A Dynamical Systems Approach to Cryptocurrency Stability
AA Dynamical Systems Approach to CryptocurrencyStability
Carey Caginalp, , ∗ Department of Mathematics, University of Pittsburgh, USA301 Thackeray Hall, Pittsburgh PA 15260, USA Economic Science Institute, Chapman UniversityOne University Drive, Orange CA 92866, USA ∗ To whom correspondence should be addressed; E-mail: carey [email protected].
Recently, the notion of cryptocurrencies has come to the fore of public interest.These assets that exist only in electronic form, with no underlying value, offerthe owners some protection from tracking or seizure by government or credi-tors. We model these assets from the perspective of asset flow equations devel-oped by Caginalp and Balenovich, and investigate their stability under variousparameters, as classical finance methodology is inapplicable. By utilizing theconcept of liquidity price and analyzing stability of the resulting system of or-dinary differential equations, we obtain conditions under which the system islinearly stable. We find that trend-based motivations and additional liquid-ity arising from an uptrend are destabilizing forces, while anchoring throughvalue assumed to be fairly recent price history tends to be stabilizing.
Introduction
Blockchain technology enables large numbers of participants to make electronic transactionsdirectly without intermediaries, and has led, in recent years to a new form of payment, andessentially to a new set of currencies called cryptocurrencies. During 2017 the spectacular nine-fold rise in the price of Bitcoin focused the spotlight of public attention on cryptocurrenciesthat evolved into a new asset class. Following the pattern of other nascent assets, speculatorsdominated trading and pushed prices toward a bubble.1 a r X i v : . [ q -f i n . M F ] M a y s with some other asset bubbles of the past, notably the dot-com frenzy of the late 1990s,the emergence of a new technology clouded judgements about the basic value of the asset.In almost all cases, unlike traditional equities, cryptocurrencies have no tangible value, andeven their creation is often shrouded in mystery ( ). For Bitcoin, the creation of additionalunits is relegated to a process termed mining, named after the old technology of mining gold.In this electronic version, computing power is used to solve complex mathematical problems,and once solutions are found, the miner is rewarded with some units of the cryptocurrency. It ispeculiar that this brand new technology is coupled with the organizational hierarchy of centuriesago that featured the hegemony of gold miners and traders who held power over the economiclives of people without any accountability. Thus, the advances in technology are coupled witha regressive organizational structure. Major currencies such as the US dollar are controlled byofficials appointed by elected representatives; assets such as common stocks are governed by aboard of directors that are elected by the shareholders whose rights are assured by law, albeitthrough a circuitous process. Changes made by cryptocurrencies are often made at an ad hocmeeting of the developers or miners, reminiscent of tribal chiefs meetings of more primitiveeras.In certain types of securities, i.e. exchange-traded funds (ETFs), a certain group of tradersknown as authorized participants have the capacity to demand the underlying shares, preventingthe price of the ETF from straying too far from its fundamental value. Securities held in abrokerage account are insured up to a value of several million dollars by the federal governmentagainst circumstances such as hacking or company insolvency.Conversely, for cryptocurrencies, many of these rules have been lacking and a few have beenrecently instituted. First, these cryptocurrencies have no underlying assets. Further, there existsno mechanism by which one can redeem any value from a bank or other institution. Second,even the country of residence (let alone a business address) of the creators is merely specu-lative, making it unclear to which court one could possibly appeal in the case of a grievance.The original ”developer” for Bitcoin, for example, is known only by a pseudonym. Third, theindividual investor has no influence over something as simple as the number of Bitcoins inexistence. Instead, these decisions are generally relegated to the heads of electronic Bitcoinmining operations, whose interests may be disjoint from investors’. Bitcoin is currently set tobe capped at 21 million units, but there is no legal obstacle to prevent an increase at the whimof the miners. Fourth, many individual investors and some academicians tacitly assume thatthe laws and protections afforded by the purchase of securities through stock exchanges such asNYSE must also apply to cryptocurrencies. Recently, the Securities and Exchange Commission(SEC) has shuttered some initial coin offerings (ICOs) (
4, 34 ) and announced future regulationsto attempt to inject more clarity into the marketplace. Finally, Bitcoin is vulnerable to hackingor simply forgetfulness. Many holders of Bitcoin have their information stored in exchanges,i.e. platforms that handle transaction and storage of the cryptocurrency. These are hijacked witha disturbing regularity, and litigation can be pending years later, as in the infamous Mt. Goxhacking of 2014 ( ).Cryptocurrencies offer both opportunities and risks to society. On the one hand, cryptocur-2encies and technology underpinning them – if designed appropriately – could be used to maketransactions faster, safer and cheaper, alongside other societal benefits (
15, 22 ). A less apparentfeature is that they can make it more difficult (though not theoretically impossible; see (
5, 18 ))for totalitarian governments to expropriate savings, either directly or indirectly through cur-rency inflation, thereby depriving savers of a large fraction of their assets. In this way, a propercryptocurrency could lead to greater economic freedom, and render more difficult the financingof a dictatorship. Indeed, this can be modelled by a choice of two alternatives: either their homecurrency or cryptocurrency that cannot easily be seized (
20, 39 ).The risks presented by existing cryptocurrencies are multi-faceted. The difficulty in tracingtransactions facilitate illicit activity and its financing. A less obvious – and possibly the mostsignificant – risk arises from the instability of prices of major cryptocurrencies. As the marketcapitalization (number of units times the price of each unit) of the cryptocurrencies rises, thereis growing risk that a sharp drop in the price of a cryptocurrency could have a cascading effecton other sectors of world economy, particularly if borrowing is involved. During the periodOctober 2017 to April 2018, the price of a Bitcoin unit rose from $6,000 to $20,000 and backto $6,000. The market capitalization of all cryptocurrencies during that time period increasedfrom $170 billion to $330 billion, peaking together with Bitcoin in December 2017. Whileattention is often focused on the rise and fall of the trading prices of these assets, the magnitudeof the problem of stability has increased significantly during this six month period. As peoplebecome more accustomed to using these instruments, the market capitalization may increase toseveral trillion – i.e., a few percent of the $ 75 trillion Gross World Product – and many of thechallenges will be critical.Generally, the features of a financial instrument that might make it attractive to speculatorsare undesirable to those who seek to use it as a currency in daily transactions. Speculatorssee a greater opportunity in a volatile market, as they can use technical analysis and expertise toprofit at the expense of the layperson. Conversely, large fluctuations on a day-to-day basis createobstacles for common purchases or the pricing of service contracts ( ). Without stability in themarketplace, the cryptocurrencies may simply become ”a mechanism for a transfer of wealthfrom the late-comers to the early entrants and nimble traders” ( ). Thus, a set of questionsof critical importance deals with the potential stability (or lack thereof) of Bitcoin or othercryptocurrencies, which is the main topic of our paper.The turbulence arising from the collapse of the housing bubble was a major challenge formarkets, but from a scientific perspective, it could be addressed largely with classical methods(
19, 32, 35 ). However, classical methods are not readily adaptable to studying cryptocurrencies,as discussed below. We use a modern approach whereby an equilibrium price can be determinedand the stability properties established within a dynamical system setting (
6, 9, 10, 17, 23, 24, 28,30, 35–37, 42 ). 3
Modelling prices and stability
Most of classical finance such as the Black-Scholes option pricing model has its origin in thebasic equation P dP = µdt + σdW (1)for the change in the relative price P − dP in terms of the expected return, µ, the standarddeviation of the return, σ, and independent increments of Brownian motion, dW. It is widelyacknowledged that this equation does not arise from compelling microeconomic considerations,nor empirical data. But rather, it is mathematically convenient and elegant for expressing andproving theorems (see ( ) for discussion). Much of risk assessment is based upon this modelwith an increasing array of adjustments.The limitations of this basic model are apparent, for example, if one examines the standarddeviation of daily relative changes in the S&P 500 index, which is typically around . . This leads to the conclusion that a . drop is a sixth standard deviation event, i.e., it occursonce every billion trading days, while empirical data shows it is on the order of a few times perthousand ( ).Thus identifying risk on a large time scale based on the variance of a small time scale canvastly underestimates risk.Furthermore, the modeling of asset prices is generally based on the underlying assumptionof infinite arbitrage. While there may be some investors who are prone to cognitive errors or biasin assessing value, the impact of their trades will be marginalized by more savvy investors whomanage a large pool of money. Of course the inherent assumption is that there is some value toan asset, based for example on the projection of the dividend stream, replacement value, etc.,and that the shareholder has a vote that allows him ultimately to extract this value. For assetssuch as US Treasury bills, the model works quite well, as the owner is assured of receiving aparticular dollar amount from the US at a specified time.Herein lies the central problem for the application of classical theory to cryptocurrencies:there is no underlying asset value, as noted above. Cryptocurrencies constitute the opposite endof the market spectrum to US Treasury Bills, in which an arbitrageur can confidently buy or sellshort based on a clear contract that will deliver a fixed amount of cash at a predetermined time.If fact, classical game theory would conclude that since everyone knows the structure of thecryptocurrency, and understands that everyone else is also aware, then the price should neverdeviate much from zero. Furthermore, classical finance expressed through (1) would suggestthat there is some measure of risk based on the historical average of σ, which will be less helpfulthat it is for stock indexes as discussed above.Our analysis begins with the fact that despite the absence of underlying assets or backing,various groups have incentive to use it over traditional currencies. In particular there are largegroups who need to make transactions outside of the usual banking system. Among theseare (i) people with poor credit who cannot obtain a credit or even a debit card, (ii) citizens oftotalitarian countries who fear expropriation of their savings, (iii) citizens of countries with high4nflation and a much lower interest rate, (iv) people engaged in illicit activity, (v) people whoespouse utilizing a new idea or technology.Collectively, these groups constitute a core ownership of cryptocurrencies, investing a sumthat gradually grows with familiarity (
14, 16, 25 ). Meanwhile, the rising prices catch the at-tention of speculators who provide additional cash into the system, but also bring motivationsinherent in speculation, namely momentum trading, or the tendency to buy as prices rise, andanalogously sell as prices fall ( ).We assume a single cryptocurrency and that the price is established by supply and demandwithout infinite arbitrage, and apply a modern theory of asset flow ( ). This alternative ap-proach relies on the notion of liquidity price. The experimental asset markets presented a puzzleto the economics community by demonstrating the endogenous price bubbles in which pricessoared well above any possible expectation of outcome ( ). Caginalp and Balenovich ( )observed that in addition to the trading price and fundamental value (defined clearly by the ex-perimental setup), there was an additional important quantity with the same units: the total cashin the system divided by the number of shares. Denoting this by liquidity value or price, L, theyadapted earlier versions ( ) of the asset flow model.This approach leads to a system of ordinary differential equations, as summarized below,whereupon equilibrium points can be evaluated and their stability established as a function ofthe basic parameters. For brevity, we first present the full model which will be a nonlinear evolutionary system that isbased on ( ) but with some key differences for cryptocurrencies. We can then consider simplermodels in which some features are marginalized by setting parameters to zero and obtaining × or × systems, enabling us to understand the key factors in stability.We denote the trading price by P ( t ) , the number of units by N ( t ) , the amount of cashavailable by M ( t ) , and the liquidity price by L ( t ) = M ( t ) /N ( t ) . With B as the fraction ofwealth in the cryptocurrency, i.e., B = N P/ ( N P + M ) , the supply and demand are given by S = (1 − k ) B, D = k (1 − B ) respectively, where k is the transition rate from cash to theasset. Using a standard price equation ( ) we write τ P dPdt = DS − . (2)It follows that B (1 − B ) − = N P/M = P/L , so that the price equation is τ P dPdt = k − k LP − . (3)The variable k is assumed to be a linearization of a tanh type function and involves the motiva-tions of the traders which are expressed through sentiment, ζ = ζ + ζ where ζ is the trend5omponent and ζ is the value component. This construct has been studied, for example, inclosed-end funds (
1, 12, 13, 26 ) which frequently trade either at a discount or premium to theirnet asset value. Writing the term k/ (1 − k ) in terms of the ζ and ζ and linearizing we havethen k − k ˜=1 + 2 ζ + 2 ζ (4)and the price equation is then τ P dPdt = (1 + 2 ζ + 2 ζ ) LP − . (5)One defines ζ through two parameters, c , that expressed the time scale of the trend follow-ing and q the amplitude of this factor, as ζ ( t ) = q c (cid:90) t −∞ e − ( t − τ ) /c P ( τ ) τ dP ( τ ) dτ dτ (6)Note that L and ζ are linear functions of one another, but we retain L as a variable so the systemis more easily generalized to incorporate a time-dependent L . The valuation is more subtle fora cryptocurrency. The only concept of value relates to fairly recent trading prices. The firstpurchase with Bitcoin was for two slices of pizza for 10,000 Bitcoins ( ). The sense of valueat that time was probably much less than 2018 when people became accustomed to prices in thethousands of dollars. We thus stipulate the definitions P a ( t ) = 1 c (cid:90) t −∞ e − ( t − τ ) /c P ( τ ) dτ, (7) ζ ( t ) = q c (cid:90) t −∞ e − ( t − τ ) /c P a ( τ ) − P ( τ ) P ( τ ) dτ, (8)i.e., ζ represents the motivation to buy based on the discount from the perceived value of thecryptocurrency. Finally, the liquidity will not be constant but will be the sum of the core group’scapital L plus the additional amounts arriving from speculators that is correlated with the recenttrend: L ( t ) = L + L c q (cid:90) t −∞ e − ( t − τ ) /c τ P ( τ ) dP ( τ ) dτ dτ (9)We assume that L is constant, but one can easily adapt the model to include temporal changesin L due to, for example, greater public acceptance of cryptocurrencies. By differentiating(6-9) and combining the resulting equations with (5) we obtain the 5x5 system of ordinary6ifferential equations: c P (cid:48) a = P − P a ,c ζ (cid:48) = q P a ( t ) − P ( t ) P a ( t ) − ζ ,τ P (cid:48) = (1 + 2 ζ + 2 ζ ) L − P,cL (cid:48) = 1 − L + q { (1 + 2 ζ + 2 ζ ) L − P } ,c ζ (cid:48) = q (cid:18) (1 + 2 ζ + 2 ζ ) LP − (cid:19) − ζ . (10)We find a unique equilibrium at ( P, P a , L, ζ , ζ ) = (1 , , , , . In other words, the onlysteady-state of the system occurs when the price, the anchoring notion of fundamental value,and liquidity price all coincide with the base liquidity value L ( ). The time scale for priceadjustment will be short as markets adjust rapidly to supply/demand changes. Much longerwill be the time scale for observing the trend and reacting to under or over-valuation, and theassessment of valuation anchored through weighted price averages. Moreover, one might expectthat the valuation is on an even longer time scale. Thus one expects three time scales such that τ (cid:28) c, c , c (cid:28) c , which we can scale as c = c = c = 1 , and we allow arbitrary τ , c in theanalysis. τ Pc P a Lζ ζ = − − − q q − q q − q q q − q − q q − PP a Lζ ζ . (11)Thus, the system is determined entirely by three parameters: q , the attention to trend; q , ameasure of the influence of delay times; and q , the influence of fundamental value, along withthe time parameters τ and c . The question of stability can be investigated by calculating theeigenvalues in the relevant parameter space, i.e. ( q, q , q ) ∈ R (the first octant), along with τ and c . In particular, the main question is whether the maximal real part of the eigenvaluesis positive, leading to instability, or if they are all negative, yielding stability. One sees thatthere is a double eigenvalue at λ = − , and the other three eigenvalues remain negative if theRouth-Hurwitz conditions ( ) below are satisfied τ + 1 c + Q > , (cid:18) Qc + 1 τ + 2 q τ + 1 τ c (cid:19) (cid:18) τ + 1 c + Q (cid:19) > c τ . (12)where we have set Q = 1 − q − q . A sufficient set of conditions for (12) to hold is the7ollowing: c + 1 τ > q + 2 q =: K, c + 1 τ > Kc − q τ . (13)However, one can observe numerically that (13) are not necessary conditions to satisfy (12).Also, if we set q = 0 , we obtain the simpler condition c + 1 τ > K (14)for stability, which we will see describes a simpler model that excludes valuation and the com-ponent of investor sentiment associated with it. We sketch various cross-sections holding oneof these parameters constant and numerically computing eigenvalues across values of the othertwo. Note in Figure 1 below that increasing q induces a stabilizing effect, while large K servesto make the system less stable. We choose various values of τ and c in Figure 2.This yields a number of results. First, as market participants focus greater attention to thedeviation of the asset from the acquired fundamental value driven from the liquidity price, thereis less room for prices to stray from equilibrium. In addition, for a fixed q , the asset wouldexperience stability given that K is large enough. Finally, for K large enough, one sees that wehave instability for a large range of q , i.e., if investors place too much emphasis on the relativetrend, the asset price becomes unstable. The shaded regions indicate the range of parametersfor which the system (11) is stable.When we set q = 0 , the model simplifies somewhat, leaving a linear interface betweenthe regions of stability and instability. We then have the following theorem. We define Q :=1 − q − q . Theorem 1
Consider the system (11). With q = 0 , one has stability of the system (11) if andonly if Q + 1 τ > (15) . Proof.
Setting q = 0 , the necessary conditions become (cid:18) Q + 1 τ (cid:19) (cid:18) τ + Qc + 1 c τ + 1 c (cid:19) > and Q + 1 τ + 1 c > (16)We prove this is equivalent to Q + τ > .(i) Assume Q + τ > . Then clearly the second inequality in (16) is satisfied. Also, onehas τ + Qc + 1 c τ + 1 c = (cid:18) Q + 1 τ (cid:19) (cid:18) c (cid:19) + 1 τ + 1 c > , (17)8atisfying the first inequality.(ii) Suppose (16) holds. Then clearly < (cid:18) Q + 1 τ + 1 c (cid:19) c + 1 τ = 1 τ + Qc + 1 c τ + 1 c , (18)implying (15). In order to isolate the effect of liquidity, we eliminate the role of investor sentiment and valueby setting the associated parameters to zero. To this end, we are left with the system τ P (cid:48) = L − P,cL (cid:48) = 1 + ( q − L − qP. (19)One readily calculates that there will be positive eigenvalues of the linearized system if and onlyif q > cτ In other words, in a system where only price and liquidity are relevant, a largeamplitude q of liquidity is destabilizing while a large time scale for the liquidity is stabilizing.The stability is illustrated in the Figure 2.Another nontrivial subcase is obtained from examining the full model (10) in the case wherewe set the value component of the sentiment, ζ , and the fundamental value equal to zero. Wethen have the system of equations τ dPdt = (1 + 2 ζ ) L − Pc dLdt = 1 − L + q (1 + 2 ζ ) L − qPc dζ dt = q (1 + 2 ζ ) LP − q − ζ (20)One then observes that the only equilibrium point is L = P = L and ζ = 0 . Recalling that Q := 1 − q − q , one has the following. Theorem 2
The system (20) incorporating valuation and sentiment (with c := c ) is stable ifand only if Q + cτ > , (21) i.e. if the perturbations from trend and valuation sentiment are sufficiently small as a relativecomparison to the timescale of reaction with respect to price. roof. By scaling, assume without loss of generality that c = c = 1 ; then we can linearize thesystem as follows: PLζ (cid:48) = − /τ /τ /τ − q q − q − q q q − PLζ =: A PLζ . (22)Leaving aside the eigenvalue of − that is present for all values of the parameters, the matrix A has eigenvalues with positive real part if and only if q + 2 q > τ . (23)After rescaling, this is the statement of the theorem.Furthermore, we have either zero or two roots with positive real parts, so that we will havea stable spiral for Q + cτ > and an unstable spiral for Q − cτ < for the equilibrium pointat (1 , , . This matches our intuition from an economics perspective since one has instabilitywhen q + 2 q > cτ , i.e., there will be stability if q + 2 q < regardless of c and τ . For q + 2 q > , one sees that instability arises when cτ is sufficiently small, i.e. traders are focusedon short term trends.The analysis above clearly shows that the potential stability of a crypto-asset may be con-tingent on several parameters that one may be able to influence. With this information, furtherresearch may be useful to examine the correlations and fit of these parameters with the effectsof news and government policy. A problem of future interest would be whether, and if so how,governmental policy might be developed to diminish the volatility in cryptocurrencies. Anotheralternative would be a decentralized cryptocurrency with a concrete value. A good index tobase this on would be either current or future gross world product (which could be estimatedvia futures markets). For a nominal fee, holders of this currency would be able to demand abasket of underlying currencies (such as dollar, euro, yen, etc.), which would keep the value ofsuch a currency relatively close to its true fundamental value. References and Notes
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Supplementary materials
N/A 13igure 1: Stability of the × system in the K − q plane for different values of the timescales c and τ . Increasing c and decreasing τ increases the region of linear stability for theequations. 14igure 2: Stability for our simplified model without the presence of fundamental value or sen-timent. The system is stable in the shaded region for the parameters q and cτ0