**Meet the editors**

Dr. Asma Salman is an associate professor of finance, serving at the College of Business Administration at the American University in the Emirates. She completed her MBA in Finance and Accounting and earned a PhD in Finance from the School of Management at Harbin Institute of Technology, Harbin, China. With research credentials from Brunel Business School at Brunel Uni-

versity, London, she has been actively engaged in scientific research and has published credible articles in international journals and conferences. One of her recent publications on "Bitcoin" has gained wide visibility across the world and then went on to achieve the Blockchain Explorer certification from IBM. Dr. Salman also served as the Dubai cohort supervisor for students under the Nottingham Business School DBA program where she was affiliated as a visiting scholar for eight years. Her research has provided her with several awards and honors around the globe. Her research interests include accounting, international finance, blockchain, and digital currencies.

Professor Muthanna G. Abdul Razzaq completed his Bachelor Degree in Business Administration and Accounting from Al Mustansirya University. He attained his Postgraduate Diploma in Cost Accounting from Baghdad University, Iraq, and then moved to the UK to pursue his PhD in Management Accounting from the University of Manchester.

Starting his career with the Ministry of Irrigation, soon his passion for education compelled him to join Baghdad University. Going a step further, in 1997 Professor Muthanna joined as the Acting Chair of the Department of Accounting at Applied Science University, Jordan, and later in 1999 as the Deputy Dean of the Faculty of Business Administration at Ajman University of Science and Technology, United Arab Emirates. His constant drive to excel inspired him to establish the American University in the Emirates in 2006, where he also serves as the President and CEO.

During his career spanning four decades he has served in various roles as Chairman of the Academic Committee, Vice President for Scientific Research, Member of the Board of Governors, Vice Chancellor for Administrative and Financial Affairs, and Member of the Board of Trustees. An accountant by profession and a professor of accounting, Professor Muthanna is a visionary who carries with him a perfect blend of administrative and academic exposure, with equal grip on accounting-related subjects. A selfless initiator, Professor Muthanna, with a broad vision, is dedicated to devoting his life to the cause of education and educational reforms.

Contents

**Preface VII**

Chapter 1 **Modeling Bitcoin Price and Bubbles 3**

**Section 2 Blockchain and Digital Returns 21**

**Section 3 Impact of Cryptocurrencies in Gaming 53**

Chapter 4 **Cryptocurrencies in the Ludic Economies: The Case of Contemporary Game Cultures 55**

Chapter 5 **On the Origin of the Value of Cryptocurrencies 75**

Chapter 6 **The Condition of the Cryptocurrency Market and Exchanges**

Tatjana Boshkov

Chapter 3 **Cryptocurrency Returns 41**

**Section 4 Cryptocurrency Exchanges 73**

Er'el Granot

**in Poland 95** Ireneusz Miciuła

Alessandra Cretarola and Gianna Figà-Talamanca

Chapter 2 **Blockchain and Digital Currency in the World of Finance 23**

Mike Cudd, Kristen Ritterbush, Marcelo Eduardo and Chris Smith

Leonardo José Mataruna-dos-Santos and Vanissa Wanick

**Section 1 Bitcoin 1**

## Contents

Preface

ty, and so forth.

This book on blockchain and cryptocurrencies shows the alternative side of finance in the digital era. It is amazing to see how blockchain technology has disrupted different sectors and how the revolutionary payment mechanisms in the form of cryptocurrencies and ex‐ changes are primarily witnessed in the financial sector. This is one of the true examples of moving from the "Internet of Use" to the "Internet of Value." Digital currencies have led the world to believe in an alternative economy. It includes both crypto and non-cryptocurren‐ cies. Bitcoins, ethereum, altcoins, and the likes have captured the curiosity of economists, philosophers, venture capitalists, and financial gurus alike and has given a new kind of power to users, which is decentralized and anonymous, creating disruption in the financial markets. Currently, some countries have offered their own digital currencies as an alterna‐ tive to fiat currencies, while others are recognizing digital currencies as an asset, a commodi‐

This book intends to feature the most talked about topic of cryptocurrencies along with its underlying technology of blockchain and how it has impacted the economy, the central banks, people, and organizations, and whether this power struggle of sorts will venture into a new digitized economy. It is divided into four main sections, with topics on Bitcoin, block‐ chain and digital returns, the impact of cryptocurrencies in gaming, and cryptocurrency ex‐ changes. This book shows many different aspects of the treatment of cryptocurrencies in various parts of the world along with the technology that underpins it, the blockchain. The diversity of the authors who sum up this book signify the importance, acceptance, and im‐

The first section is titled "Bitcoin." Authors Alessandra Cretarola and Gianna Figà-Talaman‐ ca from Italy present recent developments in Bitcoin price modeling and related applica‐ tions. They investigate the relation between the correlation parameter and possible bubble effects in the asset price. The authors then use a multivariate framework to represent the special feature of Bitcoin being traded on several exchanges and discuss conditions to rule

Section 2 deals with the "Blockchain and Digital Returns" and two chapters form part of this section. Tatjana Boshkov from Macedonia aims to provide an analysis of the use of crypto‐ currencies in general, especially Bitcoin as the technology adoption in the presence of net‐ work externalities. The chapter explores financial privacy, which is a very sensitive issue that uses digital currency (or cryptocurrency) and discusses private choices versus political rules. Authors Mike Cudd, Kristen Ritterbush, Marcelo Eduardo, and Chris Smith from the United States discuss profiling the risk and return metrics of cryptocurrencies. The crypto‐

plementation of this alternative currency and payment mechanism.

out arbitrage opportunities in this setting.

## Preface

This book on blockchain and cryptocurrencies shows the alternative side of finance in the digital era. It is amazing to see how blockchain technology has disrupted different sectors and how the revolutionary payment mechanisms in the form of cryptocurrencies and ex‐ changes are primarily witnessed in the financial sector. This is one of the true examples of moving from the "Internet of Use" to the "Internet of Value." Digital currencies have led the world to believe in an alternative economy. It includes both crypto and non-cryptocurren‐ cies. Bitcoins, ethereum, altcoins, and the likes have captured the curiosity of economists, philosophers, venture capitalists, and financial gurus alike and has given a new kind of power to users, which is decentralized and anonymous, creating disruption in the financial markets. Currently, some countries have offered their own digital currencies as an alterna‐ tive to fiat currencies, while others are recognizing digital currencies as an asset, a commodi‐ ty, and so forth.

This book intends to feature the most talked about topic of cryptocurrencies along with its underlying technology of blockchain and how it has impacted the economy, the central banks, people, and organizations, and whether this power struggle of sorts will venture into a new digitized economy. It is divided into four main sections, with topics on Bitcoin, block‐ chain and digital returns, the impact of cryptocurrencies in gaming, and cryptocurrency ex‐ changes. This book shows many different aspects of the treatment of cryptocurrencies in various parts of the world along with the technology that underpins it, the blockchain. The diversity of the authors who sum up this book signify the importance, acceptance, and im‐ plementation of this alternative currency and payment mechanism.

The first section is titled "Bitcoin." Authors Alessandra Cretarola and Gianna Figà-Talaman‐ ca from Italy present recent developments in Bitcoin price modeling and related applica‐ tions. They investigate the relation between the correlation parameter and possible bubble effects in the asset price. The authors then use a multivariate framework to represent the special feature of Bitcoin being traded on several exchanges and discuss conditions to rule out arbitrage opportunities in this setting.

Section 2 deals with the "Blockchain and Digital Returns" and two chapters form part of this section. Tatjana Boshkov from Macedonia aims to provide an analysis of the use of crypto‐ currencies in general, especially Bitcoin as the technology adoption in the presence of net‐ work externalities. The chapter explores financial privacy, which is a very sensitive issue that uses digital currency (or cryptocurrency) and discusses private choices versus political rules. Authors Mike Cudd, Kristen Ritterbush, Marcelo Eduardo, and Chris Smith from the United States discuss profiling the risk and return metrics of cryptocurrencies. The crypto‐ currency returns over different observation periods are examined and compared to those of conventional currencies and equities for relevance and proportion.

Section 3 covers the "Impact of Cryptocurrencies in Gaming" where Leonardo José Mataru‐ na-dos- Santos from the United Arab Emirates and Vanissa Wanick from the United King‐ dom explore the cryptocurrencies in ludic economies where games have been described as having their own economic models. The authors give an overview of the current state of the art economic models within games and eSports. This aims to situate and analyze the applica‐ tion of these business models derived from games, eSports, and the future of ludic econo‐ mies.

Section 4 deals with "Cryptocurrency Exchanges" and two chapters also form part of this section.

Author Er'el Granot from Israel explains why Bitcoin and other cryptocurrencies have re‐ ceived a lot of criticism during the last nine years. The author then analyzes the origin of the value of Bitcoin and other cryptocoins from the perspective of the Austrian School of Eco‐ nomics. It is explained that Bitcoin does not contradict the regression theorem. First, because the initial value estimation can be a random event, and second, the Bitcoin network (even now) has a non-monetary value.

Author Ireneusz Miciuła from Poland explores the global development of cryptocurrencies and discusses the functioning of a financial system based on cryptocurrencies and its signifi‐ cance for economies. The author presents the development of the global cryptocurrency market and analyzes the history of the most popular cryptocurrency, the Bitcoin. This last chapter ends the book on a rather high note of how a proper financial system through cryp‐ tocurrencies can be implemented in economies.

Last, but certainly not the least, the successful completion of this book has been the result of various "behind-the-scene" members who have supported it right from its inception until the publication process. We would like to express our sincere gratitude to all the authors for their diverse contributions. Also, thanks go to Author Service Manager Ms. Romina Skomer‐ sic for her endless support during the publishing process.

> **Dr. Asma Salman** Associate Professor of Finance College of Business Administration American University in the Emirates United Arab Emirates

**Bitcoin**

**Section 1**

**Professor Muthanna G. Abdul Razzaq** President and CEO American University in the Emirates United Arab Emirates

**Section 1**

## **Bitcoin**

currency returns over different observation periods are examined and compared to those of

Section 3 covers the "Impact of Cryptocurrencies in Gaming" where Leonardo José Mataru‐ na-dos- Santos from the United Arab Emirates and Vanissa Wanick from the United King‐ dom explore the cryptocurrencies in ludic economies where games have been described as having their own economic models. The authors give an overview of the current state of the art economic models within games and eSports. This aims to situate and analyze the applica‐ tion of these business models derived from games, eSports, and the future of ludic econo‐

Section 4 deals with "Cryptocurrency Exchanges" and two chapters also form part of this

Author Er'el Granot from Israel explains why Bitcoin and other cryptocurrencies have re‐ ceived a lot of criticism during the last nine years. The author then analyzes the origin of the value of Bitcoin and other cryptocoins from the perspective of the Austrian School of Eco‐ nomics. It is explained that Bitcoin does not contradict the regression theorem. First, because the initial value estimation can be a random event, and second, the Bitcoin network (even

Author Ireneusz Miciuła from Poland explores the global development of cryptocurrencies and discusses the functioning of a financial system based on cryptocurrencies and its signifi‐ cance for economies. The author presents the development of the global cryptocurrency market and analyzes the history of the most popular cryptocurrency, the Bitcoin. This last chapter ends the book on a rather high note of how a proper financial system through cryp‐

Last, but certainly not the least, the successful completion of this book has been the result of various "behind-the-scene" members who have supported it right from its inception until the publication process. We would like to express our sincere gratitude to all the authors for their diverse contributions. Also, thanks go to Author Service Manager Ms. Romina Skomer‐

**Dr. Asma Salman**

United Arab Emirates

President and CEO

United Arab Emirates

Associate Professor of Finance College of Business Administration American University in the Emirates

**Professor Muthanna G. Abdul Razzaq**

American University in the Emirates

conventional currencies and equities for relevance and proportion.

mies.

VIII Preface

section.

now) has a non-monetary value.

tocurrencies can be implemented in economies.

sic for her endless support during the publishing process.

**Chapter 1**

Provisional chapter

**Modeling Bitcoin Price and Bubbles**

Modeling Bitcoin Price and Bubbles

Alessandra Cretarola and Gianna Figà-Talamanca

conditions to rule out arbitrage opportunities in this setting.

Keywords: Bitcoin, market attention, arbitrage, option pricing, bubbles

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

DOI: 10.5772/intechopen.79386

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

We use the following rule throughout the paper: the term BitCoin refers to the whole system network while Bitcoin refers

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Alessandra Cretarola and Gianna Figà-Talamanca

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.79386

Abstract

1. Introduction

to the digital currency.

1

#### **Modeling Bitcoin Price and Bubbles** Modeling Bitcoin Price and Bubbles

Alessandra Cretarola and Gianna Figà-Talamanca Alessandra Cretarola and Gianna Figà-Talamanca

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.79386

#### Abstract

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.

DOI: 10.5772/intechopen.79386

Keywords: Bitcoin, market attention, arbitrage, option pricing, bubbles

#### 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

1 We use the following rule throughout the paper: the term BitCoin refers to the whole system network while Bitcoin refers to the digital currency.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

currency, not subject to the control of central authorities and without inflation; furthermore, transactions in the network are pseudonymous and irreversible.

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

Modeling Bitcoin Price and Bubbles http://dx.doi.org/10.5772/intechopen.79386 5

Two of the main crackdowns were China enforcement in December 2013 and Mt. Gox bank-

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 https://www.gov. uk/government/publications/uk-national-risk-assessment-of-money-laundering-and-terrorist-

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

Despite the flaws in the system, Bitcoin has achieved a notwithstanding rise in recent years.

Figure 1. Bitcoin price (top) and returns (bottom) from January 2012 to December 2017.

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

writing, Bitcoin was experiencing a fall in its exchange rate with main fiat currencies.

ruptcy in February 2014.

financing).

by this feature of the network.

https://blockchain.info/en/charts).

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, [2–4]). Moreover, many contributions claim that Bitcoin price is driven by attention or sentiment about the Bitcoin system itself; see [5–8]. 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:

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.

currency, not subject to the control of central authorities and without inflation; furthermore,

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, [2–4]). Moreover, many contributions claim that Bitcoin price is driven by attention or sentiment about the Bitcoin system itself; see [5–8]. 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

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

transactions in the network are pseudonymous and irreversible.

the dynamic relation between speculation activity and price.

The Bitcoin network has several attractive properties for its users: • No central bank authority for money supply and no regulator;

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

• 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

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.

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

the network nodes agree on a deceptive action.

2. The Bitcoin network

4 Blockchain and Cryptocurrencies

usual currencies in the cryptocurrency.

• Transaction are anonymous;

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 https://www.gov. uk/government/publications/uk-national-risk-assessment-of-money-laundering-and-terroristfinancing).

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 https://blockchain.info/en/charts).

Figure 1. Bitcoin price (top) and returns (bottom) from January 2012 to December 2017.

#### 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.

#### 3.1. The model specification

Consider a probability space ð Þ Ω; F; P endowed with a filtration F ¼ Ft f g ; t ≥ 0 satisfying usual assumptions of right continuity and completeness.

Let us denote the Bitcoin price process as S ¼ St f g ; t ≥ 0 and assume that it depends on an attention factor denoted by A ¼ At f g ; t ≥ 0 . The dynamics of the two processes are described by the following equation:

$$\begin{cases} \frac{dS\_t}{S\_t} = \mu\_S A\_t dt + \sigma\_S \sqrt{A\_t} dW\_t, \mathbf{S}\_0 = \mathbf{s}\_0 \in \mathbb{R}^+ \\\\ dA\_t = \mu\_A A\_t dt + \sigma\_A A\_t dZ\_t, A\_0 = \mathbf{a}\_0 \in \mathbb{R}^+, \end{cases} \tag{1}$$

Theorem 2.1. The random vector R ¼ ð Þ R1; R2;…Rn , given X ¼ ð Þ X1; X2; …Xn , is normally distrib-

� �Xi, for i ¼ 1, 2, …n,

� �:

Proof. In order to prove the theorem it suffices to remind that, for i = 1, 2, …, n, the random

2X2; ::; σS

p dWu, conditional on knowing Xi, is normally distributed with zero mean

Var Xi ½ �þ σS

2E Xi ½ �,

8 < :

ð Þ� are the probability density function of X1 and Xi given Xi�1,

� �.

E X1 2 � �

E½ � X1 2 !

i ¼ σ2 AΔ: ,

ð Þ� . Under our assumptions, such densities are not given within known

1 ¼ log

ri � μS � σ2

2σ2 Sxi

S 2 � �xi

9 =

ðÞ¼ � LN αi ð Þ ; νi , with

;, (5)

2Xn

(3)

7

Modeling Bitcoin Price and Bubbles http://dx.doi.org/10.5772/intechopen.79386

(4)

(6)

2 2

2X1; σS

and variance Xi, and that the increments of the Brownian motion W are independent.

2 2 � �E Xi ½ �,

> 2 2 � �2

where E Xi ½ �, Var Xi ½ � can be computed in closed form as a function of μA, σA, Δ (see for example [10]). The above outcomes are applied in order to derive the likelihood of the vector ð Þ R; X .

Proposition 2.2. The joint probability density of the vector ð Þ R; X is given by g : R � Rþ ! R with

It is worth to remark that the probability density gð Þ� in Eq. (5) depends on suitable choices for

distribution; however, by applying outcomes in [13], we can approximate them as log-normals

ðÞ¼ � LNð Þ α; ν and, for i ¼ 2, 3, …n, f Xi∣Xi�1

CA, ν2

A 2 � �Δ, ν2

1

2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E X1 2 q � �

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2πσ2 Sxi q exp �

mi ¼ μS � σS

Σ ¼ Diag σS

As for the unconditional distribution, it is easy to obtain, for i ¼ 1, 2, …, n,

E Ri ½ �¼ μS � σS

Var Ri ½ �¼ μS � σS

Indeed, by simply applying Bayes' rule, we get the following result:

f Xi∣Xi�1

The proof follows from Bayes' rule and application of Theorem 2.1.

with means and variances given as closed expressions of μA; σA

α1 ¼ log E½ � X1

0 B@

αi ¼ logð Þþ Xi�1 μA � σ2

ð Þ xi Yn i¼1

uted with mean m and covariance matrix Σ where

variable ÐiΔ

where f X1

f X1

respectively.

ð Þ� and f Xi∣Xi�1

Precisely, we have that f X1

ð Þ i�1 Δ

ffiffiffiffiffiffi Au

g rð Þ¼ ; x f X1

ð Þ� and f Xi∣Xi�1

ð Þ x1 Yn i¼2

where μA, μS, σA > 0, σS > 0 are constant parameters and ð Þ¼ W; Z Wt ð Þ ;Zt f g ; t ≥ 0 is an ð Þ F; P standard Brownian motion in R2. Assume that Ft ¼ σð Þ Wu;Zu; u ≤ t , for each t ≥ 0:

It is well known that the above dynamics for the attention factor is a geometric Brownian motion, the solution of which is given by At ¼ A0 exp μA � σA2 2 � �t þ σAZt � � for t ≥ 0 which has a log-normal distribution; integrating the price process is straightforward to get

$$\mathcal{S}\_t = \mathcal{S}\_0 \exp\left( \left( \mu\_S - \frac{\sigma\_S^2}{2} \right) \int\_0^t A\_u du + \sigma\_S \int\_0^t \sqrt{A\_u} dW\_u \right), t \ge 0. \tag{2}$$

#### 3.2. Statistical properties and model fitting

We collect in this subsection the properties of the logarithmic returns obtained by the price process defined in Eq. (1).

Consider the discrete process f g ð Þ AiΔ; SiΔ ; i ¼ 1; 2;…; n obtained by sampling the price process and the attention factor at times ti ¼ iΔ, i ¼ 1, 2, …n with constant observation step Δ; denote the logarithmic changes of the process by Ri ¼ log SiΔ Sð Þ i�1 Δ , Pi ¼ log AiΔ Að Þ i�1 Δ and define Xi ≔ Ð iΔ ð Þ i�1 Δ Audu.

Note that Ri, i ¼ 1, 2, …n represent the logarithmic returns of asset S for the sampling dates and that Xi, i ¼ 1, 2, …n the cumulative attention in the time interval ½ � ð Þ i � 1 Δ, iΔ . Then it is straightforward to prove the following:

Theorem 2.1. The random vector R ¼ ð Þ R1; R2;…Rn , given X ¼ ð Þ X1; X2; …Xn , is normally distributed with mean m and covariance matrix Σ where

$$\begin{aligned} m\_i &= \left(\mu\_S - \frac{\sigma\_S^2}{2}\right) \mathbf{X}\_{i\prime} \quad \text{for } i = 1, 2, \ldots \text{n}, \\\\ \Sigma &= \text{Diag}\left(\sigma\_S^2 \mathbf{X}\_1, \sigma\_S^2 \mathbf{X}\_2, \ldots, \sigma\_S^2 \mathbf{X}\_n\right). \end{aligned} \tag{3}$$

Proof. In order to prove the theorem it suffices to remind that, for i = 1, 2, …, n, the random variable ÐiΔ ð Þ i�1 Δ ffiffiffiffiffiffi Au p dWu, conditional on knowing Xi, is normally distributed with zero mean and variance Xi, and that the increments of the Brownian motion W are independent.

As for the unconditional distribution, it is easy to obtain, for i ¼ 1, 2, …, n,

3. An attention-based model

6 Blockchain and Cryptocurrencies

3.1. The model specification

the following equation:

other assets that are proven to depend on market attention.

usual assumptions of right continuity and completeness.

dSt St

motion, the solution of which is given by At ¼ A0 exp μA � σA2

St ¼ S0exp μS � σS

3.2. Statistical properties and model fitting

logarithmic changes of the process by Ri ¼ log SiΔ

process defined in Eq. (1).

forward to prove the following:

0 @

8 ><

>:

¼ μSAtdt þ σS

standard Brownian motion in R2. Assume that Ft ¼ σð Þ Wu;Zu; u ≤ t , for each t ≥ 0:

a log-normal distribution; integrating the price process is straightforward to get

2 2 � � ðt

0

We collect in this subsection the properties of the logarithmic returns obtained by the price

Consider the discrete process f g ð Þ AiΔ; SiΔ ; i ¼ 1; 2;…; n obtained by sampling the price process and the attention factor at times ti ¼ iΔ, i ¼ 1, 2, …n with constant observation step Δ; denote the

Note that Ri, i ¼ 1, 2, …n represent the logarithmic returns of asset S for the sampling dates and that Xi, i ¼ 1, 2, …n the cumulative attention in the time interval ½ � ð Þ i � 1 Δ, iΔ . Then it is straight-

Audu þ σS

ðt

0

Sð Þ i�1 Δ , Pi ¼ log AiΔ

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

Consider a probability space ð Þ Ω; F; P endowed with a filtration F ¼ Ft f g ; t ≥ 0 satisfying

Let us denote the Bitcoin price process as S ¼ St f g ; t ≥ 0 and assume that it depends on an attention factor denoted by A ¼ At f g ; t ≥ 0 . The dynamics of the two processes are described by

> ffiffiffiffiffi At

dAt ¼ μAAtdt þ σAAtdZt, A0 ¼ a0 ∈ Rþ,

where μA, μS, σA > 0, σS > 0 are constant parameters and ð Þ¼ W; Z Wt ð Þ ;Zt f g ; t ≥ 0 is an ð Þ F; P -

It is well known that the above dynamics for the attention factor is a geometric Brownian

p dWt, S0 ¼ s0 ∈ Rþ

2 � �

> ffiffiffiffiffiffi Au p dWu

� �

t þ σAZt

1

Að Þ i�1 Δ and define Xi ≔ Ð

(1)

for t ≥ 0 which has

iΔ

Audu.

ð Þ i�1 Δ

A, t ≥ 0: (2)

$$\begin{aligned} E[\mathbf{R}\_i] &= \left(\mu\_S - \frac{\sigma\_S^2}{2}\right) E[\mathbf{X}\_i], \\\\ Var[\mathbf{R}\_i] &= \left(\mu\_S - \frac{\sigma\_S^2}{2}\right)^2 Var[\mathbf{X}\_i] + \sigma\_S^2 E[\mathbf{X}\_i]. \end{aligned} \tag{4}$$

where E Xi ½ �, Var Xi ½ � can be computed in closed form as a function of μA, σA, Δ (see for example [10]). The above outcomes are applied in order to derive the likelihood of the vector ð Þ R; X . Indeed, by simply applying Bayes' rule, we get the following result:

Proposition 2.2. The joint probability density of the vector ð Þ R; X is given by g : R � Rþ ! R with

$$g(r, \mathbf{x}) = f\_{\mathbf{X}\_1}(\mathbf{x}\_1) \prod\_{i=2}^n f\_{\mathbf{X}\_i | \mathbf{X}\_{i-1}}(\mathbf{x}\_i) \prod\_{i=1}^n \frac{1}{\sqrt{2\pi\sigma\_S^2 \mathbf{x}\_i}} \exp\left\{-\frac{r\_i - \left(\mu\_S - \frac{\sigma\_S^2}{2}\right) \mathbf{x}\_i}{2\sigma\_S^2 \mathbf{x}\_i}\right\},\tag{5}$$

where f X1 ð Þ� and f Xi∣Xi�1 ð Þ� are the probability density function of X1 and Xi given Xi�1, respectively.

The proof follows from Bayes' rule and application of Theorem 2.1.

It is worth to remark that the probability density gð Þ� in Eq. (5) depends on suitable choices for f X1 ð Þ� and f Xi∣Xi�1 ð Þ� . Under our assumptions, such densities are not given within known distribution; however, by applying outcomes in [13], we can approximate them as log-normals with means and variances given as closed expressions of μA; σA � �.

Precisely, we have that f X1 ðÞ¼ � LNð Þ α; ν and, for i ¼ 2, 3, …n, f Xi∣Xi�1 ðÞ¼ � LN αi ð Þ ; νi , with

$$\begin{aligned} \alpha\_1 &= \log \left( \frac{\mathbb{E}[\mathbf{X}\_1]^2}{\sqrt{\mathbb{E}\left[\mathbf{X}\_1^2\right]}} \right), \quad \nu\_1^2 = \log \left( \frac{\mathbb{E}\left[\mathbf{X}\_1^2\right]}{\mathbb{E}\left[\mathbf{X}\_1\right]^2} \right), \\\alpha\_i &= \log(\mathbf{X}\_{i-1}) + \left(\mu\_A - \frac{\sigma\_A^2}{2}\right)\Delta, \quad \nu\_i^2 = \sigma\_A^2 \Delta. \end{aligned} \tag{6}$$

We apply the outcomes above in order to estimate model parameters according to the maximum-likelihood method (see for example [14, 15]) where the likelihood is approximated by applying the Levy approximation [13].

Parameter estimates are obtained as

$$\hat{\mathfrak{l}}\left(\hat{\boldsymbol{\mu}}\_{A},\hat{\boldsymbol{\mu}}\_{\mathcal{S}},\hat{\boldsymbol{\sigma}}\_{A},\hat{\boldsymbol{\sigma}}\boldsymbol{\varepsilon}\right) = \mathbf{arg}\max\_{\boldsymbol{\mu}\_{A},\boldsymbol{\mu}\_{\mathcal{S}},\boldsymbol{\sigma}\_{A},\boldsymbol{\sigma}\_{\mathcal{S}}}\log\mathfrak{l}\left(\boldsymbol{\mu}\_{A},\boldsymbol{\mu}\_{\mathcal{S}},\boldsymbol{\sigma}\_{A},\boldsymbol{\sigma}\boldsymbol{\varepsilon};\boldsymbol{\tau},\mathbf{x}\right),\tag{7}$$

of a riskless asset (also called the savings account), whose price process B ¼ Bt f g ; t ∈½ � 0; T is

where r : ½ �! 0; T R is a bounded, deterministic function representing the instantaneous riskfree 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 S is nonempty. Precisely, it is possible to prove that it contains more than a single element.

Lemma 3.1. Every equivalent martingale measure Q for S is characterized by its density process with

γudZu � 1

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 γ can be interpreted as the risk perception associated to the future direction or future possible movements of the Bitcoin market. 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 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 P, 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 γ � 0 in Eq. (10). Intuitively, under the minimal martingale measure, say Pb, the drift of the Brownian motion driving the Bitcoin price

2 ðt 0

du !, (10)

0 γu

μSAu � r uð Þ σS ffiffiffiffiffiffi Au p � �2

du � 1 2 ðt 0 γu 2

2du < þ∞, P-a.s.

ðt 0

r sð Þds � �, t∈ ½ � 0; T , (9)

Modeling Bitcoin Price and Bubbles http://dx.doi.org/10.5772/intechopen.79386 9

0

Bt ¼ exp ðt

respect to the initial probability measure P as follows:

μSAu � r uð Þ σS ffiffiffiffiffiffi Au p dWu �

where γ ¼ γt f g ; t ∈½ � 0; T is an F-adapted process such that Ð T

given by

dQ dP � � � � Ft

¼ exp �

ðt 0

where

$$\begin{split} \log \ell(\mu\_A, \mu\_S, \sigma\_A, \sigma\_S; \mathbf{r}, \mathbf{x}) &= \sum\_{i=1}^n \log \frac{1}{\sqrt{2\pi \mathbf{x}\_i \sigma\_S^2}} - \frac{\left(r\_i - \left(\mu\_S - \frac{\sigma\_S^2}{2}\right) \mathbf{x}\_i\right)^2}{2\mathbf{x}\_i \sigma\_S^2} \\ &+ \sum\_{i=1}^n \log \frac{1}{\mathbf{x}\_i \sqrt{2\pi \mathbf{w}\_i^2}} - \frac{\left(\log \mathbf{x}\_i - a\_i\right)^2}{2\mathbf{w}\_i^2} \end{split} \tag{8}$$

#### 3.3. Empirical application on Bitcoin prices

The first step in our procedure is to identify possible measures of investors' attention. As already mentioned in the introduction, we consider the total trading volume on Bitcoin available from https://blockchain.info as well as the search volume index (SVI) for Google searches on the topic "bitcoin" provided by https://trends.google.it/trends/.

The trading volume of exchange is a classical measure of the attractiveness of a traded asset for an investor; besides, in [16], the authors find evidence that the latter captures the attention of retail/uniformed investors.

We consider daily data from January 1, 2015, to June 30, 2017, for the total volume and the SVI Index. As for the daily value of the Bitcoin, we have considered the average mean across main exchanges represented by the Index in https://blockchain.info.

In Table 1, the outcomes for parameter estimates, obtained by maximizing the approximate likelihood given the observed time series, are summed up.

Table 1. Parameter estimates for the model in Eq. (1) fitted on daily observations from January 2015 to June 2017.

#### 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 T > 0 and assume the existence of a riskless asset (also called the savings account), whose price process B ¼ Bt f g ; t ∈½ � 0; T is given by

We apply the outcomes above in order to estimate model parameters according to the maximum-likelihood method (see for example [14, 15]) where the likelihood is approximated

i¼1

þ Xn i¼1

� � ¼ argmaxμA,μS,σA,σS logℓ μA; μS; σA; σS;r; x � �, (7)

ffiffiffiffiffiffiffiffiffiffiffi 2πvi

2 p � ð Þ logxi � αi

ri � μS � σ2

2vi 2

� �2

2xiσ2 S

S 2 � �

2

xi

(8)

log 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2πxiσ2 S q �

> log 1 xi

The first step in our procedure is to identify possible measures of investors' attention. As already mentioned in the introduction, we consider the total trading volume on Bitcoin available from https://blockchain.info as well as the search volume index (SVI) for Google searches

The trading volume of exchange is a classical measure of the attractiveness of a traded asset for an investor; besides, in [16], the authors find evidence that the latter captures the attention of

We consider daily data from January 1, 2015, to June 30, 2017, for the total volume and the SVI Index. As for the daily value of the Bitcoin, we have considered the average mean across main

In Table 1, the outcomes for parameter estimates, obtained by maximizing the approximate

A = Vol 0.9571 1.1346 0.0218 0.0829 A = SVI 1.3584 1.0687 0.0743 0.1559

μbA σbA μbS σbS

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 T > 0 and assume the existence

Table 1. Parameter estimates for the model in Eq. (1) fitted on daily observations from January 2015 to June 2017.

by applying the Levy approximation [13].

μbA; μbS; bσA; bσS

3.3. Empirical application on Bitcoin prices

retail/uniformed investors.

logℓ μA; μS; σA; σS;r; x � � ¼ Xn

on the topic "bitcoin" provided by https://trends.google.it/trends/.

exchanges represented by the Index in https://blockchain.info.

likelihood given the observed time series, are summed up.

4. A closed formula for Bitcoin option prices

Parameter estimates are obtained as

8 Blockchain and Cryptocurrencies

where

$$B\_t = \exp\left(\int\_0^t r(s)ds\right), \quad t \in [0, T]. \tag{9}$$

where r : ½ �! 0; T R is a bounded, deterministic function representing the instantaneous riskfree 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 S is nonempty. Precisely, it is possible to prove that it contains more than a single element.

Lemma 3.1. Every equivalent martingale measure Q for S is characterized by its density process with respect to the initial probability measure P as follows:

$$\frac{dQ}{dP}\bigg|\_{\mathcal{F}\_t} = \exp\left(-\int\_0^t \frac{\mu\_S A\_u - r(u)}{\sigma\_S \sqrt{A\_u}} d\mathcal{W}\_u - \int\_0^t \gamma\_u dZ\_u - \frac{1}{2}\int\_0^t \left(\frac{\mu\_S A\_u - r(u)}{\sigma\_S \sqrt{A\_u}}\right)^2 du - \frac{1}{2}\int\_0^t \gamma\_u \,^2\_u du\right), \tag{10}$$

where γ ¼ γt f g ; t ∈½ � 0; T is an F-adapted process such that Ð T 0 γu 2du < þ∞, P-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 γ can be interpreted as the risk perception associated to the future direction or future possible movements of the Bitcoin market. 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 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 P, 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 γ � 0 in Eq. (10). Intuitively, under the minimal martingale measure, say Pb, the drift of the Brownian motion driving the Bitcoin price process is modified to make S an F-martingale, while the drift of the Brownian motion which is strongly orthogonal to S is not affected by the change measure from P to Pb. More precisely, by Girsanov's theorem, the R2 -valued process W c;Zb � � ¼ W ct;Zbt � �; t ∈½ � 0; T n o defined by

$$
\widehat{\boldsymbol{W}}\_t \coloneqq \boldsymbol{W}\_t + \int\_0^t \frac{\mu\_S \boldsymbol{A}\_u - \boldsymbol{r}(\boldsymbol{u})}{\sigma\_S \sqrt{\boldsymbol{A}\_u}} d\boldsymbol{u}, \quad \widehat{\boldsymbol{Z}}\_t \coloneqq \boldsymbol{Z}\_t. \tag{11}
$$

CBSð Þ t;s; x ≔ sN ð Þ� d1ð Þ t;s; x Ke�

d1ð Þ¼ t;s; x log s

d2ð Þ¼ t;s; x log s

x p , or more explicitly

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

ðy �∞ e �z2

The following result provides the risk-neutral price of the option under the minimal martingale measure Pb. The proof is straightforward and may be derived by using similar arguments

Proposition 3.3. The risk-neutral price Ct at time t of a European call option written on the Bitcoin

where the function CBS is defined in Eq. (15); the functions d1ð Þ� , d2ð Þ� are, respectively, given in Eqs. (16)-(17); and f Xt,T ð Þ� denotes the density function of Xt,T, for each t∈ ½ Þ 0; T , provided that it exists.

Hence, the resulting risk-neutral pricing formula when evaluated in St corresponds to the expected value of Black & Scholes price as defined in [20] at time t∈ ½ Þ 0; T of a European call option written on S, with strike price K and maturity T, in a financial market where the

In order to appreciate the performance of the pricing formula in Eq. (19), we compute model prices for option traded on the online platform http://www.deribit.com on July, 30, 2017, by plugging in the estimated parameters. The outcomes are compared with the Black & Scholes benchmark (see [20]) as a reference price, computed by plugging the volatility parameter estimated on the same time series of the trading volume/SVI index, and with the bid-ask prices provided in the website. Best overall pricing values are obtained when market attention is

ffiffiffiffiffiffi Xt,T T�t q . Ð T

t r uð Þdu ðþ∞ 0

1 ffiffiffiffiffiffi 2π p

ffiffiffi

N ð Þ¼ y

with price S expiring in T and with strike price K is given by the formula

N d1 t; St ð Þ ð Þ ; x f Xt,T ð Þx dx � Ke�

K � � þ Ðt

K � � þ Ðt

σS ffiffiffi

σS ffiffiffi

where

and d2ð Þ¼ t;s; x d1ð Þ� t;s; x σS

to those developed in [19].

¼ St

Ct ¼ EP C b BS t; St ð Þj ; Xt,T St � �

> ðþ∞ 0

volatility is random and given by σS

4.1.  Under any equivalent martingale measure, the discounted Bitcoin price process S~ ¼ S~t; t∈ ½ � 0; T � � given by S~t ≔ St Bt , for each t ∈½ � 0; T behaves like a martingale. Precisely, on the probability space Ω; F; Pb � �, the pair S~; A � � has the following dynamics:

$$\begin{cases} d\tilde{S}\_t = \sigma\_\mathcal{S} \sqrt{A\_t} \tilde{S}\_t d\widehat{W}\_{t\prime} \tilde{S}\_0 = \mathbf{s}\_0 \in \mathbb{R}^+, \\\\ dA\_t = \mu\_A A\_t dt + \sigma\_A A\_t dZ\_{t\prime} A\_0 = a\_0 \in \mathbb{R}^+. \end{cases} \tag{12}$$

Equivalently, we can write the discounted Bitcoin price process S~ as

$$\tilde{S}\_t = \mathbf{s}\_0 \exp\left(\sigma\_S \int\_0^t \sqrt{A\_u} d\widehat{W}\_u - \frac{\sigma\_S^2}{2} \int\_0^t A\_u du\right), \quad t \in [0, T]. \tag{13}$$

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

$$dS\_t = r(t)S\_t dt + \sigma\_S \sqrt{A\_t} S\_t d\widehat{W}\_t \quad \text{ } \mathbb{S}\_0 = \mathbb{s}\_0 \in \mathbb{R}^+,\tag{14}$$

where r tð Þ is the risk-free interest rate at time t.

Remark 3.2. Note that, under any equivalent martingale measure that keeps the drift of the attention factor dynamics linear in A (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 σS ¼ 1; 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 FT-measurable random variable representing the payoff of a European call option on the Bitcoin with price S with date of maturity T and strike price K, which can be traded on the underlying digital market. Recall that Xt,T ¼ XT � Xt, for each t ∈½ Þ 0; T , refers to the variation of the integrated attention process X defined over the interval ½ � t; T . Then, denote by EPb �jFt ½ � the conditional expectation with respect to the σ-field Ft under the probability measure Pb and so on. Define the function CBS : ½ Þ� 0; T Rþ � Rþ ! R as follows:

#### Modeling Bitcoin Price and Bubbles http://dx.doi.org/10.5772/intechopen.79386 11

$$\mathcal{C}^{\rm BS}(t,s,\mathbf{x}) \coloneqq \mathbf{s} \mathcal{N}(d\_1(t,s,\mathbf{x})) - \mathbf{K} e^{-\int\_0^t r(u) du} \mathcal{N}(d\_2(t,s,\mathbf{x})),\tag{15}$$

where

process is modified to make S an F-martingale, while the drift of the Brownian motion which is strongly orthogonal to S is not affected by the change measure from P to Pb. More precisely, by

c;Zb � �

μSAu � r uð Þ σS ffiffiffiffiffiffi Au

¼ W ct;Zbt � �

ct, S~0 ¼ s0 ∈ Rþ,

dAt ¼ μAAtdt þ σAAtdZt, A0 ¼ a0 ∈ Rþ:

cu � σ2 S 2 ðt 0 Audu

� �

ffiffiffiffiffi At p StdW

Remark 3.2. Note that, under any equivalent martingale measure that keeps the drift of the attention factor dynamics linear in A (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 σS ¼ 1; 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

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

FT-measurable random variable representing the payoff of a European call option on the Bitcoin with price S with date of maturity T and strike price K, which can be traded on the underlying digital market. Recall that Xt,T ¼ XT � Xt, for each t ∈½ Þ 0; T , refers to the variation of the integrated attention process X defined over the interval ½ � t; T . Then, denote by EPb �jFt ½ � the conditional expectation with respect to the σ-field Ft under the probability measure Pb and

selected pricing measure, that is, the minimal martingale measure. Let CT = (ST � K)

Bt

; t ∈½ � 0; T

p du, Zbt ≔ Zt, (11)

defined by

, for each t ∈½ � 0; T behaves like a

, t∈ ½ � 0; T : (13)

ct, S0 ¼ s0 ∈ Rþ, (14)

(12)

+ be the

, the pair S~; A � � has the following

n o

ðt 0

ffiffiffiffiffi At p S~tdW

ffiffiffiffiffiffi Au p dW

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

W ct ≔ Wt þ

unted Bitcoin price process S~ ¼ S~t; t∈ ½ � 0; T � � given by S~t ≔ St

dS~t ¼ σS

Equivalently, we can write the discounted Bitcoin price process S~ as

dSt ¼ r tð ÞStdt þ σS

so on. Define the function CBS : ½ Þ� 0; T Rþ � Rþ ! R as follows:

ðt 0

martingale. Precisely, on the probability space Ω; F; Pb � �

8 < :

S~t ¼ s0 exp σS

where r tð Þ is the risk-free interest rate at time t.

Girsanov's theorem, the R2

10 Blockchain and Cryptocurrencies

is an F; Pb � �

dynamics:

martingale measure.

$$d\_1(t, \mathbf{s}, \mathbf{x}) = \frac{\log\left(\frac{\mathbf{s}}{\mathbf{K}}\right) + \int\_0^t r(\boldsymbol{\mu})d\boldsymbol{u} + \frac{\sigma\_\mathbf{s}^2}{2}\mathbf{x}}{\sigma\_\mathbf{S}\sqrt{\mathbf{x}}} \tag{16}$$

and d2ð Þ¼ t;s; x d1ð Þ� t;s; x σS ffiffiffi x p , or more explicitly

$$d\_2(t, s, \mathbf{x}) = \frac{\log\left(\frac{s}{K}\right) + \int\_0^t r(\mu)d\mu - \frac{\sigma\_s^2}{2}\mathbf{x}}{\sigma\_\delta \sqrt{\mathbf{x}}}.\tag{17}$$

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

$$\mathcal{N}(y) = \frac{1}{\sqrt{2\pi}} \int\_{-\infty}^{y} e^{-\frac{z^2}{2}} dz, \quad \forall \ y \in \mathbb{R}. \tag{18}$$

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

Proposition 3.3. The risk-neutral price Ct 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

$$\begin{split} \mathbf{C}\_{t} &= \hat{E}\hat{P}\Big[\mathbf{C}^{\rm BS}(t, \mathbf{S}\_{t}, \mathbf{X}\_{t,T})|\mathbf{S}\_{t}\Big] \\ &= \mathbf{S}\_{t} \int\_{0}^{+\infty} \mathcal{N}(d\_{1}(t, \mathbf{S}\_{t}, \mathbf{x})) f\_{\mathbf{X}\_{t}\top}(\mathbf{x}) d\mathbf{x} - \mathbf{K} e^{-\int\_{t}^{\top} r(\mathbf{u}) d\mathbf{u}} \int\_{0}^{+\infty} \mathcal{N}(d\_{2}(t, \mathbf{S}\_{t}, \mathbf{x})) f\_{\mathbf{X}\_{t}\top}(\mathbf{x}) d\mathbf{x}, \end{split} \tag{19}$$

where the function CBS is defined in Eq. (15); the functions d1ð Þ� , d2ð Þ� are, respectively, given in Eqs. (16)-(17); and f Xt,T ð Þ� denotes the density function of Xt,T, for each t∈ ½ Þ 0; T , provided that it exists.

Hence, the resulting risk-neutral pricing formula when evaluated in St corresponds to the expected value of Black & Scholes price as defined in [20] at time t∈ ½ Þ 0; T of a European call option written on S, with strike price K and maturity T, in a financial market where the volatility is random and given by σS ffiffiffiffiffiffi Xt,T T�t q .

#### 4.1. A numerical application

In order to appreciate the performance of the pricing formula in Eq. (19), we compute model prices for option traded on the online platform http://www.deribit.com on July, 30, 2017, by plugging in the estimated parameters. The outcomes are compared with the Black & Scholes benchmark (see [20]) as a reference price, computed by plugging the volatility parameter estimated on the same time series of the trading volume/SVI index, and with the bid-ask prices provided in the website. Best overall pricing values are obtained when market attention is

Wt ¼ rZt þ ffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffi At

dAt ¼ μAAtdt þ σAAtdZt, A0 ¼ a0 ∈ Rþ,

where higher positive values for the correlation appear to boost the asset value.

dSt ¼ μSAtStdt þ σS

(

the underlying Bitcoin market model.

the sign of the correlation parameter r.

we introduce the following definition from [23].

martingale under the chosen risk-neutral measure.

Bitcoin price dynamics for r ¼ 0 (black), r ¼ 0:5(green), and r ¼ 1 (blue).

where we have set r ≔ ffiffiffiffiffiffiffiffiffiffiffiffiffi

described by

concepts).

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

By simulating trajectories for the asset price S according to the model in Eq. (1) for several values of the correlation parameter, it seems that the latter is related to the presence of bubble effect; in fact, in Figure 2, we plot examples of trajectories for r ¼ 0, � 0:5, 0:5, 1, respectively

Indeed, we will show formally that the possibility of Bitcoin speculative bubbles is related to

The mathematical theory of financial bubbles is developed, among others, in [23–25]. Precisely,

Definition 4.1. The Bitcoin price process S has a bubble on the time interval ½ � 0; T if S is a strict F-local

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

Figure 2. Simulated trajectories with n = 250 daily observations for the attention process (red) and the corresponding

p Stð Þ rdZt þ rdVt , S0 ¼ s0 ∈ Rþ,

1 � r2 p . The aim is to investigate the existence of asset-price bubbles in

1 � r2 p Vt, t ∈½ � 0; T : (20)

Modeling Bitcoin Price and Bubbles http://dx.doi.org/10.5772/intechopen.79386

(21)

13

Table 2. Comparison between model prices computed according to formula in Eq. (19), Black & Scholes formula in [20], and the bid and ask prices provided in http:\\www.deribit.comfor options traded on July, 30, 2017, and expiring on August 25, 2017, and on September 28, 2017.

measured by volume; in the case of the SVI Google index, near-term options are very close to the mid-value of the bid-ask, while next-term options are overpriced. One possible explanation is that investors that get information about Bitcoin on search engines are more likely to be uninformed/retail investors that are self-exciting and may add spurious noise to the Bitcoin price volatility leading to an increase in call option prices (Table 2).

#### 5. The presence of model stock bubbles

Motivated by empirical evidences (see for example [21, 22]), we discuss a generalization of the model introduced in Section 3.1, which is capable to describe speculative bubbles in Bitcoin markets.

Precisely, we fix a finite time horizon T > 0 and assume that the underlying Brownian motions W and Z are correlated with constant correlation coefficient r ∈ ð Þ �1; 1 , that is, < W,Z>t ¼ rt for each t ∈½ � 0; T . If V ¼ Vt f g ; t∈ ½ � 0; T is an additional ð Þ F; P -Brownian motion that is Pindependent of Z, then we can write

$$\mathcal{W}\_t = \rho Z\_t + \sqrt{1 - \rho^2} V\_{t\nu} \quad t \in [0, T]. \tag{20}$$

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

$$\begin{cases} d\mathcal{S}\_t = \mu\_S A\_t \mathcal{S}\_t dt + \sigma\_S \sqrt{A\_t} \mathcal{S}\_t (\rho d\mathcal{Z}\_t + \overline{\rho} dV\_t), & \mathcal{S}\_0 = \mathbf{s}\_0 \in \mathbb{R}^+, \\ dA\_t = \mu\_A A\_t dt + \sigma\_A A\_t d\mathcal{Z}\_t, & A\_0 = a\_0 \in \mathbb{R}^+, \end{cases} \tag{21}$$

where we have set r ≔ ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � r2 p . 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. (1) for several values of the correlation parameter, it seems that the latter is related to the presence of bubble effect; in fact, in Figure 2, we plot examples of trajectories for r ¼ 0, � 0:5, 0:5, 1, respectively where higher positive values for the correlation appear to boost the asset value.

Indeed, we will show formally that the possibility of Bitcoin speculative bubbles is related to the sign of the correlation parameter r.

The mathematical theory of financial bubbles is developed, among others, in [23–25]. Precisely, we introduce the following definition from [23].

Definition 4.1. The Bitcoin price process S has a bubble on the time interval ½ � 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).

measured by volume; in the case of the SVI Google index, near-term options are very close to the mid-value of the bid-ask, while next-term options are overpriced. One possible explanation is that investors that get information about Bitcoin on search engines are more likely to be uninformed/retail investors that are self-exciting and may add spurious noise to the Bitcoin

Table 2. Comparison between model prices computed according to formula in Eq. (19), Black & Scholes formula in [20], and the bid and ask prices provided in http:\\www.deribit.comfor options traded on July, 30, 2017, and expiring on

T-K Market bid Market ask Model volume Model Google SVI Benchmark BS

Aug-2200 0.1662 0.2318 0.2029 0.2282 0.1967 Aug-2300 0.1670 0.2072 0.1737 0.2032 0.1655 Aug-2400 0.1390 0.1845 0.1469 0.1802 0.1369 Aug-2500 0.1142 0.1638 0.1228 0.1591 0.1112 Aug-2600 0.0922 0.1376 0.1014 0.1399 0.0887 Aug-2700 0.0749 0.1202 0.0828 0.1226 0.0695 Aug-2800 0.0572 0.1047 0.0684 0.107 0.0535 Aug-2900 0.0442 0.0983 0.0549 0.0931 0.0405 Sept-2200 0.1991 0.2648 0.2546 0.3204 0.2173 Sept-2300 0.1766 0.2432 0.2321 0.3019 0.1906 Sept-2400 0.1890 0.2230 0.2113 0.2844 0.1662 Sept-2500 0.1375 0.2042 0.1919 0.2679 0.1439 Sept-2600 0.1207 0.1828 0.1741 0.2523 0.1239 Sept-2700 0.1120 0.1668 0.1576 0.2377 0.1060 Sept-2800 0.0953 0.1504 0.1463 0.2239 0.0903 Sept-2900 0.0848 0.1422 0.1325 0.2109 0.0764

Motivated by empirical evidences (see for example [21, 22]), we discuss a generalization of the model introduced in Section 3.1, which is capable to describe speculative bubbles in Bitcoin

Precisely, we fix a finite time horizon T > 0 and assume that the underlying Brownian motions W and Z are correlated with constant correlation coefficient r ∈ ð Þ �1; 1 , that is, < W,Z>t ¼ rt for each t ∈½ � 0; T . If V ¼ Vt f g ; t∈ ½ � 0; T is an additional ð Þ F; P -Brownian motion that is P-

price volatility leading to an increase in call option prices (Table 2).

5. The presence of model stock bubbles

independent of Z, then we can write

August 25, 2017, and on September 28, 2017.

12 Blockchain and Cryptocurrencies

markets.

Figure 2. Simulated trajectories with n = 250 daily observations for the attention process (red) and the corresponding Bitcoin price dynamics for r ¼ 0 (black), r ¼ 0:5(green), and r ¼ 1 (blue).

Recall that the absence of arbitrage opportunities is "essentially" equivalent to the existence of a probability measure Q, equivalent to the initial probability P, under which the discounted price process satisfies the martingale property.

Remark 4.2. 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.

Then, to exclude arbitrage opportunities from the market, we define the process L ¼ Lt f ; t ∈ ½ �g 0; T by setting

$$L\_t \coloneqq \frac{dQ}{dP}\Big|\_{\mathcal{F}\_t} = \exp\left(-\int\_0^t \lambda\_u dV\_u - \frac{1}{2}\int\_0^t \lambda\_u^2 du - \int\_0^t \gamma\_u dZ\_u - \frac{1}{2}\int\_0^t \gamma\_u^2 du\right), \qquad t \in [0, T]. \tag{22}$$

where λ ¼ λt f g ; t∈ ½ � 0; T and γ ¼ γt f g ; t ∈½ � 0; T are F-adapted processes satisfying the integrability conditions Ð T 0 λ2 udu < ∞ P-a.s. and Ð T 0 γ2 udu < ∞ P-a.s., respectively. The (local) martingale property of the discounted Bitcoin price process S under Q implies the following condition:

$$
\mu\_S A\_l = \sigma \varsigma \sqrt{A}\_l (\lambda\_l \overline{\rho} + \gamma\_l \rho), \qquad t \in [0, T], \qquad P-\text{a.s.} \tag{23}
$$

dMt ¼ � μS

8 < :

property of the discounted stock price S under Q0

6. Toward a multiexchange generalization

dSt St

8 < :

where φ : ½ �! �l; 0 Rþ.

parameter vary.

returns and market attention is positive.

the presence of bubbles in this setting.

and only if r > 0.

rσS

ffiffiffiffiffi At

dAt ¼ μAAtdt þ σAAtdZt, A0 ¼ a0:

is a true ð Þ F; P -martingale thanks to [27]. We have the following result, which allows to detect

Proposition 4.3. In the model outlined in Eq. (24), the Bitcoin price process S has a bubble on ½ � 0; T if

The proof is based on the application of some of Sin's results given in [27], where the existence of risk-neutral measures for the Hull-White stochastic volatility model [19] and for similar frameworks is determined by the possibility of explosion in finite time for solutions of certain auxiliary stochastic differential equations. Precisely, it is possible to show that the martingale

and only if r ≤ 0: Hence, a bubble arises if and only if the correlation parameter between stock

Let us generalize the model introduced in Eq. (1) by assuming a possible delay τ for the attention factor to affect the Bitcoin price dynamics. Assume that the attention factor has been

> ffiffiffiffiffiffiffiffiffi At�τ

dAt ¼ μPAtdt þ σAAtdZt, At ¼ φð Þt for t∈½ � �l; 0 ,

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. (27)

In Figure 3, we plot simulated trajectories of the price process in Eq. (27) by letting the delay

A = Vol 1 day 0.4881 1.0459 0.0282 0.0924 A = SVI 7 days 1.0964 0.9946 0.1005 0.1885

Table 3. Parameter estimates for model in Eq. (27) fitted on daily observations from January 2015 to June 2017.

τ μbA σbA μbS σbS

p dWt, S0 ¼ s0 ∈ Rþ,

observed or is described by a deterministic function for t ∈½ � �l; 0 with l ≥ τ. We get

¼ μSAt�τdt þ σS

on the same daily data considered in Section 2 are summed up in Table 3.

p MtdVt, M0 ¼ 1,

, given in Eq. (25) with γ ¼ 0, is fulfilled if

(26)

15

Modeling Bitcoin Price and Bubbles http://dx.doi.org/10.5772/intechopen.79386

(27)

To ensure that L provides the density process of a probability measure equivalent to P, we require that E L½ �¼ T 1, meaning that L is an ð Þ F; P -martingale. The processes λ and γ are interpreted, respectively, as the risk premium and the risk perception associated to the future direction or future possible movements of the Bitcoin market. For each choice of the process γ, the process λ is fixed by Eq. (23), that is,

$$\lambda\_t = \frac{1}{\overline{\rho}} \left( \frac{\mu\_S \sqrt{A\_t}}{\sigma\_S} - \rho \gamma\_t \right), \qquad t \in [0, T]. \tag{24}$$

and we can consider the corresponding family of equivalent (local) martingale measures Qγ for S parameterized by the process γ. To check if there are stock bubbles in the underlying market model, we study under which conditions the discounted Bitcoin price is a strict ð Þ F; P -local martingale with respect to an equivalent local martingale measure Qγ: By applying Girsanov's theorem, the dynamics of the model under Qγ is described by the following equations:

$$\begin{cases} d\mathcal{S}\_t = \sigma\_{\mathcal{S}} \sqrt{A\_t} \mathcal{S}\_t \{\rho d\tilde{Z}\_t + \overline{\rho} d\tilde{V}\_t\}, \qquad \mathcal{S}\_0 = \mathbf{s}\_0 \in \mathbb{R}^+, \\\ d A\_t = \left(\mu\_A - \sigma\_A \chi\_t\right) A\_t dt + \sigma\_A A\_t d\tilde{Z}\_t, \qquad A\_0 = \mathbf{a}\_0 \in \mathbb{R}^+, \end{cases} \tag{25}$$

where the R2-valued process V~ ;Z~ Þ ¼ V~ t;Z~t � �; t ∈½ � 0; T � � � defined by V~ t ≔ Vt þ Ðt 0 λudu,. Z~t ≔ Zt þ Ðt 0 γudu, is an F; Qγ ð Þ-standard Brownian motion.

Now, suppose that the risk perception process is zero, that is, γ � 0. Then, the change of measure from P to Q0 is well-defined since the associated density process M ¼ Mt f g ; t∈ ½ � 0; T satisfying

$$\begin{cases} dM\_t = -\frac{\mu\_S}{\overline{\rho}\sigma\_S} \sqrt{A\_t} M\_t dV\_t & M\_0 = 1, \\ dA\_t = \mu\_A A\_t dt + \sigma\_A A\_t dZ\_t & A\_0 = a\_0. \end{cases} \tag{26}$$

is a true ð Þ F; P -martingale thanks to [27]. We have the following result, which allows to detect the presence of bubbles in this setting.

Proposition 4.3. In the model outlined in Eq. (24), the Bitcoin price process S has a bubble on ½ � 0; T if and only if r > 0.

The proof is based on the application of some of Sin's results given in [27], where the existence of risk-neutral measures for the Hull-White stochastic volatility model [19] and for similar frameworks is determined by the possibility of explosion in finite time for solutions of certain auxiliary stochastic differential equations. Precisely, it is possible to show that the martingale property of the discounted stock price S under Q0 , given in Eq. (25) with γ ¼ 0, is fulfilled if and only if r ≤ 0: Hence, a bubble arises if and only if the correlation parameter between stock returns and market attention is positive.

#### 6. Toward a multiexchange generalization

Let us generalize the model introduced in Eq. (1) by assuming a possible delay τ for the attention factor to affect the Bitcoin price dynamics. Assume that the attention factor has been observed or is described by a deterministic function for t ∈½ � �l; 0 with l ≥ τ. We get

$$\begin{cases} \frac{dS\_t}{S\_t} = \mu\_S A\_{t-\tau} dt + \sigma\_S \sqrt{A\_{t-\tau}} dW\_{t\prime} S\_0 = \mathbf{s}\_0 \in \mathbb{R}^+,\\ dA\_t = \mu\_P A\_t dt + \sigma\_A A\_t dZ\_{t\prime} A\_t = \boldsymbol{\varphi}(t) \text{ for } t \in [-l, 0]. \end{cases} \tag{27}$$

where φ : ½ �! �l; 0 Rþ.

Recall that the absence of arbitrage opportunities is "essentially" equivalent to the existence of a probability measure Q, equivalent to the initial probability P, under which the discounted

Remark 4.2. 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.

Then, to exclude arbitrage opportunities from the market, we define the process L ¼ Lt f ; t ∈

ðt 0

� �

where λ ¼ λt f g ; t∈ ½ � 0; T and γ ¼ γt f g ; t ∈½ � 0; T are F-adapted processes satisfying the inte-

0 γ2

gale property of the discounted Bitcoin price process S under Q implies the following condition:

To ensure that L provides the density process of a probability measure equivalent to P, we require that E L½ �¼ T 1, meaning that L is an ð Þ F; P -martingale. The processes λ and γ are interpreted, respectively, as the risk premium and the risk perception associated to the future direction or future possible movements of the Bitcoin market. For each choice of the process γ,

> � rγt !

and we can consider the corresponding family of equivalent (local) martingale measures Qγ for S parameterized by the process γ. To check if there are stock bubbles in the underlying market model, we study under which conditions the discounted Bitcoin price is a strict ð Þ F; P -local martingale with respect to an equivalent local martingale measure Qγ: By applying Girsanov's

� �, S0 ¼ s0 ∈ Rþ,

� �; t ∈½ � 0; T � � � defined by V~ t ≔ Vt þ Ðt

� �Atdt þ σAAtdZ~t, A0 ¼ a0 ∈ Rþ,

Now, suppose that the risk perception process is zero, that is, γ � 0. Then, the change of measure from P to Q0 is well-defined since the associated density process M ¼ Mt f g ; t∈ ½ � 0; T satisfying

theorem, the dynamics of the model under Qγ is described by the following equations:

p St rdZ~t þ rdV~ t

γudZu � 1

2 ðt 0 γ2 udu

tð Þ λtr þ γtr , t∈ ½ � 0; T , P � a:s:: (23)

udu < ∞ P-a.s., respectively. The (local) martin-

, t∈ ½ � 0; T , (24)

, t∈ ½ � 0; T , (22)

(25)

0 λudu,.

price process satisfies the martingale property.

¼ exp �

0 λ2

the process λ is fixed by Eq. (23), that is,

μSAt ¼ σS

ðt 0

λudVu � 1

udu < ∞ P-a.s. and Ð T

ffiffiffiffi A p

λt ¼ 1 r

> ffiffiffiffiffi At

0 γudu, is an F; Qγ ð Þ-standard Brownian motion.

dAt ¼ μA � σAγt

dSt ¼ σS

where the R2-valued process V~ ;Z~ Þ ¼ V~ t;Z~t

(

Z~t ≔ Zt þ Ðt

μS ffiffiffiffi A p t σS

2 ðt 0 λ2 udu �

½ �g 0; T by setting

Lt ≔ dQ dP � � � � Ft

14 Blockchain and Cryptocurrencies

grability conditions Ð T

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. (27) on the same daily data considered in Section 2 are summed up in Table 3.

In Figure 3, we plot simulated trajectories of the price process in Eq. (27) by letting the delay parameter vary.

Table 3. Parameter estimates for model in Eq. (27) fitted on daily observations from January 2015 to June 2017.

Figure 3. Simulated trajectories of n = 250 daily observations of the attention factor (red) and the Bitcoin price according to model in Eq. (27) when the delay parameter is τ ¼ 1 day (black) and τ ¼ 10 days (blue).

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. (27) but allowing different parameters naturally arises.

In particular, considering for instance two exchanges, we have

$$\begin{cases} \frac{dS^1\_t}{S^1\_t} = \mu^1\_S A\_{t-\tau\_1} dt + \sigma^1\_S \sqrt{A\_{t-\tau\_1}} dW\_t & S^1\_0 = s^1\_0 \in \mathbb{R}^+, \\\ d\underline{S}^2\_t = \mu^2\_S A\_{t-\tau\_2} dt + \sigma^2\_S \sqrt{A\_{t-\tau\_2}} dW\_t & S^2\_0 = s^2\_0 \in \mathbb{R}^+, \\\ dA\_t = \mu\_A A\_t dt + \sigma\_A A\_t d\mathbb{Z}\_t & A\_t = \varphi(t) \text{ for } t \in [-l, 0]. \end{cases} \tag{28}$$

attention is measured by the trading volume, and in Table 5, when attention is measured by

Table 5. Model fitting with delay parameter: outcomes for Bitstamp and Gdax exchanges when attention is measured by

Exchange τ μA σA μS σS Bitstamp 7 days 1.0934 0.9946 0.0992 0.1782 Gdax 7 days 1.0964 0.9946 0.1160 0.2087

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

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

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. Without entering technical

> S σ2 S

It is evident that these values are not equal if we plug parameter estimates in Eq. (30); hence, arbitrage opportunities are not ruled out at least from a theoretical point of view. We will

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

ffiffiffiffiffiffiffiffiffiffiffi At�τ2

p , t ≥ 0: (29)

Modeling Bitcoin Price and Bubbles http://dx.doi.org/10.5772/intechopen.79386 17

details and assuming r ¼ 0 for the sake of simplicity, this is true if and only if

ffiffiffiffiffiffiffiffiffiffiffi At�τ1 p ¼ μ2

the Google SVI index.

the SVI index.

7. Conclusion

parameters for the price dynamics is nonnegligible.

of averaging so arbitrage cannot be achieved in a direct way.

μ1 S σ1 S

address this issue more precisely in future research.

where φ : ½ �! �l; 0 Rþ with l > maxf g τ1; τ2 and μi A, μi S, σi A > 0, σi S > 0 for i = 1, 2 are constant parameters.

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 2015 to June 2017 for Bitstamp, Kraken, Cex.io, Gdax, and The Rock exchanges we get cross-correlation values larger than 0.999.

We fit model in Eq. (28) for the Bitstamp and Gdax exchanges on daily observations of Bitcoin price from January 2015 to June 2017 obtaining the outcomes reported in Table 4, when the

Table 4. Model fitting with delay parameter: outcomes for Bitstamp and Gdax exchanges when attention is measured by the trading volume.

Table 5. Model fitting with delay parameter: outcomes for Bitstamp and Gdax exchanges when attention is measured by the SVI index.

attention is measured by the trading volume, and in Table 5, when attention is measured by the Google 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. Without entering technical details and assuming r ¼ 0 for the sake of simplicity, this is true if and only if

$$\frac{\mu\_{\rm S}^{1}}{\sigma\_{\rm S}^{1}}\sqrt{A\_{t-\tau\_{1}}} = \frac{\mu\_{\rm S}^{2}}{\sigma\_{\rm S}^{2}}\sqrt{A\_{t-\tau\_{2}}} \quad t \ge 0. \tag{29}$$

It is evident that these values are not equal if we plug parameter estimates in Eq. (30); hence, arbitrage opportunities are not ruled out at least from a theoretical point of view. We will address this issue more precisely in future research.

#### 7. Conclusion

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. (27) but allowing different parameters natu-

Figure 3. Simulated trajectories of n = 250 daily observations of the attention factor (red) and the Bitcoin price according

In particular, considering for instance two exchanges, we have

SAt�τ1 dt þ σ1

to model in Eq. (27) when the delay parameter is τ ¼ 1 day (black) and τ ¼ 10 days (blue).

SAt�τ2 dt þ σ2

S

S

ffiffiffiffiffiffiffiffiffiffiffi At�τ1 p dWt, S1

ffiffiffiffiffiffiffiffiffiffiffi At�τ2 p dWt, S2

dAt ¼ μAAtdt þ σAAtdZt, At ¼ φð Þt for t ∈½ � �l; 0 ,

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 2015 to June 2017 for Bitstamp, Kraken, Cex.io, Gdax, and The Rock exchanges we

We fit model in Eq. (28) for the Bitstamp and Gdax exchanges on daily observations of Bitcoin price from January 2015 to June 2017 obtaining the outcomes reported in Table 4, when the

Table 4. Model fitting with delay parameter: outcomes for Bitstamp and Gdax exchanges when attention is measured by

Exchange τ μA σA μS σS Bitstamp 1 0.4994 1.0461 0.0281 0.0896 Gdax 2 0.4997 1.0420 0.0326 0.1036

A, μi S, σi 0 ¼ s 1 0 ∈ Rþ,

0 ¼ s 2 0 ∈ Rþ,

A > 0, σi

(28)

S > 0 for i = 1, 2 are constant

¼ μ1

¼ μ2

dS1 t S1 t

8 >>>>>><

>>>>>>:

get cross-correlation values larger than 0.999.

dS2 t S2 t

where φ : ½ �! �l; 0 Rþ with l > maxf g τ1; τ2 and μi

rally arises.

16 Blockchain and Cryptocurrencies

parameters.

the trading volume.

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.

References

Letters. 2016;16:85-92

Letters. 2017;158:3-6

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[4] Katsiampa P. Volatility estimation for bitcoin: A comparison of garch models. Economics

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[9] Blau BM. Price dynamics and speculative trading in bitcoin. Research in International

[10] Cretarola A, Figà-Talamanca G, Patacca M. Market Attention and Bitcoin Price Modeling:

[11] Kim YB, Lee SH, Kang SJ, Choi MJ, Lee J, Kim CH. Virtual world currency value fluctuation prediction system based on user sentiment analysis. PLoS One. 2015;10(8):e0132944

[12] Yermack D. Is bitcoin a real currency? An economic appraisal. In: Handbook of Digital

[13] Levy E. Pricing European average rate currency options. Journal of International Money

[15] Pawitan Y. In all Likelihood: Statistical Modelling and Inference Using Likelihood. UK:

[16] Da Z, Engelberg J, Gao P. In search of attention. The Journal of Finance. 2011;66(5):1461-1499 [17] Föllmer H, Schweizer M, Hedging of contingent claims under incomplete information. In M.H.A Davis and R.J.Elliot, editors, Applied Stochastic Analysis. New York, Gordon and

[18] Föllmer H, Schweizer M. Minimal martingale measure. In: Encyclopedia of Quantitative

[14] Davison AC. Statistical Models, Volume 11. USA: Cambridge University Press; 2003

between phenomena of the internet era. Scientific Reports. 2013;3(3415):1-7

coherence analysis. PLoS One. 2015;10(4):e0123923

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and Finance. 1992;11(5):474-491

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Finance. Wiley Online Library; 2010

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versity in Brno, Faculty of Business and Economics; 2016

Theory, Estimation and Option Pricing. Submitted; 2018

Currency. USA: Academic Press; 2015. pp. 31-43

[2] Urquhart A. The inefficiency of bitcoin. Economics Letters. 2016;148:80-82

By applying outcomes within the mathematical theory of bubbles [23–25, 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. (21) may result in an explosion of the price process.

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. (28)) in order to take into account the special feature of multiple exchanges where it is possible to trade in Bitcoins. Preliminary results indicate that arbitrage opportunities may arise in two exchanges that are characterized by different delays.

#### Acknowledgements

We gratefully acknowledge Marco Patacca for having provided the routines in Matlab® to develop the empirical sections of the paper and for useful suggestions and comments. We also acknowledge funding from Fondazione Cassa di Risparmio di Perugia (Grant 2015:0459013) and Bank of Italy (Grant 407660/16).

#### Conflict of interest

The authors declare no conflict of interest.

#### Thanks

Gianna Figà-Talamanca gratefully acknowledges interesting discussions on the topic of this chapter with colleagues of the Department of Finance and Risk Engineering where she was visiting professor while preparing this research. The authors also wish to thank Stefano Bistarelli for having introduced them to the intriguing and worth to explore world of cryptocurrencies.

#### Author details

Alessandra Cretarola1 \* and Gianna Figà-Talamanca2

\*Address all correspondence to<EMAIL_ADDRESS>

#### References

model to observed data is also suggested and, under the assumption of no correlation, a closed

By applying outcomes within the mathematical theory of bubbles [23–25, 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. (21) may result in an explosion of the price process. 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. (28)) in order to take into account the special feature of multiple exchanges where it is possible to trade in Bitcoins. Preliminary results indicate that arbitrage opportunities may arise in two exchanges

We gratefully acknowledge Marco Patacca for having provided the routines in Matlab® to develop the empirical sections of the paper and for useful suggestions and comments. We also acknowledge funding from Fondazione Cassa di Risparmio di Perugia (Grant 2015:0459013)

Gianna Figà-Talamanca gratefully acknowledges interesting discussions on the topic of this chapter with colleagues of the Department of Finance and Risk Engineering where she was visiting professor while preparing this research. The authors also wish to thank Stefano Bistarelli for having introduced them to the intriguing and worth to explore world of cryptocurrencies.

1 Department of Mathematics and Computer Science, University of Perugia, Perugia, Italy

\* and Gianna Figà-Talamanca2

\*Address all correspondence to<EMAIL_ADDRESS>
2 Department of Economics, University of Perugia, Perugia, Italy

formula for standard European option prices on Bitcoin is provided.

that are characterized by different delays.

and Bank of Italy (Grant 407660/16).

The authors declare no conflict of interest.

Acknowledgements

18 Blockchain and Cryptocurrencies

Conflict of interest

Thanks

Author details

Alessandra Cretarola1

[19] Hull J, White A. The pricing of options on assets with stochastic volatilities. The Journal of Finance. 1987;42(2):281-300

**Section 2**

**Blockchain and Digital Returns**

**Blockchain and Digital Returns**

[19] Hull J, White A. The pricing of options on assets with stochastic volatilities. The Journal of

[20] Black F, Scholes M. The pricing of options and corporate liabilities. The Journal of Political

[21] Malhotra A, Maloo M. Bitcoin–Is it a bubble? Evidence from unit root tests. SSRN Elec-

[22] Corbet S, Lucey B, Yarovya L. Datestamping the Bitcoin and Ethereum bubbles. Finance

[23] Jarrow RA, Protter PE, Shimbo K. Asset price bubbles in incomplete markets. Mathematical

[24] Protter PE. A mathematical theory of financial bubbles. In: Paris-Princeton Lectures on

[25] Protter PE. Bubbles and crashes - mathematical models of bubbles. Quantitative Finance

[27] Sin CA. Complications with stochastic volatility models. Advances in Applied Probability.

[26] Protter PE. Stochastic Integration and Differential Equations. Germany: Springer; 2005

Mathematical Finance. Switzerland: Springer; 2013. pp. 1-108

Finance. 1987;42(2):281-300

20 Blockchain and Cryptocurrencies

Economy. 1973:81(3):637-654

Research Letters. 2017:8 (In press)

Finance. 2010;20(2):145-185

Letters. 2016;4:10-13

1998;30(1):256-268

tronic Journal. 2014:13

**Chapter 2**

**Provisional chapter**

**Blockchain and Digital Currency in the World of**

**Blockchain and Digital Currency in the World of**

DOI: 10.5772/intechopen.79456

High-tech enables payment evolution and global competition. The ambiguities surrounding of the digital currency still leave enough space for the analysis of its unreserved acceptance, trust and anticipation, which are the main driver for the spread of the network. Banks should carefully consider the technology underlying these cryptocurrencies as a potential generic new way of transferring ownership of the value over the long term. The chapter provides an analysis of the use of cryptocurrencies in general, especially Bitcoin as the technology adoption in the presence of network externalities. The objective attitude is the future of the digital currency in the moment is still unsolved issue due to the existence of "critical mass". Further, the chapter explores financial privacy which is very sensitive issue in using digital currency (or cryptocurrency) and discuss about private choices versus political rules. The research has shown that the future of cryptocurrencies can be bright if some institutional-formal conditions are met due to the fact that success evolution of e-money requires building safety payments through three criteria–standardization, com-

**Keywords:** Blockchain, digital currency, network externalities, critical mass, technology,

Electronic money is not a new phenomenon. Trade over the Internet has increased the use of new technologies, thereby increasing the demand for new electronic payment methods. What really is new is electronic payment in retail and use of the Internet as new monetary market. Today, money becomes ready information on the microprocessor or in the database. Without

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.79456

patibility and innovation.

payment evolution

**1. Introduction**

**Finance**

**Finance**

Tatjana Boshkov

**Abstract**

Tatjana Boshkov

#### **Blockchain and Digital Currency in the World of Finance Blockchain and Digital Currency in the World of Finance**

DOI: 10.5772/intechopen.79456

#### Tatjana Boshkov Tatjana Boshkov

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.79456

#### **Abstract**

High-tech enables payment evolution and global competition. The ambiguities surrounding of the digital currency still leave enough space for the analysis of its unreserved acceptance, trust and anticipation, which are the main driver for the spread of the network. Banks should carefully consider the technology underlying these cryptocurrencies as a potential generic new way of transferring ownership of the value over the long term. The chapter provides an analysis of the use of cryptocurrencies in general, especially Bitcoin as the technology adoption in the presence of network externalities. The objective attitude is the future of the digital currency in the moment is still unsolved issue due to the existence of "critical mass". Further, the chapter explores financial privacy which is very sensitive issue in using digital currency (or cryptocurrency) and discuss about private choices versus political rules. The research has shown that the future of cryptocurrencies can be bright if some institutional-formal conditions are met due to the fact that success evolution of e-money requires building safety payments through three criteria–standardization, compatibility and innovation.

**Keywords:** Blockchain, digital currency, network externalities, critical mass, technology, payment evolution

#### **1. Introduction**

Electronic money is not a new phenomenon. Trade over the Internet has increased the use of new technologies, thereby increasing the demand for new electronic payment methods. What really is new is electronic payment in retail and use of the Internet as new monetary market. Today, money becomes ready information on the microprocessor or in the database. Without

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

a doubt, the purpose of such an instrument is to improve the efficiency of the traditional payment method. At this moment, there are still no clear standards in the Blockchain mechanism and therefore we do not know the boundaries, so participants can easily communicate without the presence of a regulator. Behind Blockchain technology is the universal Internet currency, which in turn raises many questions about the utilization of the advantages and risks/damages that would be arisen from the application.

cash, e-bag, e-currency, digital currency, digital money or digital cash. Without a doubt, the purpose of such an instrument is to improve the efficiency of the traditional payment method. Bitcoin is a digital currency whose value varies according to the worldwide customer acceptance. This is primarily due to the fact that, unlike the standard currencies we use, such as the dollar or the euro, which are regulated by central banks, for Bitcoin there is no regulation. Therefore, transactions with Bitcoin are considered more private and anonymous due to the open system and no existence of a regulatory body and/or intermediary in the performance of transactions. Transactions are carried out using cryptographic protection, and their execution is done through a network of public electronic books called "ledgers". For verification of transactions, it is necessary to have specific hardware and software that users can set up and after a certain number of transactions they receive a proportion of Bitcoin. In this way, it is also performed

Blockchain and Digital Currency in the World of Finance http://dx.doi.org/10.5772/intechopen.79456 25

From the aspect of the development of e-payment method, digital currency is not physically printed by the Central Bank. For now, digital currency is considered with its own rules of the game. In the literature, all those who support the use of Bitcoin underscore the characteristic as a currency that does not cause financial crises. Namely, the view is that banks can print more money to cover their national debt, thus devaluing their currencies, Bitcoin does not

Electronic payment method exists from the 1960s, i.e. from the development of Electronic Funds Transfer (EFT), which became more sophisticated and applicable in a growing number of countries [2]. EFT implies the application of computer and telecommunication technology in payment. This method was used by banks and other financial institutions to exchange and transfer a large amount of money on a national and international level. The basis for the operation of EFT is that the money moves through a network as a substitute for cash or checks to execute a transaction. In this way, the time for paying should be shortened and the transaction costs reduced. The use of EFT has significantly increased with the emergence and acceptance of ATMs, which allow money transfer at the point of sale (EFTPOS). EFT is

In the early 1980s, thanks to the development of network technology, the costs of telecommunications and data processing were reduced, and electronic payments became more useful with the appearance of credit and debit cards, which for several years (after their appearance) became the most popular electronic small transaction tool. Also, the development of encryption has played a major role in successful card payments. This innovation is considered as a

The growth and acceptance of card payments had negative consequences for the traditional way of payment. Many countries have made a move from the use of paper instruments, such as cash and checks, to the use of electronic instruments. For the first time in many countries, the number of checks payments has been reduced. Namely, checks as a very popular payment

an additional commissioning of this digital currency.

**2.1. Development of e-payment and digital currency**

considered as first degree in the electronization of transactions.

instrument loose the market role, thereby reducing their use [3].

second degree in the electronization of transactions.

function in such a way.

High-tech enables payment evolution and global competition. But still the ambiguities surrounding the use of the digital currency leave enough space for the analysis of its unreserved acceptance, trust and anticipation, which are the main driver for the spread of the network. More precisely, the spread of the network requires interdependence of demand, which means the Network, must reach the minimum required volume before it reaches a balance. The minimum volume of the network is called "critical mass". Therefore, the objective attitude is the future of the digital currency in the moment is still unsolved issue due to the existence of "critical mass".

This chapter underlines the technology adoption in the presence of network externalities. Payment innovations that involve the creation of a network between the manufacturer and the consumer are product that inevitably involves network externalities that must touch the critical mass of the user before it starts to use it successfully. Network externalities exist due to the average consumer benefits from such an instrument, only if other consumers and traders use the same payment instrument.

Further, the chapter explores financial privacy which is very sensitive issue in using digital currency (or cryptocurrency). The analysis explores what are the private choices versus political rules. Success evolution of e-money requires building safety payments through three criteria—standardization, compatibility and innovation.

The diffusion that digital currency brings in the modern era expands the antitrust issues related to network externalities and global competition between most explored world currencies. This is the reason to include a review of social costs and benefits, as possible risks of using digital currency. These mean that in order to remain compatible with each other, all users should use software that meets the same rules. Therefore, all users and developers have a strong incentive to protect this consensus and set up a regulator.

At the end, the chapter examines the question—are there prospects of taking hand in hand the technology revolution and monetary evolution without risks in the real world?!

#### **2. Overview of the IT revolution and innovations related to money**

The online trade increased the use of new technologies, and thus increased the demand for new electronic payment methods. This began especially in the mid-1990s with the information revolution, the decline in computer prices and the networking of the same. Cohen introduces the term "change the geography of money" [1]. This term occurs as a result of the electronic payment in retail and use of the Internet as a new monetary market. Due to the information revolution, a new electronic payment method has been introduced, known as electronic cash, e-bag, e-currency, digital currency, digital money or digital cash. Without a doubt, the purpose of such an instrument is to improve the efficiency of the traditional payment method.

Bitcoin is a digital currency whose value varies according to the worldwide customer acceptance. This is primarily due to the fact that, unlike the standard currencies we use, such as the dollar or the euro, which are regulated by central banks, for Bitcoin there is no regulation. Therefore, transactions with Bitcoin are considered more private and anonymous due to the open system and no existence of a regulatory body and/or intermediary in the performance of transactions.

Transactions are carried out using cryptographic protection, and their execution is done through a network of public electronic books called "ledgers". For verification of transactions, it is necessary to have specific hardware and software that users can set up and after a certain number of transactions they receive a proportion of Bitcoin. In this way, it is also performed an additional commissioning of this digital currency.

#### **2.1. Development of e-payment and digital currency**

a doubt, the purpose of such an instrument is to improve the efficiency of the traditional payment method. At this moment, there are still no clear standards in the Blockchain mechanism and therefore we do not know the boundaries, so participants can easily communicate without the presence of a regulator. Behind Blockchain technology is the universal Internet currency, which in turn raises many questions about the utilization of the advantages and

High-tech enables payment evolution and global competition. But still the ambiguities surrounding the use of the digital currency leave enough space for the analysis of its unreserved acceptance, trust and anticipation, which are the main driver for the spread of the network. More precisely, the spread of the network requires interdependence of demand, which means the Network, must reach the minimum required volume before it reaches a balance. The minimum volume of the network is called "critical mass". Therefore, the objective attitude is the future of the digital currency in the moment is still unsolved issue due to the existence of "critical mass".

This chapter underlines the technology adoption in the presence of network externalities. Payment innovations that involve the creation of a network between the manufacturer and the consumer are product that inevitably involves network externalities that must touch the critical mass of the user before it starts to use it successfully. Network externalities exist due to the average consumer benefits from such an instrument, only if other consumers and traders

Further, the chapter explores financial privacy which is very sensitive issue in using digital currency (or cryptocurrency). The analysis explores what are the private choices versus political rules. Success evolution of e-money requires building safety payments through three

The diffusion that digital currency brings in the modern era expands the antitrust issues related to network externalities and global competition between most explored world currencies. This is the reason to include a review of social costs and benefits, as possible risks of using digital currency. These mean that in order to remain compatible with each other, all users should use software that meets the same rules. Therefore, all users and developers have

At the end, the chapter examines the question—are there prospects of taking hand in hand the

The online trade increased the use of new technologies, and thus increased the demand for new electronic payment methods. This began especially in the mid-1990s with the information revolution, the decline in computer prices and the networking of the same. Cohen introduces the term "change the geography of money" [1]. This term occurs as a result of the electronic payment in retail and use of the Internet as a new monetary market. Due to the information revolution, a new electronic payment method has been introduced, known as electronic

technology revolution and monetary evolution without risks in the real world?!

**2. Overview of the IT revolution and innovations related to money**

risks/damages that would be arisen from the application.

criteria—standardization, compatibility and innovation.

a strong incentive to protect this consensus and set up a regulator.

use the same payment instrument.

24 Blockchain and Cryptocurrencies

From the aspect of the development of e-payment method, digital currency is not physically printed by the Central Bank. For now, digital currency is considered with its own rules of the game. In the literature, all those who support the use of Bitcoin underscore the characteristic as a currency that does not cause financial crises. Namely, the view is that banks can print more money to cover their national debt, thus devaluing their currencies, Bitcoin does not function in such a way.

Electronic payment method exists from the 1960s, i.e. from the development of Electronic Funds Transfer (EFT), which became more sophisticated and applicable in a growing number of countries [2]. EFT implies the application of computer and telecommunication technology in payment. This method was used by banks and other financial institutions to exchange and transfer a large amount of money on a national and international level. The basis for the operation of EFT is that the money moves through a network as a substitute for cash or checks to execute a transaction. In this way, the time for paying should be shortened and the transaction costs reduced. The use of EFT has significantly increased with the emergence and acceptance of ATMs, which allow money transfer at the point of sale (EFTPOS). EFT is considered as first degree in the electronization of transactions.

In the early 1980s, thanks to the development of network technology, the costs of telecommunications and data processing were reduced, and electronic payments became more useful with the appearance of credit and debit cards, which for several years (after their appearance) became the most popular electronic small transaction tool. Also, the development of encryption has played a major role in successful card payments. This innovation is considered as a second degree in the electronization of transactions.

The growth and acceptance of card payments had negative consequences for the traditional way of payment. Many countries have made a move from the use of paper instruments, such as cash and checks, to the use of electronic instruments. For the first time in many countries, the number of checks payments has been reduced. Namely, checks as a very popular payment instrument loose the market role, thereby reducing their use [3].

#### **2.2. e-Payment and transaction costs**

In classical trade payments require at least one buyer and one seller, both having to have accounts in banks that are connected through clearing houses. Payments with traditional instruments such as checks require intervention of a financial intermediary like bank. Payment with e-money is similar to the traditional scheme—there are two parties—one or two banks. However, the whole process becomes more efficient and easier. The transaction does not require any code and cannot exceed the previously defined amount. If the amount that is on the chip is fully spent, the card can be automatically refilled at the merchant, without charging any fees, thanks to the special POS mechanism [4].

Their main role is to support online e-commerce, enable transactions, reduce their costs, or replace the payment of money and coins in retail. The second reason for the emergence of e-money is the information revolution, which is characterized by the integration of electronic information processing and telecommunication technologies, which reduces the geographical differences by means of which information can be transmitted to the whole world. The information revolution has changed the financial sector, making payment modes more secure and more efficient, giving an additional reason for the emergence of new monetary innovations [9]. Unlike the information revolution, the emergence of e-money is a new way of processing information for transferring purchasing power. Many financial innovations are not a new form of money, but a different way of using existing money in transactions [10]. Regardless of the consequences of the mentioned technological development, the nature of the money is still identical i.e. money serves as a means of exchange, as an asset and as a value. The nature of the money will never change, so the money will remain only an intermediary in the exchange of goods and services. e-Money card is a different payment method that allows electronic transfer of the value from the card to the terminal or from the card in the wallet, both in real time and through networks [11]. It is considered that e-money is the most important achievement that transfers the predetermined monetary value so it can be used for more transactions of lesser value. e-Pocket consists of a microcomputer that contains information about the monetary value that can be used. It is a higher degree of technological development compared to magnetic tape cards. Also, the e-pouch is more secure, which can reduce deception because cards with a chip can be more difficult to abuse than magnetic tape cards.

Blockchain and Digital Currency in the World of Finance http://dx.doi.org/10.5772/intechopen.79456 27

Although cash is a quick and efficient payment method, the disadvantages of its use are numerous. Keeping cash is followed with many costs, including fraud, money loss, deposit-

The purpose of e-money is replacing the cash in transactions of small values, thus avoiding its shortcomings, for example French experience with Moneo. Moneo is designed to reduce the cost of keeping cash and purchasing power to be temporarily transferred in a more efficient manner. This structure should be applied to various retail transactions of lesser value in order