Price variation bitcoin

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The goal of this chapter is to present recent developments about Bitcoin1 price modeling and related applications. Precisely, we consider a bivariate model in continuous time to describe the behavior of Bitcoin price and of the investors’ attention on the overall network. 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 European-style 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.

1. Introduction

Bitcoin is a digital currency built on a peer-to-peer 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 2009 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 cyber-threats; 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.

2. The Bitcoin network

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;

• Transactions are 24/7 and without any country border;

• Transaction cost are almost negligible with respect to traded amount;

• Transaction are anonymous;

• The security of each transaction is guaranteed by cryptography and digital signature;

• The security of the whole network is guaranteed by construction unless more than 50% of the network nodes agree on a deceptive action.

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 2013 and Mt. Gox bankruptcy in February 2014.

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 2011, payable in Bitcoins. The website was finally shutdown by 2013 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. However, according to a research performed by the UK government, the highest score related to money laundering is still cash, followed by the bank, accountancy and legal service providers (see).

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.

In Figure 1, we report Bitcoin price and returns from January 2012 to December 2017 (source).

3. An attention-based model

The model we suggest in what follows is motivated by findings in [5, 6, 8, 11] where it is showed that Bitcoin price is related to investors’ attention measured by the trading volume and/or the number of searches in engines such as Google and Wikipedia. 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.

4. A closed formula for Bitcoin option prices

In this section, we show how to characterize the price of European call options on Bitcoins in the underlying market model. Let us fix a finite time horizon and assume the existence of a riskless asset (also called the savings account), whose price process is given by

where is a bounded, deterministic function representing the instantaneous risk-free interest rate. 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 is nonempty. Precisely, it is possible to prove that it contains more than a single element.

Lemma 3.1.Every equivalent martingale measureforis characterized by its density process with respect to the initial probability measureas follows:

whereis an-adapted process such that, -a.s.

The proof can be deduced from that of Lemma 1.4 in [10], where they also account for a possible delay between the attention factor and its effect on Bitcoin prices trend. The process can be interpreted as the risk perception associated to the future direction or future possible movements of the Bitcoin market. Since is the only tradable asset, the risk perception is not fixed and this explains the nonuniqueness of the martingale measure in this market framework that turns out to be incomplete. Consequently, given any European-type contingent claim, it is not possible in general to find a self-financing 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 no-arbitrage price of any derivative is uniquely determined by the unique martingale measure. On the other hand, in incomplete markets, we deal with a family of martingale measures and have at our disposal a set of possible prices, which are all compatible with the “no-arbitrage condition.” One common approach to option pricing in incomplete markets in the mathematical financial literature is to select one specific martingale measure (which can be also called pricing measure) under which the discounted traded assets are martingales and to compute option prices via expectation under this measure via risk-neutral evaluation formulas. One simple example of a candidate equivalent martingale measure is the so-called minimal martingale measure (see [17, 18]), which minimizes the relative entropy, of the objective measure , with respect to any risk-neutral 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. This is the probability measure which corresponds to the choice in Eq. (10). Intuitively, under the minimal martingale measure, say , the drift of the Brownian motion driving the Bitcoin price process is modified to make an -martingale, while the drift of the Brownian motion which is strongly orthogonal to is not affected by the change measure from to . More precisely, by Girsanov’s theorem, the -valued process defined by

is an -standard Brownian motion. Under any equivalent martingale measure, the discounted Bitcoin price process given by for each behaves like a martingale. Precisely, on the probability space , the pair has the following dynamics:

Equivalently, we can write the discounted Bitcoin price process as

Clearly, under the minimal martingale measure , the Bitcoin price process satisfies

where is the risk-free interest rate at time .

Remark 3.2.Note that, under any equivalent martingale measure that keeps the drift of the attention factor dynamics linear in(in particular, under the minimal martingale measure), the model proposed in [10] nests the Hull-White stochastic volatility model, which corresponds to the particular case where; see [19]. Indeed, the authors only referred to a risk-neutral 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 risk-neutral 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. Let CT = (ST − K)⁺ be the -measurable random variable representing the payoff of a European call option on the Bitcoin with price with date of maturity and strike price , which can be traded on the underlying digital market. Recall that , for each , refers to the variation of the integrated attention process defined over the interval . Then, denote by the conditional expectation with respect to the -field under the probability measure and so on. Define the function as follows:

where

and , or more explicitly

Here, stands for the standard Gaussian cumulative distribution function, that is,

The following result provides the risk-neutral price of the option under the minimal martingale measure . The proof is straightforward and may be derived by using similar arguments to those developed in [19].

Proposition 3.3. The risk-neutral priceat timeof a European call option written on the Bitcoin with priceexpiring inand with strike priceis given by the formula

where the functionis defined inEq. (15); the functions, are, respectively, given inEqs. (16)-(17); anddenotes the density function of, for each, provided that it exists.

Hence, the resulting risk-neutral pricing formula when evaluated in corresponds to the expected value of Black & Scholes price as defined in [20] at time of a European call option written on , with strike price and maturity , in a financial market where the volatility is random and given by .