Self-sustained price bubbles driven by Bitcoin innovations and adaptive behavior
aa r X i v : . [ q -f i n . T R ] D ec SELF-SUSTAINED PRICE BUBBLES DRIVEN BY BITCOIN INNOVATIONS AND ADAPTIVEBEHAVIOR
MISHA PEREPELITSA AND ILYA TIMOFEYEVA
BSTRACT . We show that infinite divisibility of a trading commodity leads to a self-sustained price bubble when traders useadaptive investment strategies. The adaptive strategy can be viewed as a psychological response of a trader to the situationwhen the trader’s estimation of future prices does not match the actual, realized price.We use a multi-agent model to illustrate the price bubble formation and to quantify its main statistical properties such asthe return, the volatility, and the systematic risk of the price bubble to crash. We discuss the plausibility for bubbles to driveprices of digital currencies.
1. I
NTRODUCTION
With an introduction of Bitcoin in 2010, digital currencies became a global social phenomenon and they are gainingpopularity with investors. One of the distinctive features of digital currencies is their ultimate divisibility, as virtuallyany fraction amount of it can be traded. Currently, the minimum trading amount for Bitcoin is set at 10 − units, butthe design of the currency allows it to be lowered by any order of magnitude. This, technically, solves the problemof liquidity, as no matter how high the price is, one can always buy some of it, say, with 1 dollar. And it seems thatthe divisibility indeed comes handy, as the price for Bitcoin skyrocketed in the last decade. It’s growth is attributedto many factors such as the novelty effect, the demand from the black market, global access to the currency throughthe internet, the usage of bitcoin as storage in markets with hyper-inflated national currencies, and, not least, the factthat bitcoin became a sort of a social cult of a decentralized currency, see Dodd (2017). Apart from the black marketdemand, which alone can not contribute to this phenomenal growth, all other factors point at a price bubble, whichseems to be a consensus among the economists, see an article by Wolff-Mann (2018).In this paper we show that this unique property of a traded commodity to be infinitely divisible makes possible theemergence of self-sustained price bubbles. By this we mean that a group of investors can “pump” the price up at apositive rate, without ever running out of cash. More than that, we show it may happen without any collusion betweenthe investors, if each of them follows a psychologically appealing, adaptive investment strategy.It is hard to estimate if a speculative bubble of this kind is behind the growth of the Bitcoin. Several mechanismsa likely to be at work here. For example, the bubble might have started as an adaptive speculative bubble that latermorphed into a spontaneously organized, global Ponzi scheme, or the other way around. Other mechanisms of pricebubbles in digital currencies are possible, see for example Griffin and Shams (2020), or, for generic models of bubbles,Sornette (2003). It is certainly not the goal of this work to fit the parameters of the model to the time series of Bitcoinprices and to estimate its statistical significance. Rather than that, our intention was to show the potentiality of thissort of bubble to develop in the market of digital currencies.Let us mention that the adaptive behavior of investors is not limited to trading of digital currencies but rather isan ubiquitous phenomenon. Thus the model for the price bubble dynamics, considered in this paper, has a widerapplicability. In particular, it shows that for “classical commodities” there must be a continuous in-flow of investmentsfor a price bubble to persist, and provides a analytical tool to obtain quantitative information on the operation of aprice bubble.The paper organized as follows. In section 2 we describe the model and its main properties and provide numericalestimates of the return and the volatility of the price. Date : January 1, 2021.
In section 3, we give numerical estimates of the systematic risk for the price bubble to crash, measured by twodifferent metrics, one based on the low cash-to-stock portfolios of traders, and the other, on low current-cash-to-initial-cash ratio. We obtain two different estimates of the expected stock price adjusted by the risk of a crash.We argue that the second metric is a more relevant measure of risk for the adaptive speculative bubbles. This givesanother evidence of a bubble being self-sustained. On the other hand, we argue that overestimation of the risk by thefirst metric and the high volatility of the returns may contribute to longevity of speculative bubbles generated by thismechanism in efficient markets.Additionally, we describe a relation between the volatility of the price and the systematic risk, see formula (7) onpage 7. It allows for the direct estimation of the systematic risk from an empirical values of the volatility of the pricetime series. Several interesting statistical laws of a price bubble in a fixed “psycho-state” are derived from this formula,such as the distribution of the times of a crash, and the maximum expected price that can be attained in a life-time ofa bubble.In the section 4 we give a psychological interpretation of two main parameters of our model and consider a possibil-ity that these parameters have cyclic dynamics in the domain of the psychological variables, thus defining an intrinsiclife span of a speculative bubble. This view of a price dynamics as an output of a cyclic model suggests that the pricedynamics generated by this model can be in agreement with the no-arbitrage principle.Models of adaptive economic behavior were developed by many authors in many different contexts. In strategicdecision making they were advanced as models of convergence by Cross (1983), models of learning in games byFudenberg and Levine (1998), and reinforcement learning models by Roth and Erev (1995), to mention just a fewcontributions. Our work is close to the methodology of multi-agent dynamic simulations of Stigler (1964), Kimand Markowitz (1989), Arthur et al. (1997), Levy et al. (1994, 1995, 2000), Lux (1995, 1998), Lux and Marchesi(1999, 2000). Price bubbles in efficient markets is a well-known phenomenon. They were discussed, for example, byMandelbrot (1966), Blanchard and Watson (1982) and Sornette (2003).2. A
SYNCHRONOUS STOCHASTIC PRICE PUMP
Adaptive behavior is an universal paradigm. Not surprisingly it is widely used as an investment strategy. Considerfor a example an investor who has investment in a stock and has certain cash reserve and would like to keep the stock-to-cash ratio at certain level, so that the investor is regularly buying or selling stocks to be at that target level. We let ourinvestor, perhaps a bitcoin enthusiast, to be somewhat uncertain about the investment goals, and in particular, to haveinfinite investment horizon. This might be rather natural assumption in the case of the digital currencies, since theyhave been introduced just recently. The investor might consider the market as a complex, exogenous environment withconstantly changing characteristics. As an adaptive strategy the investor may increase his investment in stock if themarket outperformed the investor’s expectations as measured by the stock-to-cash ratio and decrease the investment,otherwise. For example, a trader may increase the target stock-to-cash ratio by a factor of α > , in the former case,and decrease it by the factor of β < , in the latter. Parameters α and β express a degree of the psychological responseof the investor to the market performance and can be thought of as measures of investor’ optimism and pessimism, or,greed and fear of losing.What will happen if all other investors in the market follow similar strategies, trying to adapt to the changingmarket environment that they themselves create? One might expect some sort of anomaly to arise. The scenario wasconsidered by Perepelitsa and Timofeyev (2019) based on the analysis of a multi-agent model, called asynchronousstochastic price pump (ASPP) that we describe below.2.1. To introduce the model we start with a market of only two agents trading shares of a single asset.The agents, do not know the asset fundamental value (and maybe there isn’t), but expect that the price can grow atmoderate rates for long periods of time, and they always prefer increasing values to decreasing.The agents meet at time intervals that we label with n = , , .. At period n , the first agent’s portfolio has ( s n , b n ) , dollar value of stock and bond investments, and the second: ( s n , b n ) . The stock price per share at period n is P n . Thenext period investment portfolio ( s n + i , b n + i ) and price P n + determined trough the following steps.Evaluating the last change in their portfolios, agents set target stock-to-bond ratios for the next period, that wedenote by k n + i . The new price P n + is set in such a way that the dollar amount of funds that agent 1 wishes to movefrom stocks to bonds, to be at the target ratio k n + , equals the dollar amount agent 2 wants to move from bonds to ELF-SUSTAINED PRICE BUBBLES DRIVEN BY BITCOIN INNOVATIONS AND ADAPTIVE BEHAVIOR 3 stock to be at ratio k n + . This balance is expressed as(1) P n + P n s n − k n + b n = k n + b n − P n + P n s n . Here we assume that any fractional amount of a share can be exchanged, and for simplicity of the exposition, considerthe case with zero interest in the bond account. With the newly set price the agents re-balance portfolios and evaluatethe market performance by comparing ratios P n + s ni / ( P n b ni ) to target values k n + i . If P n + s ni / ( P n b ni ) > k n + i , from theagent i point of view, the market performed better than expected, and if P n + s ni / ( P n b ni ) < k n + i it performed worse.Since while one agent is selling shares the other one is buying, they have different opinions on the market performance.The agents update the target portfolio ratios according to the rule(2) k n + i = α k n + i P n + s ni P n b ni > k n + i , k n + i P n + s ni P n b ni = k n + i , β k n + i P n + s ni P n b ni < k n + i . If the agent identifies a growing market she increases her stock-to-bond ratio by a fixed amount, while the other agentdecreases her ratio. The feedback reflects the fact that when faced with a series of bad investments the agent willreduce the equity part of the portfolio, while with the investment growing better than expected, the agent will takeriskier position.It easy to see that there is an equilibrium when no shares are traded and price doesn’t change: s ni = s i , b ni = b i , k ni = s i / b i , P n = P . Small deviations will set off a non-trivial dynamics. In this process, agents will alternativelychange their preferences by factors α and β , so that, on average, the stock-to-bond ratios change at the rate ( αβ ) / per period. The gross return, P n + / P n , after a short transient period will settle at the average value of ( αβ ) / perperiod, but will keep oscillating around that value, as in Figure 1. The stock price goes up when agents have biastoward the increase of the risk preferences, expressing the optimistic outlook, quantified by the condition αβ > . The story of this process is simple: agents are trading stock and and move their stock-to-bond ratios up and downwith an upward average trend. This drives up the average demand and consequently the price.2.2.
Properties of N -agent ASPP. For a mulit-agent model, we populate the market with many copies of agentsconsidered above and let them trade anonymously, by submitting their buy-sell orders to a market maker. Thus, weconsider a set of N agents, described by their portfolios ( s i , b i ) , i = . . . N , of dollar values of a stock and cash accounts,and let k i stand for agent i stock-to-bond ratio, either actual, or a future, target value. P will denote a current priceper share and P the new price determined by agents’ demand. Let { i l : l = . . . m } be the set of “active” agents, i.e.the ones setting the new price. The set of active traders is determined each trading period by a random draw from thepopulation.If x i l is the dollar amount that agent i l wants to invest in stock, then PP s i l + x i l b i l − x i l = k i l . The demand-supply balance is m ∑ l = x i l = , which can be solved for P : PP = m ∑ l = k i l b i l + k i l ! m ∑ l = s i l + k i l ! − . Once the price is set, agents move corresponding amounts between cash and stock accounts, re-balancing their port-folios. The update mechanism for new stock-to-bond ratios is given by (2). The interaction is repeated the following
MISHA PEREPELITSA AND ILYA TIMOFEYEV trading periods g r o ss r e t u r n F IGURE
1. Gross return for first 25 trading periods in a 2-agent ASPP model, with α = . , and β = . . Circles indicate the values of gross returns for alternating ups and downs of stock price,with a net return per period approaching value p αβ = . , (solid line).trading periods with new, randomly selected sets of active agents. In this trading scheme the total amount of cash inthe system and number of shares of stock remain constant.There is an equilibrium state, when all agents have balanced portfolios, s i / b i = k i , i = , . . . , N . In this case stockprice doesn’t grow, P n = P n − . When the parameters are out of the equilibrium, even by a small degree, the systemexhibits non-trivial, divergent dynamics, such as shown in Figure 3.In this divergent regime, after a certain transient period, the logarithm of the price start fluctuating as normallydistributed white noise with linearly growing in time mean, see Perepelitsa and Timofeyev (2019). The distribution ofreturns depends on the mechanism for selecting a random group of active agents. If the number of active agents, m , is fixed the returns in the stationary regime follow log-normal distribution. If number m is itself random, for examplechosen from an uniform distribution, the returns have significantly more mass at the tails and at 1, than a correspondinglog-normal distribution.To estimate the values of σ in (4) we performed numerical simulation of ASPP model, with N =
500 and m = , , , on 200,000 trajectories for each pair of test values of ( α , β ) . The values of σ for the case with m = α − β andthe corresponding linear regression line. Linear regression coefficients are listed in Table 2. It can be seen from thattable that formula c ( m )( α − β ) is a good approximation for volatility σ , as a function of α and β . In all three cases,there is a positive correlation of about 0 . α \ β ABLE σ as a function of α and β for ASPP model with N =
500 and m = , in a stationaryregime reached after 2 years of trading time. ELF-SUSTAINED PRICE BUBBLES DRIVEN BY BITCOIN INNOVATIONS AND ADAPTIVE BEHAVIOR 5 - F IGURE
2. Values of sigma from Table 1 plotted against α − β . c dm=10 0.2066 0.0081m=20 0.1457 0.0055m=40 0.1084 0.0027T ABLE
2. Linear regression coefficients ( c , d ) in σ = c ( α − β ) + d for several values of m . Combining these findings and assuming for the simplicity of presentation that there is one trade per day (total 360per year), and using t n = n /
360 to label n th trading period on the time axis measured in years, we obtain that dailyprices P ( t n ) are independently distributed with(3) ln ( P ( t n )) ∈ N ( n ln r , σ / √ ) , r = ( αβ ) m N , σ = c ( m )( α − β ) . In this formulas, N is the total number of investors and m is the number of investors participating in one trade per day,and c ( m ) is a positive number, depending on m . Accordingly, for the daily returns, R ( t n ) = P ( t n ) / P ( t n − ) , we have(4) ln ( R ( t n )) ∈ N ( ln r , σ ) where r and σ as in (3).Let’s illustrate this behavior on a particular model setup, in which there is total of N =
500 investors, with m = α = . β = .
96 Figure 3 illustrates typical characteristic features of a stochastic price pump. The top panel shows timeseries of the logarithm of the price of a share with a “healthy” growth rate 1 .
66 per year (66% increase). The panelbelow shows that the returns are not constant but fluctuate as a white noise with volatility of about 0 .
07 (7% daily pricechanges). This price growth is achieved by a modest amount of trading, with daily trading volume not exceeding 0 . − and by year 20, of order 10 − . This property is a consequence of the exponentialgrowth of the price combined with the fact that the trading conserves the amount of cash in the system.
MISHA PEREPELITSA AND ILYA TIMOFEYEV years l og o f s t o ck p r i c e years -0.4-0.200.20.4 l og g r o ss r e t u r n years t r ad i ng v o l u m e years t r ad i ng v o l u m e F IGURE
3. Four typical time series describing one simulation of ASPP for a period of 20 years.From top to bottom: logarithm of the stock price, logarithm of the daily gross returns, daily tradingvolume as percentage of the total cash in the system, daily trading volume in units of shares.This example shows that investors acting adaptively can generate a long-lasting, positive price dynamics, and anecessary condition for an ASPP to function for prolonged time is an infinite divisibility of traded commodity, whichis one of the main characteristic properties of digital currencies.Our main interest in this paper will be the systematic risk of ASPP and its relation to the volatility, which we discussin the next section. 3. S
YSTEMATIC RISK
Multi-agent models offer a convenient tool for estimating systematic risk of sharp price drops (crashes), since theinformation about investment portfolio of each agent is a part of the description of the model. Here, by systematic riskwe understand the risk of a breakdown of the system that generates the price dynamics, in this case, the breakdownof ASPP. We consider two metrics for measuring this risk. The first is based on the proportion of agents with lowcash-to-stock ratios. Let us denote by q ( t ) the proportion of agents with cash-to-stock ratio less than, say 10%, at time ELF-SUSTAINED PRICE BUBBLES DRIVEN BY BITCOIN INNOVATIONS AND ADAPTIVE BEHAVIOR 7 t . Higher values of q ( t ) mean higher likelihood of a crash, as a larger proportion of agents may spontaneously initiatestock sellout. We will use a hazard rate function(5) h q ( t ) = q ( t ) − q ( t ) to quantify the probability of a crash. According to this formula, the hazard rate changes monotonically from 0 , when q ( t ) = , to + ∞ , when q ( t ) = . Recall that the hazard rate function is defined so that for an infinitesimal time intervalof length dt , h q ( t ) dt is the probability of a crash to occur during interval [ t , t + dt ] , given that it didn’t occur beforetime t . Then, the probability that no crash occurs up to time t , equals(6) S q ( t ) = e − R t h q ( s ) ds . Thus, the “discounted” stock price at time t equals P ( t ) S q ( t ) , assuming, for simplicity, that a crash brings price to zero.Alternatively, the systematic risk can be computed based off the proportion of agents with low ratio of cash reservesat time t to the initial cash reserves, say less than 60%. We denote this proportion by r ( t ) , and the corresponding hazardrate and the survival functions by h r ( t ) and S r ( t ) , defined by (5) and (6) where h q is replaced by h r . Figure 5 shows plots of the expected price, and the prices discounted by two hazard rates, h q and h r , for the samemodel as used in the simulation in Figure 3. Measured by the first metric, the price collapses to zero, while in thesecond metric the price continues to grow (eventually it will go down as well).It might seem that the first metric seems to be more relevant measure of a sudden sellout. Note however, that a lowcash-to-stock ratio is not caused by traders spending all cash buying stock, but rather on the anomalous price growth.For this reason the second metric is more likely to be used by adaptive speculators with long investment horizon. Asexplained in the previous section the latter is model of main interest in this paper.Another difference between two metrics, that we would like to emphasis here, is that the first depends on the initialinvestment portfolios, and can be arbitrary high at time t = t = h q ( t ) , whereas the actual risk in an adaptive speculation model given by the hazardrate h r ( t ) with significantly smaller values, or the high volatility which is positively correlated with the return, seeformulas for r and σ in (3). The latter might be the reason why short term investors stay away from them.As can be seen from the previous examples, for fixed values of α and β , the volatility of the price is constant. Atthe same time, the hazard function increases, implying that the volatility can not be a correct measure of the systematicrisk. However, they are not completely unrelated. To show this, let us define a cumulative volatility at t years as V ( t ) = Z t σ ( s ) ds . For a time series of daily stock price P ( t n ) , the cumulative volatility V ( t n ) = ∑ t m ≤ t n [ ln R ( t m ) − ln R ( t m − )] / , where R ( t n ) = P ( t n ) / P ( t n − ) is the gross return.Figure 6 shows a scatter plot of daily values of V ( t n ) and the hazard rate h r ( t n ) , and the best fit linear approximation(linear regression). This numerical evidence is strongly suggestive of a linear relation between V ( t ) and h r ( t ) , or,equivalently, a linear relation(7) σ = γ dh r dt . Table 3 lists the statistical values of parameter γ for several values of ( α , β ) . This formula provides an importantpractical tool for estimation of the systematic risk from the volatility, since the latter can be computed from time seriesof the price.From this formula we can find the hazard rate h t ( t ) = σ t γ , and the PDF for the time of a crash. The latter has aWeibull distribution σ t γ exp (cid:18) − σ γ t (cid:19) . MISHA PEREPELITSA AND ILYA TIMOFEYEV years popu l a t i on p r opo r t i on F IGURE
4. Proportions of traders with low cash-to-stock portfolios (left) and low cash-to-initial-cash ratio (right) as functions of time in one simulation of the model for the period of 20 years. years s t o ck p r i c e F IGURE
5. Expected stock price discounted by the risk of a crash. Top graph is the expected pricewithout discounting. The graph in the middle is the expected price discounted by hazard rate h r ( t ) . The graph at the bottom is the expected price discounted by hazard rate h q ( t ) . Whith that, the expected discounted stock price becomes equal P d ( t ) = P ( ) exp (cid:18) r t − σ γ t (cid:19) , r = m N ln ( αβ ) . It can be seen that there is a maximum value max P d ( t ) of the price, attained when t = r γ / σ . It places a limit onthe price that a bubble can reach in a fixed “psycho-state” ( α , β ) . ELF-SUSTAINED PRICE BUBBLES DRIVEN BY BITCOIN INNOVATIONS AND ADAPTIVE BEHAVIOR 9 hazard rate c u mm u l a t i v e v o l a t ili t y F IGURE
6. Scatter plot for cumulative volatility V ( t n ) and hazard rate h r ( t n ) , in one simulation ofthe model for the first 20 years. α \ β ABLE
3. Values of γ in formula (7). Each value is an average of the linear regression coefficientsof V ( t n ) on h r ( t n ) , for 1000 20-year path simulations of ASPP model with N =
500 agents in thesystem, m =
10 active agents. 4. C
ONCLUSIONS
In the last section we listed several factors that can contribute to a prolonged life span of the price bubble. But willthe bubble perpetuate its growth forever? What will happen in an ideal case when no crashes due to systematic riskoccur, when there are no arbitrageurs out there, and investors have very long, or even infinite, investment horizon?The answer to these questions lies in the interpretation of parameters ( α , β ) of the model, that we identify as twopsychological motivators, greed and fear, receptively. Here, we are assuming that a coherent psychological state in thepopulation is achieved through the copying behavior and inter-connectedness of the investors by information networks,so that all investors share the same or almost the same values of α and β . These parameters are combined together inthe formula for the return (3), which shows that the price pump can work both ways, as a price inflator or a deflator.Being of psychological nature ( α , β ) will change according to some intrinsic mechanisms, that combine personalpsychological response of each investor to the changes in the composition of his/her portfolio and the degree ofinfluence of this response on the behavior of other investors. As an approximation, one can assume that ( α , β ) co-evolve in such a way that α (greed) is self-reinforcing and generates β (fear), and fear, in its turn, inhibits greed, byincreasing investor’s sensitivity to idiosyncratic noise.According to this model, the evolution of a price bubble will proceed in a “predator-prey” type cycle in the domainof psychological variables, triggered by an exogenous shock. Along the cycle, the return and volatility of the pricewill changes according to formulas from (3), producing a pattern of increasing returns with low volatility, followed by a period of growing volatility, during which the return reaches its maximum and starts to decrease, which eventuallyends with sharp decrease on return and low volatility. At the end of the cycle the system will reach some “backgroundlevel” of ( α , β ) , at which they balance each other. The system will remain in this “no bubble” phase until anotherexternal factor pushes it on a road to a new bubble.This type of dynamics can produce complex patterns of price variations. Since the period of a bubble cycle is anon-constant (random) variable, there can be little predictability of the future price, even more so with crashes due tosystematic risk. This suggests an intriguing idea that the price fluctuations due to adaptive speculative market behaviormight be consistent with the no-arbitrage principle. R EFERENCES[1] Arthur, W.B., Holland, J.H., LeBaron, B, Palmer, R.G., Tayler, P. (1997). Asset Pricing under Endogenous Expectations in an Artificial StockMarket. in W.B. Arthur, S. Durlauf and D. Lane. eds. The Economy as an Evolving Complex System II. Addison-Wesley.[2] Blanchard. O.J., Watson, M.W. (1982). Bubbles, rational expectations and speculative markets. in
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NIVERSITY OF H OUSTON , 4800 C
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EMAIL : MPEREPEL @ CENTRAL . UH . EDU , ILYA @ MATH . UH . EDU L S 4m pp