This drop-in volatility was probably partly due to the good economic conditions of the moment and with a resolutely accommodating orientation of monetary policy see also ECB In February , however, market volatility suddenly increased after the release of relative figures inflation and employment in the United States.
These figures suggested that the Fed could normalize its policy faster than expected, which caused the stock market indices to fall. This episode, therefore, seems to be part of the high frequency component of volatility, that is, an increase in volatility almost unpredictable and without major consequences for the real economy.
In October , a further correction occurred on the stock markets and volatility increased again. This event indicates that the markets remain particularly responsive to specific announcements in the context of a rate range recovery Fed director, declining net asset purchases by the ECB, trade and political tensions Brexit, Italy.
When the new information coming into the market, it can be disrupted and this affects the anticipation of shareholders for the evolution of the return. The resulting plots of the smoothed volatilities are shown in Fig. We take our analysis in the Bitcoin cryptocurrency, but the other is reported in Appendix 1 Fig.
This has proved that the algorithm used for estimated volatility is a good choice. The basic SVOL model mis-specified can induce substantial parameter bias and error in inference about vt, Geweke a , b showed that the basic SVOL has the same problem with the largest outlier, summer The vt for the model SVOL reveals a big outlier on period crises.
The corresponding plots of innovation are shown in Fig. Appendix B Fig. The Quantile—Quantile plot can be used to determine whether two data sets come from populations with a common distribution. In this graphical technique, quantiles of the first data set against the quantiles of the second data set are plotted and a degree reference line is used to interpret.
If the two data sets come from a population with the same distribution the points should fall approximately along this reference line. The greater the departure from this reference line, the greater the evidence for the conclusion that the two data sets have come from populations with different distributions. The advantages of Asymmetric basic SV are able to capture some aspects of the financial market and the main properties of their volatility behavior Danielsson ; Eraker et al.
The following figure illustrates the Q—Q plot drawn to evaluate the fitted Scaled t-distribution graphically. In general, Q—Q plot demonstrates linear patterns which confirms that the two distributions considered in the graph are similar. Therefore, it can be said that the fitted Scaled t distributions with respective parameters exhibit a good fit for the return distribution for all indexes. The algorithm used to estimate the volatility using the SV-t model ensures much higher normality for bitcoin than the SVOL.
The MCMC is more efficient for all parameters used in these two models. A certain threshold, all parameters are stable and converge to a certain level. Appendix C and D Figs. The results for both simulated show that the algorithm of the SV-t model is fast and converges rapidly with acceptable levels of numerical efficiency.
Then our sampling provides strong evidence of convergence of the chain. Bitcoin is an innovative payment network and a new kind of money, then is a digital payment currency that utilizes crypto-currency and peer-to-peer technology to create and manage monetary transactions.
We have applied these MCMC methods to the study of various indexes and crypto-currency. We wanted to study the behavior of this cryptocurrency, and it turned out that it behaves like the stock market indices of the different international markets facing different shocks Efe Caglar The empirical results show that the SV-t model can describe extreme values to a certain extent and it is more appropriate to accommodate outlier.
Our result proves the efficiency of Markov Chain for our sample and the convergence and stability for all parameter to a certain level. Our results are consistent with the work of Dyhrberg and contradict those of Baur et al. This view is endorsed by many researchers, e. Glaser et al. To our knowledge, no paper has processed bitcoin by conducting a comparative study using stochastic volatility models.
Our results prove that this cryptocurrency does not behave differently from stock market indices although it is traded on a virtual market. Observing the market, several investors are wary of this cryptocurrency while justifying themselves by several considerations. This paper has made certain contributions, but several extensions are still possible, and it can find the best results if you opt for extensions of SVOL like the model of Singleton , knight et al.
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Reserve Bank St. Louis Rev. Second Quarter , pp — Download references. Correspondence to A. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. See Fig. Reprints and Permissions. Hachicha, A. Rev Quant Finan Acc 56, — Download citation. Published : 06 July Issue Date : February Without loss of generality, we assume that the interest rate is fixed and equal to zero.
In this setting, the discounted Bitcoin price trend and the market attention factor dynamics are described by. The aim is to investigate the existence of asset-price bubbles in the underlying Bitcoin market model. By simulating trajectories for the asset price S according to the model in Eq. The mathematical theory of financial bubbles is developed, among others, in [ 23 , 24 , 25 ]. Precisely, we introduce the following definition from [ 23 ].
Definition 4. The Bitcoin price process S has a bubble on the time interval 0 T if S is a strict F -local martingale under the chosen risk-neutral measure. The term strict F -local martingale refers to the fact that S is an F -local martingale, but not a true F -martingale under the chosen risk-neutral measure.
Further, since S is nonnegative, we must have that S is an F -supermartingale we refer to [ 26 ] for rigorous definitions and related concepts. Remark 4. Note that stock bubbles arise if S has an equivalent local martingale measure but not an equivalent martingale measure. Arbitrage appears only if no equivalent local martingale measure exists.
The local martingale property of the discounted Bitcoin price process S under Q implies the following condition:. We have the following result, which allows to detect the presence of bubbles in this setting. Proposition 4. In the model outlined in Eq. Precisely, it is possible to show that the martingale property of the discounted stock price S under Q 0 , given in Eq.
Hence, a bubble arises if and only if the correlation parameter between stock returns and market attention is positive. Let us generalize the model introduced in Eq. We get. Analogous results as those in Section 2 can be derived by similar computations, and model parameters, for a fixed delay, can be estimated by means of the maximum likelihood method. In order to estimate the delay parameter, we maximize the profile likelihood as defined in [ 15 ].
Details of this procedure can be found in [ 10 ]. The estimation results of model in Eq. Parameter estimates for model in Eq. In Figure 3 , we plot simulated trajectories of the price process in Eq. The different delays result in a shift to the south-east between the faster and slower reacting trajectories; in the picture, this behavior is sharp since the other model parameters are kept constant.
By looking at the picture, the idea to model the price of Bitcoin in different exchanges by the same model in Eq. Note that within this model, prices for Bitcoin traded in different exchanges are perfectly correlated. Indeed, this is what happens in observed data; considering daily prices from January to June for Bitstamp, Kraken, Cex.
We fit model in Eq. Model fitting with delay parameter: outcomes for Bitstamp and Gdax exchanges when attention is measured by the trading volume. Model fitting with delay parameter: outcomes for Bitstamp and Gdax exchanges when attention is measured by the SVI index. It is evident from the outcomes in Table 4 that the model parameters are not significantly different while the delay might be quite different as if the reaction to the attention factor is faster for some exchanges and slower for others.
On the contrary, when attention is measured by the Google SVI Index, the delay is unchanged, but the difference between estimated parameters for the price dynamics is nonnegligible. By analyzing the outcomes and considering the shift effect as depicted in Figure 3 , it is tempting to conjecture that the faster reaction determines the leader exchanges and that the slower exchange will then follow.
If we could forecast that the next day price of the slower exchange will reach the price today for the faster one, we could obtain a profit by suitably investing in the two exchanges. However, it is worth noticing that the estimation of the delay parameter is obtained by maximizing the likelihood over a whole time series and is a product of averaging so arbitrage cannot be achieved in a direct way.
Nevertheless, in a multivariate setting as ours, the theory guarantees that arbitrage opportunities are ruled out if the market price of risk in the market is unique. It is evident that these values are not equal if we plug parameter estimates in Eq.
We will address this issue more precisely in future research. In this chapter, we have introduced a model in continuous time in order to describe the dynamics of Bitcoin price depending on an exogenous stochastic factor, which represents market attention on the Bitcoin system.
Market attention is measured either by the total trading volume in Bitcoins or by means of the Google Search Volume Index, which, as suggested in [ 16 ], is a direct measure of the revealed attention for uniformed retail investors. More precisely, the attention factor affects directly the instantaneous mean and volatility of logarithmic returns; in addition, it may be also correlated with the price changes.
An estimation procedure to fit the model to observed data is also suggested and, under the assumption of no correlation, a closed formula for standard European option prices on Bitcoin is provided. By applying outcomes within the mathematical theory of bubbles [ 23 , 24 , 25 , 27 ], we are able to show that Bitcoin boosts in a bubble if and only if there is a positive correlation between changes in the price and in the attention factor. This finding is reasonable and claims that a stronger positive dependence between the two processes in Eq.
Finally, we allow for a delay on the effect of market attention on the Bitcoin price, and, based on this generalized model, we introduce a multivariate setting for our model Eq. Preliminary results indicate that arbitrage opportunities may arise in two exchanges that are characterized by different delays.
The authors also wish to thank Stefano Bistarelli for having introduced them to the intriguing and worth to explore world of cryptocurrencies. Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3. Help us write another book on this subject and reach those readers. Login to your personal dashboard for more detailed statistics on your publications. Edited by Asma Salman. We are IntechOpen, the world's leading publisher of Open Access books.
Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. Downloaded: Abstract The goal of this chapter is to present recent developments about Bitcoin1 price modeling and related applications. Keywords Bitcoin market attention arbitrage option pricing bubbles. The Bitcoin network. An attention-based model. Statistical properties and model fitting We collect in this subsection the properties of the logarithmic returns obtained by the price process defined in Eq.
Then it is straightforward to prove the following: Theorem 2. Table 1. A closed formula for Bitcoin option prices. A numerical application In order to appreciate the performance of the pricing formula in Eq. Table 2. The presence of model stock bubbles. Toward a multiexchange generalization. Table 3. Table 4. Table 5. Conflict of interest.
Notes We use the following rule throughout the paper: the term BitCoin refers to the whole system network while Bitcoin refers to the digital currency. More Print chapter.
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Another rally and associated crash occurred towards the end of that year. The fifth price bubble occurred in The hot streak also helped place Bitcoin firmly in the mainstream spotlight. Governments and economists took notice and began developing digital currencies to compete with Bitcoin. Analysts debated its value as an asset even as a slew of so-called experts and investors made extreme price forecasts. As in the past, Bitcoin's price moved sideways for the next two years.
In between, there were signs of life. It was not until , when the economy shut down due to the pandemic, that Bitcoin's price burst into activity once again. The pandemic shutdown, and subsequent government policy, fed into investors' fears about the global economy and accelerated Bitcoin's rise. Upon the release of those checks the entire stock market, including cryptocurrency, saw a huge rebound from March lows and even continued past their previous all-time-highs.
These checks further amplified concerns over inflation and a potentially weakened purchasing power of the U. Money printing by governments and central banks helped to bolster the narrative of Bitcoin as a store of value as its supply is capped at 21 million.
This narrative began to draw interest among institutions instead of just retail investors, who were largely responsible for the run up in price in Its price has mostly mimicked the classic Gartner Hype Cycle of peaks due to hype about its potential and troughs of disillusionment that resulted in crashes. And so, each swell and ebb in Bitcoin's price has shone a spotlight on the shortcomings of its ecosystem and provided a fresh infusion of investor funds to develop its infrastructure.
Previous analysis of Bitcoin's price made the case that its price was a function of its velocity or its use as a currency for daily transactions and trading. But crypto trading volumes are a fraction of their mainstream counterparts and Bitcoin never really took off as a medium of daily transaction. This is partly due to the fact that the narrative around Bitcoin has changed from being a currency to a store of value, where people buy and hold for long periods of time rather than use it for transactions.
This state of affairs translated to wide price swings when investors booked profits or when an adverse industry development, such as a ban on cryptocurrency exchanges, was reported. The rise and fall of cryptocurrency exchanges, which controlled considerable stashes of Bitcoin, also influenced Bitcoin's price trajectory.
Events at Mt. Even earlier, in December , rumors of poor management and lax security practices at Mt. In recent times, the matrix of factors affecting Bitcoin price has changed considerably. Depending on whether it is positive or negative, each regulatory pronouncement increases or decreases prices for Bitcoin. Interest from institutional investors has also cast an ever-lengthening shadow on Bitcoin price workings. In the last ten years, Bitcoin has pivoted away from retail investors and become an attractive asset class for institutional investors.
This is construed as a desirable development because it brings more liquidity into the ecosystem and tamps down volatility. The use of Bitcoin for treasury management at companies also strengthened its price in MicroStrategy Inc. SQ have both announced commitments to using Bitcoin, instead of cash, as part of their corporate treasuries. Industry developments are the third major influence on Bitcoin's price.
Bitcoin halving events, in which the total supply of Bitcoin available in the market declines due to a reduction in miner rewards because of an algorithmic change, have also catalyzed price increases. Among many factors, the halving in the reward given to miners that also doubles the asset's stock-to-flow ratio seems to have a large effect on Bitcoin's price.
Finally, economic instability is another indicator of price changes for Bitcoin. Since its inception, the cryptocurrency has positioned itself as a supranational hedge against local economic instability and government-controlled fiat currency. Countries like Venezuela, which have experienced hyperinflation of their currency, have seen huge increases in the use of Bitcoin as a means of transaction as well as storing wealth. For example, capital controls announced by the Chinese government were generally accompanied by an uptick in Bitcoin's price.
The pandemic shutdown produced macroeconomic instability on a global scale and galvanized Bitcoin's price, resulting in a record rally. Predictions for the future value of bitcoin vary based on who makes the estimate. Crypto Research Report.
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