This dissertation presents the application of microstructure theory on the RMB exchange rate\nand Bitcoin market price. The existing research on the RMB exchange rate and Bitcoin market price\nmainly studied their statistical characteristics through empirical methodologies. This dissertation\nfills the research gap in microstructure theory applied to the RMB exchange rate and Bitcoin market\nprice. First, the model for the determination of the two Renminbi (RMB) exchange rates and their\ninteractions is established, and empirical analysis suggests that the interactions among the two\nexchange rates and the explanatory variables are time-varying, in particular, after the "811 RMB\nexchange rate reform", the offshore RMB exchange rate replaced the onshore RMB exchange rate as\nthe leading indicator. Second, a model describing the speculative behavior in the Bitcoin trading\nmarket is developed. This theoretical model captures the statistical characteristics of Bitcoin\nmarket prices. The fundamental value of Bitcoin system is controversial, and the mysterious and\ninnovative features of the Bitcoin system incite the speculation behaviours. The speculation leads\nto the market bubble that brought the soaring and plunges of Bitcoin market price. Finally, an\neconomic model for Bitcoin mining competition based on the Bitcoin protocol is established, which\nprovides a benchmark for further research on mining competition in economics. For any Bitcoin\nminers, the equilibrium input depends on the comparison of the miner's own marginal cost with\nthat of other miners, however, whether profit can be obtained or not depends on the miner's own\nfixed cost.
The bitcoin price has surged in recent years and it has also exhibited phases of rapid decay. In this paper we address the question to what extent this novel cryptocurrency market can be viewed as a classic or semi-efficient market. Novel and robust tools for estimation of multi-fractal properties are used to show that the bitcoin price exhibits a very interesting multi-scale correlation structure. This structure can be described by a power-law behavior of the variances of the returns as functions of time increments and it can be characterized by two parameters, the volatility and the Hurst exponent. These power-law parameters, however, vary in time. A new notion of generalized Hurst exponent is introduced which allows us to check if the multi-fractal character of the underlying signal is well captured. It is moreover shown how the monitoring of the power-law parameters can be used to identify regime shifts for the bitcoin price. A novel technique for identifying the regimes switches based on a goodness of fit of the local power-law parameters is presented. It automatically detects dates associated with some known events in the bitcoin market place. A very surprising result is moreover that, despite the wild ride of the bitcoin price in recent years and its multi-fractal and non-stationary character, this price has both local power-law behaviors and a very orderly correlation structure when it is observed on its entire period of existence.
The recent evolution of cryptocurrencies has been characterized by bubble-like behavior and extreme volatility. While it is difficult to assess an intrinsic value to a specific cryptocurrency, one can employ recently proposed bubble tests that rely on recursive applications of classical unit root tests. This paper extends this approach to the case where volatility is time varying, assuming a deterministic long-run component that may take into account a decrease of unconditional volatility when the cryptocurrency matures with a higher market dissemination. Volatility also includes a stochastic short-run component to capture volatility clustering. The wild bootstrap is shown to correctly adjust the size properties of the bubble test, which retains good power properties. In an empirical application using eleven of the largest cryptocurrencies and the CRIX index, the general evidence in favor of bubbles is confirmed, but much less pronounced than under constant volatility.
Aurelio F. Bariviera, María José Basgall, Waldo Hasperué, Marcelo Naiouf
In recent years a new type of tradable assets appeared, generically known as cryptocurrencies. Among them, the most widespread is Bitcoin. Given its novelty, this paper investigates some statistical properties of the Bitcoin market. This study compares Bitcoin and standard currencies dynamics and focuses on the analysis of returns at different time scales. We test the presence of long memory in return time series from 2011 to 2017, using transaction data from one Bitcoin platform. We compute the Hurst exponent by means of the Detrended Fluctuation Analysis method, using a sliding window in order to measure long range dependence. We detect that Hurst exponents changes significantly during the first years of existence of Bitcoin, tending to stabilize in recent times. Additionally, multiscale analysis shows a similar behavior of the Hurst exponent, implying a self-similar process.
Alessandra Cretarola, Gianna Figà‐Talamanca, Marco Patacca
In recent literature it is claimed that BitCoin price behaves more likely to a volatile stock asset than a currency and that changes in its price are influenced by sentiment about the BitCoin system itself; in Kristoufek [10] the author analyses transaction based as well as popularity based potential drivers of the BitCoin price finding positive evidence. Here, we endorse this finding and consider a bivariate model in continuous time to describe the price dynamics of one BitCoin as well as a second factor, affecting the price itself, which represents a sentiment indicator. We prove that the suggested model is arbitrage-free under a mild condition and, based on risk-neutral evaluation, we obtain a closed formula to approximate the price of European style derivatives on the BitCoin. By applying the same approximation technique to the joint likelihood of a discrete sample of the bivariate process, we are also able to fit the model to market data. This is done by using both the Volume and the number of Google searches as possible proxies for the sentiment factor. Further, the performance of the pricing formula is assessed on a sample of market option prices obtained by the website deribit.com.
Oriane Blondel, Patrícia Gonçalves, Marielle Simon
In this paper we prove the convergence to the stochastic Burgers equation\nfrom one-dimensional interacting particle systems, whose dynamics allow the\ndegeneracy of the jump rates. To this aim, we provide a new proof of the second\norder Boltzmann-Gibbs principle introduced in [Gon\\c{c}alves, Jara 2014]. The\nmain technical difficulty is that our models exhibit configurations that do not\nevolve under the dynamics - the blocked configurations - and are locally\nnon-ergodic. Our proof does not impose any knowledge on the spectral gap for\nthe microscopic models. Instead, it relies on the fact that, under the\nequilibrium measure, the probability to find a blocked configuration in a\nfinite box is exponentially small in the size of the box. Then, a dynamical\nmechanism allows to exchange particles even when the jump rate for the direct\nexchange is zero.\n
Oriane Blondel, Patrícia Gonçalves, Marielle Simon
In this paper we prove the convergence to the stochastic Burgers equation from one-dimensional interacting particle systems, whose dynamics allow the degeneracy of the jump rates. To this aim, we provide a new proof of the second order Boltzmann-Gibbs principle introduced in [7]. The main technical difficulty is that our models exhibit configurations that do not evolve under the dynamics - the blocked configurations - and are locally non-ergodic. Our proof does not impose any knowledge on the spectral gap for the microscopic models. Instead, it relies on the fact that, under the equilibrium measure, the probability to find a blocked configuration in a finite box is exponentially small in the size of the box. Then, a dynamical mechanism allows to exchange particles even when the jump rate for the direct exchange is zero.
Bitcoin, the first electronic payment system, is becoming a popular currency. We provide a statistical analysis of the log-returns of the exchange rate of Bitcoin versus the United States Dollar. Fifteen of the most popular parametric distributions in finance are fitted to the log-returns. The generalized hyperbolic distribution is shown to give the best fit. Predictions are given for future values of the exchange rate.
Hubert Lacoin, François Simenhaus, Fabio, Lucio Toninelli
Let \mathcal D be a simply connected, smooth enough domain of \mathbb R^2 . For L>0 consider the continuous time, zero-temperature heat bath dynamics for the nearest-neighbor Ising model on \mathbb Z^2 with initial condition such that \sigma_x=-1 if x\in L\mathcal D and \sigma_x=+1 otherwise. It is conjectured [23] that, in the diffusive limit where space is rescaled by L , time by L^2 and L\to\infty , the boundary of the droplet of " - " spins follows a deterministic anisotropic curve-shortening flow, where the normal velocity at a point of its boundary is given by the local curvature times an explicit function of the local slope. The behavior should be similar at finite temperature T<T_c , with a different temperature-dependent anisotropy function. We prove this conjecture (at zero temperature) when \mathcal D is convex. Existence and regularity of the solution of the deterministic curve-shortening flow is not obvious a priori and is part of our result. To our knowledge, this is the first proof of mean curvature-type droplet shrinking for a model with genuine microscopic dynamics.
We study the Glauber dynamics for the zero-temperature Ising model in dimension d=4 with "plus" boundary condition.Let T+ be the time needed for an hypercube of size L entirely filled with "minus" spins to become entirely "plus". We prove that T+ is O(L^2(log L)^c) for some constant c, not depending on the dimension. This brings further rigorous justification for the so-called "Lifshitz law" T+ = O(L^2) [5, 3] conjectured on heuristic grounds. The key point of our proof is to use the detail knowledge that we have on the three-dimensional problem: results for fluctuation of monotone interfaces at equilibrium and mixing time for monotone interfaces dynamics extracted from [2], to get the result in higher dimension.
We study the Glauber dynamics for the zero-temperature Ising model in\ndimension d=4 with "plus" boundary condition.Let T+ be the time needed for an\nhypercube of size L entirely filled with "minus" spins to become entirely\n"plus". We prove that T+ is O(L^2(log L)^c) for some constant c, not depending\non the dimension. This brings further rigorous justification for the so-called\n"Lifshitz law" T+ = O(L^2) [5, 3] conjectured on heuristic grounds. The key\npoint of our proof is to use the detail knowledge that we have on the\nthree-dimensional problem: results for fluctuation of monotone interfaces at\nequilibrium and mixing time for monotone interfaces dynamics extracted from\n[2], to get the result in higher dimension.\n
Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models. Only recently have some of these predictions become accessible to mathematical proof. One of the new developments is the discovery of a one-parameter family of random curves called stochastic Loewner evolution or SLE. The SLE curves appear as limits of interfaces or paths occurring in a variety of statistical physics models as the mesh of the grid on which the model is defined tends to zero. The main purpose of this article is to list a collection of open problems. Some of the open problems indicate aspects of the physics knowledge that have not yet been understood mathematically. Other problems are questions about the nature of the SLE curves themselves. Before we present the open problems, the definition of SLE will be motivated and explained, and a brief sketch of recent results will be presented.