Cryptocurrencies 
Bitcoin vs. Ethereum: What's the Difference?Cryptocurrencies differentialby Nikot В» 31.12.2019 .
The goal of this chapter is to present recent developments about Bitcoin1 price modeling and related applications. The attention index affects Bitcoin price through a suitable dependence on the drift and diffusion coefficients and a possible correlation between the sources of randomness represented by the driving Brownian motions. The model is fitted on historical data of Bitcoin prices, by considering the total trading volume and the Google Search Volume Index as proxies for the attention measure. Moreover, a closed formula is computed for Europeanstyle derivatives on Bitcoin. Finally, we discuss two possible extensions of the model. Precisely, we investigate the relation between the correlation parameter and possible bubble effects in the asset price; further, we consider a multivariate framework to represent the special feature of Bitcoin being traded on several exchanges and we discuss conditions to rule out arbitrage opportunities in this setting. Blockchain and Cryptocurrencies. Bitcoin is a digital currency built on a peertopeer network and on the blockchain, a public ledger where all transactions are recorded and made available to all nodes. Opposite to traditional banking transactions, based on trust for counterparty, Bitcoin relies on cryptography and on a consensus protocol for the network. The entire system is founded on an open source software created in by a computer scientist known under the pseudonym Satoshi Nakamoto, whose identity is still unknown see [ 1 ]. Hence, Bitcoin is an independent digital currency, not subject to the control of central authorities and without inflation; furthermore, transactions in the network are pseudonymous and irreversible. Bitcoin and the underlying blockchain technology have gained much attention in the last few years. Research on Bitcoin often deals with cybersecurity and legitimacy issues such as the analysis of double spending possibilities and other cyberthreats; recently, high returns and volatility have attracted research toward the analysis of Bitcoin price efficiency as well as its dynamics see, among others, [ 2 , 3 , 4 ]. Moreover, many contributions claim that Bitcoin price is driven by attention or sentiment about the Bitcoin system itself; see [ 5 , 6 , 7 , 8 ]. Possible driving factors for the sentiment about the Bitcoin system are the volume of Google searches or Wikipedia requests as in [ 5 ], or more traditional indicators as the number or volume of transactions, as suggested in [ 6 ]. In [ 9 ], the author suggests a time series model in order to identify the dynamic relation between speculation activity and price. In this chapter, after having introduced the basic concepts underlying Bitcoin, we sum up and describe to a broader audience the recent outcomes of the research reported in [ 10 ], by avoiding unnecessary technicalities. Some new insights are also given by looking at possible extensions in order to take into account the presence of bubble effects or the special feature of Bitcoin being traded in different online platforms exchanges that will be further investigated in our future research. We recall that Bitcoin was first introduced as an electronic payment system between peers by Satoshi Nakamoto pseudonym in [ 1 ]. Opposite to traditional transactions, which are based on the trust in financial intermediaries, this system relies on the network, on the fixed rules and on cryptography. Bitcoins can be purchased on appropriate websites that allow to change usual currencies in the cryptocurrency. The Bitcoin network has several attractive properties for its users: No central bank authority for money supply and no regulator;. As a digital payment system, Bitcoins may be used to pay for several online services and goods. Special applications have been designed for smartphones and tablets for transactions in Bitcoins and some ATMs have appeared all over the world see Coin ATM radar to change traditional currencies in Bitcoins. Accepting Bitcoins as a payment method is also related to an advertisement opportunity for companies. However, the high returns achieved in the last few years have transformed Bitcoin in a speculative asset affecting its use as a form of payment. The Bitcoin system has been subject to many cracks but has proven to be very resilient as the value of the cryptocurrency was able to rise again after all the falls. Nevertheless, at the time of writing, Bitcoin was experiencing a fall in its exchange rate with main fiat currencies. Two of the main crackdowns were China enforcement in December and Mt. Gox bankruptcy in February Besides technical and regulation issues, the Bitcoin system also faces reputational concerns. In fact, the ambiguity of anonymous transactions has blamed the network of allowing several criminal activities such as buying illegal goods, money laundering or the financing of terrorism actions. As a representative example, we recall that The Silk Road was a website that started selling narcotics and illegal drugs in , payable in Bitcoins. The website was finally shutdown by and the owner was arrested and sentenced to life in prison. Again, anonymous transactions make it possible to use huge quantities of money, exchanged in Bitcoins, without declaring its origin, hence allowing for possible money laundering. It is worth noticing that while counterparties are represented by secret addresses and are anonymous, all transactions are recorded and might be traced. Investigation is hence favored by this feature of the network. Despite the flaws in the system, Bitcoin has achieved a notwithstanding rise in recent years. Bitcoin is treated as a financial stock as suggested in [ 12 ] and the suggested model may be applied in principle to other assets that are proven to depend on market attention. The dynamics of the two processes are described by the following equation:. 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. The above outcomes are applied in order to derive the likelihood of the vector R X. Proposition 2. We apply the outcomes above in order to estimate model parameters according to the maximumlikelihood method see for example [ 14 , 15 ] where the likelihood is approximated by applying the Levy approximation [ 13 ]. In Table 1 , the outcomes for parameter estimates, obtained by maximizing the approximate likelihood given the observed time series, are summed up. Parameter estimates for the model in Eq. In this section, we show how to characterize the price of European call options on Bitcoins in the underlying market model. To be reasonable, the market model must avoid arbitrage opportunities, that is, investment strategies that do not require an initial investment and that do not expose to any risk and lead to a positive value with positive probability. From a mathematical point of view, this means to check that the set of equivalent martingale measures for the Bitcoin price process S is nonempty. Precisely, it is possible to prove that it contains more than a single element. Lemma 3. Every equivalent martingale measure Q for S is characterized by its density process with respect to the initial probability measure P as follows:. The proof can be deduced from that of Lemma 1. Since S is the only tradable asset, the risk perception is not fixed and this explains the nonuniqueness of the martingale measure Q in this market framework that turns out to be incomplete. Consequently, given any Europeantype contingent claim, it is not possible in general to find a selffinancing strategy whose terminal value exactly replicates the payoff of the claim. We recall that the notion of completeness is related to the uniqueness of the martingale measure. Indeed, in complete markets, the noarbitrage price of any derivative is uniquely determined by the unique martingale measure. One simple example of a candidate equivalent martingale measure is the socalled minimal martingale measure see [ 17 , 18 ] , which minimizes the relative entropy, of the objective measure P , with respect to any riskneutral measure. In this setting, its economic interpretation is that agents do not wish to be compensated for the risk associated with the fluctuations of the stochastic attention factor, which corresponds to the hypothesis of [ 19 ] in the stochastic volatility framework. Remark 3. Indeed, the authors only referred to a riskneutral framework without describing the dynamics under the physical measure and consequently characterizing the existence of any equivalent martingale measure. Now, we compute the fair price of a Bitcoin European call option via the riskneutral evaluation approach, so it can be expressed as expected value of the terminal payoff under the selected pricing measure, that is, the minimal martingale measure. Here, N stands for the standard Gaussian cumulative distribution function, that is,. The proof is straightforward and may be derived by using similar arguments to those developed in [ 19 ]. Proposition 3. The riskneutral price C t at time t of a European call option written on the Bitcoin with price S expiring in T and with strike price K is given by the formula. In order to appreciate the performance of the pricing formula in Eq. Best overall pricing values are obtained when market attention is measured by volume; in the case of the SVI Google index, nearterm options are very close to the midvalue of the bidask, while nextterm options are overpriced. Comparison between model prices computed according to formula in Eq. Motivated by empirical evidences see for example [ 21 , 22 ] , we discuss a generalization of the model introduced in Section 3. 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 assetprice 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 riskneutral 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 riskneutral 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. 5 Best Cryptocurrencies to Invest in 2020, time: 30:58
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