Cryptocurrency and Efficient Market Hypothesis. Drunken Risibility.

According to the traditional definition, a currency has three main properties: (i) it serves as a medium of exchange, (ii) it is used as a unit of account and (iii) it allows to store value. Along economic history, monies were related to political power. In the beginning, coins were minted in precious metals. Therefore, the value of a coin was intrinsically determined by the value of the metal itself. Later, money was printed in paper bank notes, but its value was linked somewhat to a quantity in gold, guarded in the vault of a central bank. Nation states have been using their political power to regulate the use of currencies and impose one currency (usually the one issued by the same nation state) as legal tender for obligations within their territory. In the twentieth century, a major change took place: abandoning gold standard. The detachment of the currencies (specially the US dollar) from the gold standard meant a recognition that the value of a currency (specially in a world of fractional banking) was not related to its content or representation in gold, but to a broader concept as the confidence in the economy in which such currency is based. In this moment, the value of a currency reflects the best judgment about the monetary policy and the “health” of its economy.

In recent years, a new type of currency, a synthetic one, emerged. We name this new type as “synthetic” because it is not the decision of a nation state, nor represents any underlying asset or tangible wealth source. It appears as a new tradable asset resulting from a private agreement and facilitated by the anonymity of internet. Among this synthetic currencies, Bitcoin (BTC) emerges as the most important one, with a market capitalization of a few hundred million short of $80 billions.

Bitcoin Price Chart from Bitstamp

There are other cryptocurrencies, based on blockchain technology, such as Litecoin (LTC), Ethereum (ETH), Ripple (XRP). The website https://coinmarketcap.com/currencies/ counts up to 641 of such monies. However, as we can observe in the figure below, Bitcoin represents 89% of the capitalization of the market of all cryptocurrencies.

Cryptocurrencies. Share of market capitalization of each currency.

One open question today is if Bitcoin is in fact a, or may be considered as a, currency. Until now, we cannot observe that Bitcoin fulfills the main properties of a standard currency. It is barely (though increasingly so!) accepted as a medium of exchange (e.g. to buy some products online), it is not used as unit of account (there are no financial statements valued in Bitcoins), and we can hardly believe that, given the great swings in price, anyone can consider Bitcoin as a suitable option to store value. Given these characteristics, Bitcoin could fit as an ideal asset for speculative purposes. There is no underlying asset to relate its value to and there is an open platform to operate round the clock.

Bitcoin returns, sampled every 5 hours.

Speculation has a long history and it seems inherent to capitalism. One common feature of speculative assets in history has been the difficulty in valuation. Tulipmania, the South Sea bubble, and more others, reflect on one side human greedy behavior, and on the other side, the difficulty to set an objective value to an asset. All speculative behaviors were reflected in a super-exponential growth of the time series.

Cryptocurrencies can be seen as the libertarian response to central bank failure to manage financial crises, as the one occurred in 2008. Also cryptocurrencies can bypass national restrictions to international transfers, probably at a cheaper cost. Bitcoin was created by a person or group of persons under the pseudonym Satoshi Nakamoto. The discussion of Bitcoin has several perspectives. The computer science perspective deals with the strengths and weaknesses of blockchain technology. In fact, according to R. Ali et. al., the introduction of a “distributed ledger” is the key innovation. Traditional means of payments (e.g. a credit card), rely on a central clearing house that validate operations, acting as “middleman” between buyer and seller. On contrary, the payment validation system of Bitcoin is decentralized. There is a growing army of miners, who put their computer power at disposal of the network, validating transactions by gathering together blocks, adding them to the ledger and forming a ’block chain’. This work is remunerated by giving the miners Bitcoins, what makes (until now) the validating costs cheaper than in a centralized system. The validation is made by solving some kind of algorithm. With the time solving the algorithm becomes harder, since the whole ledger must be validated. Consequently it takes more time to solve it. Contrary to traditional currencies, the total number of Bitcoins to be issued is beforehand fixed: 21 million. In fact, the issuance rate of Bitcoins is expected to diminish over time. According to Laursen and Kyed, validating the public ledger was initially rewarded with 50 Bitcoins, but the protocol foresaw halving this quantity every four years. At the current pace, the maximum number of Bitcoins will be reached in 2140. Taking into account the decentralized character, Bitcoin transactions seem secure. All transactions are recorded in several computer servers around the world. In order to commit fraud, a person should change and validate (simultaneously) several ledgers, which is almost impossible. Additional, ledgers are public, with encrypted identities of parties, making transactions “pseudonymous, not anonymous”. The legal perspective of Bitcoin is fuzzy. Bitcoin is not issued, nor endorsed by a nation state. It is not an illegal substance. As such, its transaction is not regulated.

In particular, given the nonexistence of saving accounts in Bitcoin, and consequently the absense of a Bitcoin interest rate, precludes the idea of studying the price behavior in relation with cash flows generated by Bitcoins. As a consequence, the underlying dynamics of the price signal, finds the Efficient Market Hypothesis as a theoretical framework. The Efficient Market Hypothesis (EMH) is the cornerstone of financial economics. One of the seminal works on the stochastic dynamics of speculative prices is due to L Bachelier, who in his doctoral thesis developed the first mathematical model concerning the behavior of stock prices. The systematic study of informational efficiency begun in the 1960s, when financial economics was born as a new area within economics. The classical definition due to Eugene Fama (Foundations of Finance_ Portfolio Decisions and Securities Prices 1976-06) says that a market is informationally efficient if it “fully reflects all available information”. Therefore, the key element in assessing efficiency is to determine the appropriate set of information that impels prices. Following Efficient Capital Markets, informational efficiency can be divided into three categories: (i) weak efficiency, if prices reflect the information contained in the past series of prices, (ii) semi-strong efficiency, if prices reflect all public information and (iii) strong efficiency, if prices reflect all public and private information. As a corollary of the EMH, one cannot accept the presence of long memory in financial time series, since its existence would allow a riskless profitable trading strategy. If markets are informationally efficient, arbitrage prevent the possibility of such strategies. If we consider the financial market as a dynamical structure, short term memory can exist (to some extent) without contradicting the EMH. In fact, the presence of some mispriced assets is the necessary stimulus for individuals to trade and reach an (almost) arbitrage free situation. However, the presence of long range memory is at odds with the EMH, because it would allow stable trading rules to beat the market.

The presence of long range dependence in financial time series generates a vivid debate. Whereas the presence of short term memory can stimulate investors to exploit small extra returns, making them disappear, long range correlations poses a challenge to the established financial model. As recognized by Ciaian et. al., Bitcoin price is not driven by macro-financial indicators. Consequently a detailed analysis of the underlying dynamics (Hurst exponent) becomes important to understand its emerging behavior. There are several methods (both parametric and non parametric) to calculate the Hurst exponent, which become a mandatory framework to tackle BTC trading.

Hyperbolic Brownian Sheet, Parabolic and Elliptic Financials. (Didactic 3)

Financial and economic time series are often described to a first degree of approximation as random walks, following the precursory work of Bachelier and Samuelson. A random walk is the mathematical translation of the trajectory followed by a particle subjected to random velocity variations. The analogous physical system described by SPDE’s is a stochastic string. The length along the string is the time-to-maturity and the string configuration (its transverse deformation) gives the value of the forward rate f(t,x) at a given time for each time-to-maturity x. The set of admissible dynamics of the configuration of the string as a function of time depends on the structure of the SPDE. Let us for the time being restrict our attention to SPDE’s in which the highest derivative is second order. This second order derivative has a simple physical interpretation : the string is subjected to a tension, like a piano chord, that tends to bring it back to zero transverse deformation. This tension forces the “coupling” among different times-to-maturity so that the forward rate curve is at least continuous. In principle, the most general formulation would consider SPDE’s with terms of arbitrary derivative orders. However, it is easy to show that the tension term is the dominating restoring force, when present, for deformations of the string (forward rate curve) at long “wavelengths”, i.e. for slow variations along the time-to-maturity axis. Second order SPDE’s are thus generic in the sense of a systematic expansion.

In the framework of second order SPDE’s, we consider hyperbolic, parabolic and elliptic SPDE’s, to characterize the dynamics of the string along two directions : inertia or mass, and viscosity or subjection to drag forces. A string that has “inertia” or, equivalently, “mass” per unit length, along with the tension that keeps it continuous, is characterized by the class of hyperbolic SPDE’s. For these SPDE’s, the highest order derivative in time has the same order as the highest order derivative in distance along the string (time-to-maturity). As a consequence, hyperbolic SPDE’s present wave-like solutions, that can propagate as pulses with a “velocity”. In this class, we find the so-called “Brownian sheet” which is the direct generalization of Brownian motion to higher dimensions, that preserves continuity in time-to-maturity. The Brownian sheet is the surface spanned by the string configurations as time goes on. The Brownian sheet is however non-homogeneous in time-to-maturity.

If the string has no inertia, its dynamics are characterized by parabolic SPDE’s. These stochastic processes lead to smoother diffusion of shocks through time, along time-to-maturity. Finally, the third class of SPDE’s of second-order, namely elliptic partial differential equations. Elliptic SPDE’s give processes that are differentiable both in x and t. Therefore, in the strict limit of continuous trading, these stochastic processes correspond to locally riskless interest rates.

The general form of SPDE’s reads

A(t,x) ∂²f(t,x)/∂t² + 2B(t,x) ∂²f(t,x)/∂t∂x + C(t,x) ∂²f(t,x)/∂x² = F(t,x,f(t,x), ∂f(t,x)/∂t, ∂f(t,x)/∂x, S) —– (1)

where f (t, x) is the forward rate curve. S(t, x) is the “source” term that will be generally taken to be Gaussian white noise η(t, x) characterized by the covariance

Cov η(t, x), η(t′, x′) = δ(t − t′) δ(x − x′) —– (2)

where δ denotes the Dirac distribution. Equation (1) is the most general second-order SPDE in two variables. For arbitrary non-linear terms in F, the existence of solutions is not warranted and a case by case study must be performed. For the cases where F is linear, the solution f(t,x) exists and its uniqueness is warranted once “boundary” conditions are given, such as, for instance, the initial value of the function f(0,x) as well as any constraints on the particular form of equation (1).

Equation (1) is defined by its characteristics, which are curves in the (t, x) plane that come in two families of equation :

Adt = (B + √(B2 − AC))dx —– (3)

Adt = (B − √(B2 − AC))dx —– (4)

These characteristics are the geometrical loci of the propagation of the boundary conditions.

Three cases must be considered.

• When B² > AC, the characteristics are real curves and the corresponding SPDE’s are called “hyperbolic”. For such hyperbolic SPDE’s, the natural coordinate system is formed from the two families of characteristics. Expressing (1) in terms of these two natural coordinates λ and μ, we get the “normal form” of hyperbolic SPDE’s :

∂²f/∂λ∂μ = P (λ,μ) ∂f/∂λ +Q (λ,μ) ∂f/∂μ + R (λ,μ)f + S(λ,μ) —– (5)

The special case P = Q = R = 0 with S(λ,μ) = η(λ,μ) corresponds to the so-called Brownian sheet, well studied in the mathematical literature as the 2D continuous generalization of the Brownian motion.

• When B² = AC, there is only one family of characteristics, of equation

Adt = Bdx —– (6)

Expressing (1) in terms of the natural characteristic coordinate λ and keeping x, we get the “normal form” of parabolic SPDE’s :

∂²f/∂x² = K (λ,μ)∂f/∂λ +L (λ,μ)∂f/∂x +M (λ,μ)f + S(λ,μ) —– (7)

The diffusion equation, well-known to be associated to the Black-Scholes option pricing model, is of this type. The main difference with the hyperbolic equations is that it is no more invariant with respect to time-reversal t → −t. Intuitively, this is due to the fact that the diffusion equation is not conservative, the information content (negentropy) continually decreases as time goes on.

• When B² < AC, the characteristics are not real curves and the corresponding SPDE’s are called “elliptic”. The equations for the characteristics are complex conjugates of each other and we can get the “normal form” of elliptic SPDE’s by using the real and imaginary parts of these complex coordinates z = u ± iv :

∂²f/∂u² + ∂²f/∂v² = T ∂f/∂u + U ∂f/∂v + V f + S —– (8)

There is a deep connection between the solution of elliptic SPDE’s and analytic functions of complex variables.

Hyperbolic and parabolic SPDE’s provide processes reducing locally to standard Brownian motion at fixed time-to-maturity, while elliptic SPDE’s give locally riskless time evolutions. Basically, this stems from the fact that the “normal forms” of second-order hyperbolic and parabolic SPDE’s involve a first-order derivative in time, thus ensuring that the stochastic processes are locally Brownian in time. In contrast, the “normal form” of second-order elliptic SPDE’s involve a second- order derivative with respect to time, which is the cause for the differentiability of the process with respect to time. Any higher order SPDE will be Brownian-like in time if it remains of order one in its time derivatives (and higher-order in the derivatives with respect to x).