$$

$$X=\varphi (P),{\varphi }^{\prime }()>0,\varphi (P_0)=0,$$

$$c_v(t)=\beta I(v)S(v)e^{-\gamma (t-v)}.$$

$$X_I=\beta {\int }_0^t\frac{I_v{S}_v}{P(v)}{e}^{-\gamma (t-v)}dv.$$

$$\displaystyle\phi(P)=\beta\int_{0}^{t}\frac{I_{v}S_{v}}{P(v)}e^{-\gamma(t-v)}dv\quad\Leftrightarrow\quad P=\phi^{-1}\Big{(}\beta\int_{0}^{t}\frac{I_{v}S_{v}}{P(v)}e^{-\gamma(t-v)}dv\Big{)}.$$

$$\frac{d{X}_I}{dt}=\beta \frac{I(t)S(t)}{P(t)}-\beta \gamma {\int }_0^t\frac{I_v{S}_v}{P(v)}{e}^{-\gamma (t-v)}dv=0.$$

$$\frac{d^2{X}_I}{d{t}^2}$$

$$\frac{\ddot{S}}{\dot{S}}\frac{1}{\beta }=-I+S-\frac{\gamma }{\beta }$$

$$\frac{dI}{dt}=\beta I(t)S(t)-\beta \gamma {\int }_0^t{I}_v{S}_v{e}^{-\gamma (t-v)}dv=0.$$

$$\frac{d{X}_I}{dt}P(t)=\beta I(t)S(t)-\beta \gamma {\int }_0^t\frac{I_v{S}_{v}P(t)}{P(v)}{e}^{-\gamma (t-v)}dv=0.$$

$$\frac{d{X}_I}{dt}=\beta \frac{I(t)S(t)}{P(t)}-\beta \gamma {\int }_0^t\frac{I_v{S}_v}{P(v)}{e}^{-\gamma (t-v)}dv=0.$$

$$\displaystyle\int_{0}^{t_{1}}\Big{(}1-e^{-\gamma(t_{1}-v)}\Big{)}\beta\frac{I_{v}S_{v}}{P(v)}dv=\beta\int_{t_{1}}^{t_{2}}\frac{I_{v}S_{v}}{P(v)}e^{-\gamma(t_{2}-v)}dv=0.$$

$$\displaystyle P^{*}=\phi^{-1}\Big{(}\int_{0}^{t_{1}}\beta\frac{I_{v}S_{v}}{P(v)}dt\Big{)}.$$

$$X_{RE}=\beta {\int }_0^{t_2}\frac{I_v{S}_v}{P_{RE}(v)}{e}^{-\gamma (t-v)}dv<\beta {\int }_0^{t_2}\frac{I_v{S}_v}{P_M(v)}{e}^{-\gamma (t-v)}dv=X_M.$$

$$0=\frac{IS}{P_{RE}}-\gamma {\int }_0^{t_2}\frac{I_v{S}_v}{P_{RE}(v)}{e}^{-\gamma (t-v)}dv⪌\frac{IS}{P_M}-\gamma {\int }_0^{t_2}\frac{I_v{S}_v}{P_M(v)}{e}^{-\gamma (t-v)}dv.$$

$$\displaystyle IS\Big{(}\frac{1}{P_{RE}}-\frac{1}{P_{M}}\Big{)}-\Big{(}\gamma\int_{0}^{t_{2}}\frac{I_{v}S_{v}}{P_{RE}(v)}e^{-\gamma(t-v)}dv-\gamma\int_{0}^{t_{2}}\frac{I_{v}S_{v}}{P_{M}(v)}e^{-\gamma(t-v)}dv\Big{)}>0$$

Full text

\makegloss

Thought Viruses and Asset Prices111I thank Dominik Grafenhofer and Carl Christian von Weizsäcker for helpful and encouraging discussions. This version: 30.12.2018.

Wolfgang Kuhle

*University of Economics, Prague, Czech Republic, E-mail: wkuhle@gmx.de

Max Planck Institute for Social Law and Social Policy, Munich, Germany.*

Abstract: We use insights from epidemiology, namely the SIR model, to study how agents infect each other with “investment ideas.” Once an investment idea “goes viral,” equilibrium prices exhibit the typical “fever peak,” which is characteristic for speculative excesses as described, e.g., in Kindleberger, (2000). Using our model, we identify a time line of symptoms that indicate whether a boom is in its early or later stages. Regarding the market’s top, we find that prices start to decline while the number of infected agents, who buy the asset, is still rising. Moreover, the presence of fully rational agents (i) accelerates booms (ii) lowers peak prices and (iii) produces broad, drawn-out, market tops.

Keywords: Thought Viruses, Bounded Rationality, Rational Expectations

*JEL: D5, D52, E03, E32, E62 *

“More and more people realized the misconception on which the boom rested even as they continued to play the game.” Soros, (1994), p. 57.

1 Introduction

Market practitioners frequently argue that they observe asset prices, which are “far away” from “fundamentals.”222Soros, (1994), pp. 27-141, provides a detailed argument and a large collection of case studies. See also the “Mr. Market” parable in Graham, (1973), pp. 188-213, or Fisher, (2003), pp. 266-275. Kindleberger, (2000) presents a historical account of such boom and bust episodes. They also suggest that many such deviations come in the form of boom-bust cycles, which are hard to reconcile with traditional asset pricing models. The current paper presents a simple epidemiological model where investors “infect” each other with their investment ideas. The model generates market prices that are in line with the practitioners’ boom and bust observation. In turn, we use the model to perform comparative statics, and to establish a time-line of events that lead and lag market tops.

To model the infectious spread of investment ideas, we borrow the standard SIR model from the literature on epidemiology.333Kermack and McKendrick, (1927) present the mainstream SIR specification used here. Dietz and Heesterbeek, (2002) review the historical literature on modelling epidemics dating back to Daniel Bernoulli’s original manuscript, which was first circulated in 1760. The SIR framework has been used successfully to model the spread of infectious diseases, computer viruses, chain letters, and religions. In the current setting it governs the rate with which agents adopt and discard their views on certain assets. The SIR model’s key prediction, namely that the mass of infected agents is hump-shaped, generates the boom-bust sequence in stock prices that practitioners emphasize.

We distinguish two main scenarios. First, we study a setting where susceptible investors buy the asset once infected, and sell once cured. Second, we study a setting where cured agents form rational expectations, which can induce them to hold overpriced assets in order to wait for further appreciation. Such rational expectations on the part of cured agents make prices (i) rise faster (ii) peak earlier and (iii) generate broad, drawn-out, market tops. Without rational agents, prices rise at a slower pace, but they reach a higher maximum. Finally, we find that price peaks lead the peak in infected agents across both scenarios. Put differently, prices go into decline while sentiment is still improving.

Having derived these properties, we compare the observable parts of our predictions to a few prominent cases of boom and bust. Using search queries from Google as a proxy for the mass of infected agents, we find that both, search queries and the corresponding asset’s price, show the fever curve pattern that the current model suggests. Moreover, prices indeed tend to peak before the mass of infected agents does. Finally, slow moving boom-bust cycles, where cured agents have time to form rational expectations, have broad tops.

Literature: Dawkins, (1995) suggests that humans are susceptible to “Viruses of the mind.” Such viruses can come in the form of religions or political believes, which spread in a contagious manner.444In different contexts, Harris, (1995) and Kahneman, (2012) discuss how agents adopt and discard different, not necessarily correct, approaches to decision making. See also von Weizsäcker, (1971), who proposes a model of changing tastes.

Instead of adopting religions, our agents believe in certain investment ideas, valuation techniques, or future technological developments. Moreover, our agents work as financial missionaries when they share their investment tips with colleagues, friends, and neighbors. The analogy to an infectious process is even more straight forward in specific episodes such as the “gold fever” epidemic, where a susceptible east-coast population was infected by the first prospectors who returned from California.

Shive, (2010) finds that the SIR specification can explain the clustering of agents’ real estate investments. Feng and Seasholes, (2004), Hong et al., (2004, 2005), Ivkovic and Weisbrenner, (2007), and Brown et al., (2008) find similar evidence suggesting that neighbors emulate each other’s investment choices.555Kindleberger, (2000), p. 15, notes that “there is nothing so disturbing to one’s well-being and judgement as to see a friend get rich.” Ash, (1955) shows that a considerable fraction of people follow other’s choices even when they know that these choices are wrong.

Shiller, (2017) suggests that the SIR specification helps to study the spread of economic “narratives,” such as the idea of the “Laffer curve.” In our model, such narratives come in the form of the “BRICS” countries, asian tigers, “.com” stocks, securitized mortgage debt, or “crypto currencies.” Related, Daley and Kendall, (1964, 1965) have sparked a literature, e.g., Moreno et al., (2004), that uses epidemiological models to study the spread rumors.

Scharfstein and Stein, (1990), Bikhchandani et al., (1992) and Banerjee, (1992) provide Bayesian frameworks of herd behavior, which can be used to generate boom-bust sequences in asset prices within a traditional rational expectation framework. In this interpretation, the current SIR specification studies how certain pieces of Bayesian information permeate a given population of investors. Regarding rationality, Kindleberger, (2000), p. 15-16, points out that speculative manias are concentrated in assets which are hard to understand, such as options, futures, tulip seeds during winter, real estate and land, goods manufactured for export markets, and more generally foreign exchange. Recent episodes like the “.com” crash, the US housing crisis with its complex mortgage backed securities, and the rise and fall in crypto-currencies clearly fit this description.

Our model coexists with a large number of alternative theories of boom and bust. First, any Walrasian model, which features multiple equilibria, can explain abrupt changes in price simply as a move from one equilibrium to another. Equilibrium models of portfolio insurance, flights to quality, rational herding, informational avalanches, and game theoretic formulations, such as models of exchange rate crises, can generate drastic price changes within a rational expectations (RE) frame. One key difference between theses RE models and the present one is the speed of adjustment. Rational agents ensure that markets reprice, more or less, immediately. That is, moving from one equilibrium to another produces an instantaneous change in price. The current model generates smooth booms and smooth busts, which are in line with the observation that many boom bust phenomena unfold over time frames ranging from several month to decades.666Japanese real estate saw steep appreciation for 30 years before it reached its 1989-1991 peak. In turn, prices declined for roughly 25 years.

Section introduces the model. Section 3 considers the impact of rational expectations. Section 4 compares our theoretical results to a number of historic boom-bust episodes. Section 5 concludes.

2 The Model

We recall the SIR model,777See e.g., Hirsch et al., (2003), pp. 235-239. and add the asset market later. The SIR model studies how a disease spreads among a mass $N$ of agents. These agents belong to three groups. First, there is a group of susceptible agents $S$. Second, there is a group of infected agents $I$. Finally, there are recovered agents $R$. When infected agents meet susceptible agents, they transmit the virus at rate $\beta$. Infected agents recover at rate $\gamma$. Finally, we work with continuous time $t$, such that the epidemic is characterized by a first-order differential equation:

[TABLE]

Equation (1) describes how susceptible agents get infected. Equation (2) tracks the mass of infected agents: it adds newly infected and subtracts recovered agents. Equation (3) accounts for cured agents. Finally, (4) ensures that the overall population remains constant over time; Starting in $t=0$ we have $N_{1}$ susceptible agents, $N_{2}$ infected agents, $N_{3}$ recovered agents, and a total population $N$.

Combining (1)-(4) we note:

Lemma 1**.**

Given initial conditions $\{N_{1},N_{2},N_{3}\},N_{i}>0,i=1,2,3$, the SIR model has one stable steady state, $\{N-R_{\infty},0,R_{\infty}\}$, where $R_{\infty}=-\frac{\gamma}{\beta}ln(N-R_{\infty})+C_{R},$ and $C_{R}=N_{3}+\frac{\gamma}{\beta}\ln(N_{1})$.

Proof.

Combining (1) with (2), we can integrate

[TABLE]

Likewise (1) and (3) yield

[TABLE]

From (3), we see that $I=0$ in steady state. Combining $I=0$ with (5), (6) and (4), we have $S_{\infty}=N-R_{\infty}$ and $R_{\infty}=-\frac{\gamma}{\beta}ln(N-R_{\infty})+C_{R}$, where the last equation defines a unique $R_{\infty}$. ∎

The steady state of Lemma 1 is reached via a transition path, along which susceptible agents become infected, and infected agents get cured. This transition path exhibits a peak in infected agents:

Lemma 2**.**

If $N_{1}>\frac{\beta}{\gamma}$, then there exists a point in time $t_{I}^{*}>0,$ where the mass of infected agents $I$ peaks.

Proof.

To prove that there exists a unique peak in infected agents in a period $t_{I}^{*}$, we recall (5):

[TABLE]

Taking derivatives yields

[TABLE]

and

[TABLE]

Hence, there exists a global infection maximum at $S^{*}=\frac{\gamma}{\beta}$, where $I^{*}=-\frac{\gamma}{\beta}+\frac{\gamma}{\beta}ln(\frac{\gamma}{\beta})-C_{S}$. To relate our peak in infected agents to a period in time, we note that equation (1) ensures that $\dot{S}<0$ at all points in time, such that the condition $S(t)=S^{*}=\frac{\gamma}{\beta}$ can be inverted $t^{*}_{I}=S^{-1}(\frac{\gamma}{\beta})$. That is, the mass of infected agents peaks in time $t^{*}_{I}$, and at this point in time, we have $dI/dt=0$ and $d^{2}I/dt^{2}<0$. ∎

To prove Lemma 2, it was convenient to express the mass of infected agents as a function $I(S)$. To study how this peak in infected agents relates to stock prices, it will be necessary to account for the fact that different cohorts of infected agents buy and sell at different prices. To do so, we denote cohorts $c_{v}$ of agents by the (vintage) period $t=v$, where they were first infected:

[TABLE]

Taking into account that infected agents get cured at rate $\gamma$, and summing over all infected cohorts, yields the mass of infected agents in period $t$:

[TABLE]

Using (11), the condition for $t^{*}_{I}$ from Lemma 2 can be written as:

[TABLE]

Regarding (12) we note that inserting (11) brings us back to (2). Expression (12) will, however, be useful once we have to track dated asset purchases and sales of different cohorts.

2.1 Market

We assume that susceptible agents hold one unit of currency each. When agents get infected, they buy the asset. Once cured, agents sell their position. Regarding the asset market, we assume that there is an exogenously given (excess) supply function:

[TABLE]

which interacts with the demand of infected agents. Equation (13) may be interpreted as the supply of “.com” stocks, tulips, crypto currencies, flats, which come to the market, respectively, via new IPOs, professional florists, bitcoin farms, or the incumbent population in an in-fashion residential area.

To compute infected agents’ demand, we recall (1)-(2), and note that a cohort $c_{v}=\beta I(v)S(v)$ of agents, who were infected in period $t=v$, depreciates exponentially such that the cohort’s period $t$ size is:

[TABLE]

Regarding the period $t$ asset holdings, of all infected cohorts $v$, we have:888We suppress the holdings of agents $N_{2}$, who are infected in $t=0$. Adding these holdings, inflates notation and does not change the results. Alternatively, we may assume that $N_{2}$ is small.

[TABLE]

where $\frac{1}{P(v)}$ is the number of shares that infected agents buy in period $t=v$, when they invest their one unit of currency into the speculative asset. The variable $X_{I}$ denotes the number of shares, held by all cohorts of infected agents, in period $t$.

Combining asset supply (13) with demand (15), the equilibrium price is:

[TABLE]

Evaluation of (16) yields:

Proposition 1**.**

The asset’s price peaks in period $t^{*}_{p}$, where $t^{*}_{p}<t^{*}_{I}$. The long-run, steady state, price is $P_{0}$.

Proof.

The asset price peaks when $\dot{P}=\frac{1}{\phi^{\prime}(X_{I})}\frac{dX_{I}}{dt}=0$. From (15) it follows that this condition for $t^{*}_{p}$ can be written as

[TABLE]

Regarding the derivative $\frac{dX_{I}}{dt}$, we first note that $\lim_{t\rightarrow 0}\frac{dX_{I}}{dt}>0$. That is, the price is increasing at $t=0$. Moreover, we recall (15) and note that $\lim_{t\rightarrow\infty}X_{I}=0$999To see this note that $\beta\gamma\int_{0}^{t}\frac{I_{v}S_{v}}{P(v)}e^{-\gamma(t-v)}dv<\gamma\frac{I}{P_{0}}$ and recall that $\lim_{t\rightarrow\infty}I=0$., i.e., in the long run steady state, the demand of infected speculators is zero. Hence, the long run price is $P_{0}=\phi^{-1}(0)$.

Unique peak price: We have seen that, starting with a price $P_{0}$, the price is first increasing in time and then eventually reverts to $P_{0}$ in the long-run. Hence, there must be at least one $t^{*}_{P}$, where the price peaks. This period is implicitly defined by condition (17). To show that the peak is unique, we study the second-order condition:

[TABLE]

To interpret (19), we recall $\ddot{S}_{|(\ref{e1})}=-\beta(\dot{I}S+I\dot{S})$ and rearrange it using (2), such that:

[TABLE]

For the terms in (20), we note that (1) ensures $\frac{1}{\dot{S}}<0$, and from Lemma 2 we know $S-\frac{\gamma}{\beta}<0$ for all $t>t_{I}^{*}$. If the mass of susceptible agents is sufficiently large,101010As we have seen in Lemma 2 we need $N_{1}>\frac{\gamma}{\beta}$ for a peak in $I$ to exist. Moreover, we need only a small number of initially infected agents to start an epidemic, i.e., we can pick an arbitrarily small number $N_{2}$ for the number of initially infected agents. at the beginning of time, $\ddot{S}$ starts out negative, but before time $t_{I}^{*}$ is reached, where $S=\frac{\gamma}{\beta}$, it will turn positive, and stay positive ever after. During the period where $\ddot{S}>0$, we may have price peaks, but cannot have minima. Once, $\ddot{S}<0$ we may have minima, but no maxima. That is, in $t=0$, prices start to increase away from the starting level $P_{0}$. This price increase first accelerates until the second derivative changes signs, and the price increase decelerates until a peak is reached. After the peak, prices start to decline.

Finally, it remains to show that $t^{*}_{p}<t^{*}_{I}$. To do so we compare the condition (17), for the period where the price peaks, with the condition for $t^{*}_{I}$, where the number of infected agents peaks. The mass of infected agents peaks when:

[TABLE]

Multiplying (17), the condition for the period where the price peaks, with $P(t)$, yields

[TABLE]

The difference between (21) and (22) is the term $\frac{P(t)}{P(v)}$. Regarding this term, we note that, as long as the price is increasing between time [math] and $t$, we have $\frac{P(t)}{P(v)}>1$ for all $v<t$. Accordingly, for any arbitrary period of time $t$, for which the price is increasing, we have $\frac{dI}{dt}>\frac{dX_{I}}{dt}P(t)$. Hence, when the price peaks at $t^{*}_{p}$, we have $\frac{dX_{I}}{dt}P(t^{*}_{p})=0$ and $\frac{dI}{dt}>0$. That is, when the price peaks, the mass of infected agents is still increasing in time.

Put differently, early buyers have amassed large wealth, and when they start to sell, they have a greater weight in the market than those agents who buy in late with their one unit of currency.∎

2.2 Depressions

In the current model we have discussed how investors, who infect each other with euphoria, can bid up prices. The same argument works in reverse when depression spreads. That is, we may model an infected agent as someone who adopts the view that stocks or real estate prices will never appreciate again. In that case, the same mechanism works in reverse, creating a U-shaped price pattern. Regarding this depression, we can once again show that asset prices bottom before the selling sentiment does.

3 Rational Expectations

At the top of the market there is hesitation, as new recruits to speculation are balanced by insiders who withdraw. Kindleberger, (2000) p. 17

So far we have assumed that agents simply close their position once they recover from their false investment thesis. Put differently, when agents understood that they owned an over valued asset, they sold. Another way to model recovering agents is to assume that they fully understand the model that they operate in.

When cured agents know the entire model, they can find themselves in two different situations. First, the agent gets cured after the price has peaked. In this scenario it is clearly rational to sell immediately. Second, the agent gets cured while the market price is still increasing. In this second scenario, cured agents will keep the asset for as long as it appreciates. To do so, they have to compute the point in time when the market peaks. To find this point, they must compute the asset holdings of all other agents who are also waiting to exit at the top. Second, they must know the asset holdings of those who are currently infected, but will get cured before the price peaks. Finally, the mass of susceptible investors who will buy in the future is important, since these investors provide the liquidity that rational sellers need to exit their positions.

Two points in time are thus of particular importance. First, there is the point in time $t_{1},$ where the price peaks, and rational agents start to sell to newly infected buyers. At $t_{1}$ the sum of future net buying of newly infected agents, must be sufficient to absorb all the holdings that rational agents, who were waiting for the price to peak, have accumulated up until $t_{1}$. To calculate the net buying of newly infected agents, we have to identify the point in time $t_{2}$ where the buying of newly infected agents is exactly offset by the selling of newly cured agents. Time $t_{2}$ is the point where the last rational agent, who was waiting for the market top, must have sold out. After $t_{2}$, buying of newly infected agents falls short of the selling of newly cured agents and prices decline.

We start with $t_{2}$, the point in time where the buying of newly infected agents is exactly offset by the selling of newly cured agents:

[TABLE]

This condition is the same as (17) except for the price history, i.e., the values $P(v)$ are different since cured agents did not sell immediately.111111Period $t_{2}$ therefore does not coincide with $t_{P}^{*}$.

Period $t_{1}$, is the period when rational agents start to sell:

[TABLE]

The RHS of (24) describes the net buying of newly infected agents between $t_{1}$ and $t_{2}$. This must be equal/sufficient to absorb the assets that cured agents accumulated (LHS) between $t=0$ and $t_{1}$ with the motive to sell at the market top. Put differently, (24) ensures that the entire selling of cured agents can be absorbed through the buying of newly infected agents. And once period $t_{2}$ is reached, the holdings of cured agents, who waited for the top, have been passed on completely to newly infected agents. Finally, the price at which rational agents sell is:

[TABLE]

Where (25) reflects that there is no selling before time $t_{1}$ is reached. Taking (23)-(25) together we have:

Proposition 2**.**

(Broad-Peak-Theorem): The market price remains constant at its peaks $P^{*}$ as long as $t\in[t_{1},t_{2}]$.

Proposition 2 relies on the assumption that cured agents know the correct model and all its coefficients. Moreover, they are able to solve the model. Proposition 2 should thus be interpreted as a reference point, which we can use as a comparison for the “fever peak” which we derived in Proposition 1. The fever peak relied on the assumption that recovered agents cannot fully understand the model, and thus they cannot correctly anticipate the top. Hence, they simply close their position. Put differently, these agents just discard the wrong thesis they used to buy the asset, but they do not replace this thesis with a new thesis, which tells them, e.g., that the price may appreciate further. To close our comparison of these two cases, we make three remarks. Whenever necessary, we denote variables that correspond to the case where cured agents simply sell by a subscript M. Cases where cured agents form rational expectations are denoted by $RE$. We begin by noting that the price increase is more rapid when cured agents from rational expectations:

Remark 1**.**

For the period where $t\in(0,t_{1}]$ we have $P_{M}<P_{RE}$.

Proof.

In both scenarios, the buying is governed by the same infection process. The scenarios differ regarding selling. In the RE setting, where cured agents form rational expectations, there is no selling during the period $t\in[0,t_{1}]$. Asset holdings, and prices, are thus always higher in the RE setting than in setting $M$ where agents sell immediately after being cured. ∎

Remark 2**.**

The peak price is lower under rational expectations $P_{RE}^{*}<P_{M}^{*}$.

Proof.

Let us first recall that all rational agents have sold out, by the time that $t_{2}$ is reached. Hence, the mass of infected agents who hold the speculative asset is the same in $t_{2}$ across both scenarios RE and M. To proof our remark by contradiction, we now assume that $P_{M}<P_{RE}$ for all $t\in[0,t_{2})$. Under this assumption period $t_{2}$ asset holdings are:

[TABLE]

This however means that $P(X_{RE})<P(X_{M})$, and contradicts our assumption $P_{M}<P_{RE}$ for all $t\in[0,t_{2})$. Hence, the price $P_{M}$ must exceed $P_{RE}^{*}$ before $t_{2}$ is reached. ∎

Remark 3**.**

$t_{1}<t_{P}^{*}<t_{2}<t_{I}^{*}$.

Proof.

$t_{1}<t_{P}^{*}$: The price $P_{RE}$ peaks in period $t_{1}$ and remains at this level until $t_{2}$ is reached. To see that $t_{1}<t_{P}^{*}$, we recall Remark 1, i.e., that $P_{RE}>P_{M}$ for all $t\in(0,t_{1}]$. At the same time, Remark 2 shows that $P_{M}$ must exceed $P_{RE}$ before $t_{2}$ is reached. Hence, the date $t_{P}^{*}$, where $P_{M}$ peaks, must be such that $t_{1}<t_{P}^{*}$.

$t_{2}<t_{I}^{*}$ follows immediately from the same argument given in the proof of Proposition 2, which relies on the comparison of equations (21) and (22). That is, in period $t_{2}$, all newly cured agents bought at prices less or equal $P^{*}_{RE}$. Thus, given that their one Dollar investment appreciated, they have a higher monetary weight than the newly infected buyers. Hence, at $t_{2}$, the mass of infected agents must still be increasing to absorb the selling. An alternative way, to observe the same thing, is to show that the price is already in decline at $t_{I}^{*}$.

Inequality $t_{P}^{*}<t_{2}$: We have shown in Remark 2 that $P_{M}$ must exceed $P_{RE}^{*}$ before $t_{2}$. Does the peak in $P_{M}$ also occur before $t_{2}$? If $P_{M}>P_{RE}$ then $X_{M}>X_{RE}$. taking derivatives, at $t_{2}$, yields:

[TABLE]

For $t_{2}<t^{*}_{P}$, we must have $P_{M}>P_{RE}$ in $t_{2}$ This means that $\frac{IS}{P_{M}}<\frac{IS}{P_{RE}}$ since $P=P(\gamma\int_{0}^{t_{2}}\frac{I_{v}S_{v}}{P_{RE}(v)}e^{-\gamma(t-v)}dv)$, we also know that $\gamma\int_{0}^{t_{2}}\frac{I_{v}S_{v}}{P_{RE}(v)}e^{-\gamma(t-v)}dv<\gamma\int_{0}^{t_{2}}\frac{I_{v}S_{v}}{P_{M}(v)}e^{-\gamma(t-v)}dv$. Rearranging (27) thus gives:

[TABLE]

Hence, $\frac{dX_{M}}{dt}(t_{2})<0$, i.e., the price $P_{M}$ is in decline at $t_{2}$, and its peak must have occurred earlier. That is, $t_{P}^{*}<t_{2}$. ∎

4 Examples

In this section, we use data from Google Ngrams and Goolde trends to study some of our predictions. Across cases, there is strong support for the epidemiological fever-peak. The search queries/literature mentions for particular assets show a pronounced hump-shape pattern, which coincides with a hump shaped pattern in these assets’ prices. Moreover, in line with our model’s prediction, prices peak earlier than search queries/literature mentions.

Bitcoin: Using Google trends, we have the search history for the term “Bitcoin.” Bitcoin search peaked in the period between December 17th and 23rd at 100. Before, Bitcoin search was at a level of 3 between 2013 and April 2017. Dollar prices for Bitcoin went from 1200 in April 2017 to their 19.587 peak on 16th of December 2018. While search for Bitcoin continued to rise sharply between December 17th and 23rd, prices dropped to around 14000121212Prices are taken from Yahoo Finance. December 23rd highs and lows were roughly 15000 and 13000. on December 23, and have continued to decline alongside with search queries, to 3800 respectively 10 in December 2018.

The “.com” boom: Using Google Ngrams data for the period 1980-2008, we find that the frequency with which the terms “Nasdaq,” “MSFT” and “AMZN” are mentioned follow the hump-shaped pattern suggested by the SIR model. This pattern extends to the corresponding prices. Regarding the time line of peaks, we note that mentions of the “Nasdaq” peaked in 2001, while the Nasdaq composite index peaked in March 2000. The MSFT (Microsoft) stock peaked in late 1999, mentions of the “MSFT” ticker symbol peaked in 2004. Mentions of the AMZN (Amazon) ticker peaked in 2002 while the stock peaked in late 1999. Mentions of “new economy” and “software” both peaked in 2002.

Japanese Nikkei index: Using Google Ngrams (case insensitive), we view the term “Made in Japan” as a proxy for the sentiment towards the Japanese Nikkei industrial index. The Nikkei peak in 1989 leads the “Made in Japan” peak between 1990-1994.

5 Conclusion

We model booms and busts in asset prices as an epidemic process: agents infect each other with their investment ideas, valuation techniques, or believes in future technological developments. To model how such ideas permeate a large population of susceptible investors we have used the classic SIR specification from the literature on epidemiology.

Our model generates sentiment driven booms-bust cycles, which market practitioners emphasize. In addition to producing such price swings, our model can be used to perform comparative statics. In particular, we find that rational agents (i) accelerate booms (ii) lower peak prices and (iii) make for broad, drawn-out, market tops. Finally, regardless of whether cured agents are rational or not, prices peak before sentiment, i.e. the mass of infected agents, does.

Google data indicate that interest in speculative assets indeed exhibits boom-bust patterns akin to the SIR model’s infection peak. Moreover, the peak in price tends to precede the peak in queries. The “Bitcoin” boom with its sharp peak was more in line with our scenario, where cured agents are best advised to sell quickly. Deeper markets had broader peaks, which give well informed investors more time to trade with late stage optimists.