Modelling the bitcoin

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Modelling the Bitcoin

Conor O’Leary

A thesis submitted in partial fulfilment of the

requirements for the Master’s degree in

Actuarial Science and Mathematical Finance

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Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics

Author: Conor O’Leary

Student nr: 11719796

Email: conorleary@hotmail.com

Date: August 10, 2018

Supervisor: prof. dr. J.M. (Hans) Schumacher Second reader: prof. dr. ir. M.H. (Michel) Vellekoop

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Statement of Originality

This document is written by Student Conor O’Leary who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Abstract

The aim of this thesis is to model the Bitcoin price process taking into the account the emerging options market with Bitcoin as the underlying asset. Ultimately two models are used. There is a particular focus on the volatility of the price process and the implied volatility of the options. The volatility smiles present in the data are examined. Later, they are used as a form of evaluation for both models. In addition, there is an extensive study of potential arbitrage present in the data. The first model proposed is a stochastic volatility model derived from Hobson and Rogers (1998) with the key feature that volatility is driven by the same stochastic factor as the price process. The second approach, taking inspiration from several Jarrow and Protter papers, is to model the Bitcoin process as a strict local martingale - a bubble process using the CEV model. It is found that the Hobson and Rogers stochastic volatility model is better fit for the Bitcoin.

Keywords: Bitcoin, Bubble, Options Market, Volatility Smile, No Arbitrage, Put-Call Par-ity, Stochastic VolatilPar-ity, Strict Local Martingale.

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Acknowledgements

There are a number of people I would like to acknowledge, who through one way or another helped me throughout the process of writing this thesis. I would like to thank most notably and wholeheartedly my supervisor Hans Schumacher. Through his support and wisdom, I have been able to write a thesis, on a not so traditional topic, that I am proud of. Without a great supervisor this may not have been possible and thankfully this was never an issue for me. He truly went above and beyond what I expected.

I would like to thank everyone at the UvA. All of my fellow Actuarial students, who were always there when needed and are similarly aware of the academic demands of the past year. I wish to also thank all of my professors, especially Michel Vellekoop who has taken the time out to be the second reader for this thesis.

I would like to thank all of my friends and family. I would like to thank in particular my loving girlfriend Jordan Costello who was always there for me even when we lived thousands of miles apart. And most importantly my mother, who without her support, that comes in many forms, there is simply no way I could have written this thesis and completed my studies. Thank you all.

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List of Figures

1 Log Price over entire period . . . 7

2 Annualized Volatility, one-minute non-overlapping time intervals, one-hour window length . . . 8

3 Annualized Volatility with one-minute time intervals and one day window length 9 4 Annualized Volatility over 30 days . . . 9

5 Annualized Volatility with one-hour time intervals and one day window length 10 6 Annualized Volatility with 10-minute time intervals and one-hour window length 10 7 Volatility of volatility . . . 11

8 Implied volatility of Call and Put options . . . 12

9 Volatility and Implied Volatility . . . 13

10 Volatility Smile taken from [1] . . . 14

11 Skew Patterns taken from [1] . . . 14

12 Implied volatility vs Strike for Calls with Maturity 30-03-18 . . . 15

13 Issue date 23/05/2018, Bitcoin High and Low: [$7966,$7503] . . . 16

14 Histogram of strikes for call and put data . . . 17

15 Issue date 29/03/2018, Bitcoin High and Low: [$7934,$6826] . . . 17

16 Issue date 29-05-18, Bitcoin High and Low: [$7499,$7093] . . . 18

17 Issue date 10-04-2018, Bitcoin High and Low: [$6856,$6701] . . . 18

18 Example from Data . . . 26

19 Trades executed for Strategy 1 in concrete case . . . 27

20 Trades executed for Strategy 1 in general . . . 28

21 Trades executed for Strategy 2 in concrete cases . . . 29

22 Trades executed for Strategy 2 in general . . . 30

23 Price of Asset Simulation Results . . . 38

24 Initial Simulation . . . 38

25 Bitcoin Simulation . . . 40

26 Simulation of call options . . . 40

27 Simulation of put options . . . 41

28 Simulation Trade date: 29/05/18, Underlying: $7296 . . . 41

29 Simulation Trade date: 10/04/18, Underlying: $6779 . . . 42

30 100 price paths of the bubble process with α = 1.3 and σ = 0.2 . . . 48

31 100 price paths of the bubble process with α = 1.3 but σ = 0.8 . . . 48

32 Comparison of price paths with different α . . . 49

33 Two simulations of same trade date . . . 50

34 Two simulations of same trade date . . . 50

35 Volatility vs Price . . . 51

36 Log of Volatility vs Log of Price . . . 51 37 Price over the entire period . . . I 38 Price over 2013/2014 . . . I 39 Log Price over 2013/2014 . . . I 40 Implied volatility vs Strike for Calls with Maturity 29-06-18 . . . II 41 Implied volatility vs Strike for Puts with Maturity 29-06-18 . . . III 42 Volatility smile - Data and Simulation . . . V 43 Volatility smile - Data and Simulation . . . V 44 Volatility smile - Data and Simulation . . . V

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1 Introduction

The Bitcoin is now a world-wide phenomenon. However, it existed, in relative obscurity, for quite some time. It is deemed to be the first decentralnized digital currency and it was first introduced in the paper “Bitcoin: A Peer-to-Peer Electronic Cash System” written under the presumed pseudonym Satoshi Nakamoto in October 2008 [2]. In January 2009 the first Bitcoin software was released. The opening lines from the paper announce the original anti-establishment ethos and principles of the Bitcoin: “A purely peer-to-peer version of electronic cash would allow online payments to be sent directly from one party to another without going through a financial institution”. Bitcoin has now been embraced by traditional finance. It has been the catalyst for the array of cryptocurrencies currently available on the market. Furthermore, there is an emerging option market with Bitcoin as the underlying asset. This option market plays a pivotal role in the formation of this thesis.

The original inspiration for this thesis came from trying to answer the question: Is the Bitcoin an asset price bubble? Intuitively, there seems to be some basis for this by looking at the price of the Bitcoin. The price has increased and subsequently decreased in dramatic fashion several times. From January 2017 to December of 2017 the price of a single Bitcoin rose from just under $1000 to $20000 for a single Bitcoin. By the beginning of February 2018 the price of a single Bitcoin had dropped below $7000 and has failed to increase significantly as of writing [3]. However, by originally investigating the presence of bubbles in the Bitcoin process the research question has evolved: What is an appropriate model for the Bitcoin?

The purpose of this thesis is to test two sufficient models for the purposes of modelling the Bitcoin. The first model derived from Hobson and Rogers [4] has a focus on stochastic volatility. The second model assessed is the CEV model which is a bubble process for certain parameters. An important test, when assessing the models, is the reproduction of the volatility smiles founds in the option data. The availability of this option data is unusual for a bubble process and is used extensively in this thesis. Before assessing the two models, there is an in-depth analysis of the potential arbitrage present in the data. In particular examining the consequences of the put-call parity breaking down. This study is central to the formulation of the ideas in this thesis.

However, before moving forward, another question must be addressed . . .

1.1 How does the Bitcoin work?

This thesis is written from a financial perspective. Therefore, it is not overly concerned with the computer science underpinnings that the Bitcoin functions on. However, it is still important to briefly mention how exactly the Bitcoin works. For this purpose, there will be a quick synopsis of the explanation given in the original Bitcoin paper [2]. 1 As previously mentioned, the origins of Bitcoin seemed to have an anti-establishment ethos. The creators sought to provide a solution to the double spending problem without the need for a trusted 3rd party. The double spending problem is an issue concerning digital currency when the same single digital token can be spent more than once. For this, a peer-to-peer network

1

This paper is freely available online for a more in depth analysis. Furthermore, there is a plethora of resources available discussing how the Bitcoin and other blockchain technologies operate.

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was used - a network that partitions workload between peers. This would eventually become known as the block chain. The network timestamps transactions by hashing them into an ongoing chain of hash-based proof-of-work. Hashing in a broad sense is taking an input string of any length and returning a fixed length. In the case of the Bitcoin, the hash algorithm SHA-256 is used. This takes transactions of any length as input and returns output with a fixed 256-bits length [5]. This process forms a record that cannot be changed without redoing the proof-of-work.

Nakamoto defines an electronic coin as a chain of digital signatures. A coin is as expected transferable from one owner to the next. This is done by digitally signing the hash of the transaction and the public key of the next owner and adding these to the end of the coin. It would be easy to verify the chain of ownership by verifying the signatures. However, the problem is the payee cannot verify if one of the owners did not double-spend the coin. For fiat currency, a trusted central authority would check every transaction for double spending and would be aware of all transactions and which came first. For the purposes of this new model, that hinges on the idea of not having a central authority, there needed to be a way for the payee to know that the previous owners did not sign any earlier transactions. That is that the coin was not double spent. Therefore, the earliest transaction is the one that counts and thus all transactions need to be known. This means transactions need to be publicly announced. The order in which the coins are received must be undebatable. Which means the payee needs proof that at the time of each transaction. For this, a timestamp server is used. This works by taking a hash of a block of items to be timestamped and universally publishing the hash. This proves that the data must have existed at this time. Otherwise, it would not have been incorporated into the hash. Each timestamp includes the previous timestamp in its hash and so on. This forms the all important chain.

In order for all this to work on a peer-to-peer basis, a proof-of-work system is implemented. Proof-of-work systems in general must be elaborated upon. The proof-of-work system is derived from HashCash. This was originally developed “as a mechanism to throttle systematic abuse of un-metered internet resources such as email, and anonymous remailers”. It uses a CPU cost function that computes a token that is used as a proof-of-work [6]. For the purpose of this system, the proof-of-work scans for a value that begins with a number of zero bits. The work required is exponential in the number of zero bits required. Once the CPU effort has been expended to make it satisfy the proof-of-work, the block cannot be altered without redoing the work. Furthermore, as later blocks are chained after it, one would have to redo all the additional blocks to change it. This leads to the useful case where the majority decision making being represented by the longest chain. The longest chain clearly has the greatest proof-of-work effort invested in it. This is a form of protection against attackers. One would have to redo the proof-of-work of the block and all blocks after it and then catch up with and surpass the work of the honest nodes (the probability of a slower attacker catching up diminishes exponentially as subsequent blocks are added). When the proof-of-works are generated too fast the difficulty increases. This is to compensate for increasing hardware speed.

The final aspects that must be addressed is the idea of incentive. The first transaction in a block is a special transaction that starts a new coin owned by the creator of the block. This incentivizes nodes to support the network. In addition, it provides a solution to initially dis-tributing coins into circulation without a central authority to issue them. Also, it encourages

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nodes to stay honest. Take for example a situation where someone can assemble more CPU power than all the honest nodes. It would in fact be more profitable for him to play by the rules. They would gain more new coins than everyone else combined. Thus, to go against the rules can never be advantageous.2

1.2 Will the Bitcoin “Rule the world”?

A google search of “will bitcoin rule the world” returns over 18 million results (a google search of “bitcoin” itself returns over 320 million). The everlasting impact of the Bitcoin is highly debated topic. However, one lesser known critique of the Bitcoin but nonetheless major, is the immense electrical energy needed for the mining process. The laborious hashing process which is central to the mining process is using a vast amount of electrical energy. The Bitcoin power depleting nature has been held in check for the most part by rapid gains in the energy efficiency of mining hardware. However, these efficiency gains are slowing while Bitcoin value is rising fast. In addition, the hashing process is getting more and more difficult with increasing energy demands [7]. The potential transaction growth is immense. As previously mentioned. If the Bitcoin, as some predict, is to become the global, dominant currency, surely these energy consumption woes will need to be rectified? According to [8] (2015) Bitcoin is about 5,033 times more energy intensive, per transaction, than VISA card transactions. In 2010, Visa alone handled an average of 150 million transactions every day and it has the potential to handle 24,000 transaction a second [9]. Could the Bitcoin really scale up to this magnitude? It must be noted that these statistics are quite dated. Potentially the current validity of these statistics may be questionable. However, de Vries (2018) [10] postulates that the Bitcoin mining process on a whole consumes at least 2.55 gigawatts at time of his writing and could grow to 7.67 gigawatts in the near future. To put that into perspective it is stated Ireland uses 3.1 gigawatts. Is Bitcoin really able to handle the huge volume of transactions that are constantly occurring around the world? One could consider that mimicking a currency or a way of conducting transactions may not be the future for the Bitcoin. How many people who own Bitcoin plan on ever using it to make an actual purchase? It seems, many people are looking at Bitcoin to be an investment, an asset they keep, rather than as a means of conducting transactions. However, this still may not alleviate the energy consumption used in the mining process.

1.3 What defines a bubble?

It will be seen later that in a mathematical sense (defined by Jarrow and Protter), an asset exhibits a bubble when the market price exceeds the asset’s fundamental value. In a later section this will be examined in depth. However, bubbles are often studied in much more general terms. This section is devoted to a more general discussion of bubbles.

Intuitively, the Bitcoin appears to operate like a bubble - at least in a general sense. The concept of the Bitcoin being an asset price bubble has been previously studied. One paper that discusses the bubble dynamics of the Bitcoin (as well as the cryptocurrency Ethereum)

2_{It is interesting to note that the paper does not refer to those adding coins as “miners”. However, it is}

stated that “The steady addition of a constant of amount of new coins is analogous to gold miners expending resources to add gold to circulation”. This presumably gave rise to the idea of “Bitcoin miners”.

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concluded that the Bitcoin was certainly in the bubble phase [11]. Note this paper was published on 04/12/2017 and used data up to 09/11/2017. Therefore, not only did the pricing data not contain the highest pricing rise, it also did not contain the price fall that followed.

Of course, bubbles are not a new phenomenon. Jarrow and Protter (2013) [12] list some of the more well known asset prices bubbles that have occurred throughout history; “. . . Dutch tulip mania in 1634–1637 followed by the Mississippi bubble in 1719–1720, the related South Sea bubble of 1720, up to the 1929 U.S. stock price crash and the more recent NASDAQ price bubble of 1998–2000.” Of all of these documented bubbles, it is Tulip mania that many find the most intriguing. Possibly due to it being often referred to as the first bubble. Another reason for the fascination may be the shocking claims that tulips sold for the same price of a house at this time. During the peak of the bubble, during a 3 month period, some tulips changed hands 5-10 times and increased in value 10-fold. This period was followed by a drastic and steep price decrease. However, some have even called into questions its status as a financial bubble [13].

Steven Davidoff Solomon’s NY times article “How to Deflate a Gold Bubble (That Might Not Even Exist)” [14], perfectly encapsulates the vague nature of bubbles. The article refers to mainly anecdotal evidence as it is not necessarily quantitatively based. The new found openness of the gold market is referred to as a potential catalyst - the fact that one can now buy and sell gold online without ever physically attaining it. Speculation drives bubbles. The level of speculation involved in a market place is often highlighted by the level of media coverage devoted to it and its prevalence on the internet - the article refers to plethora of advertisements for gold and its coverage by the major news channels. One can only ponder, how easy it is for one to buy a Bitcoin and how much hype surrounds the Bitcoin online and now in mainstream news. There are two things of interest to note: For one, the price of gold dropped significantly shortly after this article was published. However, Protter and Jarrow (more on them later) concluded that there actually was not a bubble during this period for gold [15]. The term ‘bubble’ is quite allusive. What one academic confirms as a bubble another could easily refute.

1.4 Literature Used

In this section some of the prominent literature studied as part of this thesis will be reviewed. This synopsis will be kept very brief. One book that will be cited throughout this thesis is “Options, futures, and other derivatives 9th edition” [16] written by John C Hull (2015) which gives insight into the fundamental concepts concerning options pricing such as the put-call parity and volatility smiles.

Gromb and Vayanos (2010) [17] was used in the study of limits to arbitrage. The paper itself while written well, was quite general. It was difficult to relate directly the points made in the paper to the Bitcoin and its options market. However, it was useful in forming the arguments of Section 3.

Another key paper used in this thesis is “Complete Models with Stochastic Volatility” [4] by Hobson and Rogers. An important finding of this thesis is that the pricing data does not follow Geometric Brownian motion (discussed in later sections). Therefore, this paper is used

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as a framework to simulate a model with nonconstant volatility for our data. Although the model discussed in the paper was developed in 1998, long before anything like the Bitcoin existed, it was implemented here due to its attractive characteristic that it is not necessary to introduce additional sources of randomness. This is because the instantaneous volatility is driven by the same stochastic factors as the price process. Furthermore, it gives insight into how the implied volatility of options relate to the strike price. It is a well written paper that has stood the test of time. Two more papers [18, 19] which are based on [4] were used alongside it to formulate Section 4.

Section 5 of this thesis required the greatest study of relevant literature. This consisted of parts of several papers written by Protter and Jarrow and Protter (2009) [20], Jarrow and Protter and Shimbo (2010) [12], Jarrow (2012) [15], Protter (2013) [21] and Jarrow [22]. Also, the slides from a talk given by Protter [23]. Chapter 4 of [21] “Examples of Bubbles” was paramount to forming Section 5 of the thesis, particularly the simulation of bubbles. There is a quite a bit of repetition between the papers. Their definition of fundamental value - which is key to their definition of a Bubble is not consistent between papers. It must be conceded that it would be quite difficult to keep consistency between multiple papers that span a number of years - ideas change and evolve. Much of the formulation for Section 5 was derived from these papers.

1.5 Thesis Structure

The structure of the thesis is as follows. Section 2 contains exploratory data analysis regarding both the pricing data and the option data. Section 3 concerns the idea of arbitrage and investigating the put-call parity. In Section 4 a stochastic volatility model is applied to the Bitcoin data. In Section 5 there is a study of asset price bubbles and a bubble process is simulated. The paper concludes with some concluding remarks regarding the Bitcoin and the work completed in this thesis.

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2 Exploratory Data Analysis

This section is formatted as follows. First, there will be a brief discussion of the methodology used. Secondly, the data itself will be discussed. Finally, some rudimentary exploratory data analysis will be discussed for both the option price data and Bitcoin pricing data.

2.1 Methodology

To analyse the data the main software used was R (being used through R studio). R is a language and environment for statistical computing and graphics [24]. R provides a wide variety of statistical and graphical techniques and is highly extensible. One of the main strengths of R is its open source nature which can be extremely beneficial. This enables one to use packages and functions created by other users. Microsoft Excel was used in conjunction with R for the purposes of data cleansing.

2.2 The Data

The main sources of data examined for this thesis was Bitcoin pricing data and option data with the Bitcoin as the underlying asset.

The Bitcoin pricing data was sourced from www.kaggle.com [25]. The data provided is ex-tremely comprehensive. It displays the open, high, low and close for the Bitcoin US Dollar price every minute from 01-01-2012 to 27-03-2018. Therefore, an average of all 4 is taken when reporting price each minute. Due to the regular short intervals of pricing data and the fact that the Bitcoin market never closes, there were an enormous number of data points. This led to some issues downloading and examining the data. The timestamp is in Unix time. This was converted into a regular date and time format in R.

The option data was taken from wwww.deribit.com [26]. Deribit is an online Bitcoin Futures and Options trading platform, with its office based in Amsterdam. The option data refers to puts and calls, with the main focus of this thesis being the call options (the call options are much more popular). The call option data ranges from 29/11/17 to 07/06/2018. While the put option data ranges from 20/03/2018 to 07/06/2018. The data being used are the actual trades that go through the system. The average daily frequency for the call options was 154 and 93 for the put options during these time frames. It must be noted that the style of option is European rather than American. Furthermore, they are cash settled options. That is, when the option is exercised the writer of the contract (seller) pays any profit due to the holder in cash (in BTC) rather than any asset transfer taking place. For each trade the underlying asset (always Bitcoin), the Maturity, the Strike price, the type of Option (denoted by P or C), both the dollar and Bitcoin price, the implied volatility, the quantity purchased and the time and date of purchase are provided. Of course, the identity of the trader is not provided. Furthermore the direction of the trades is not known. Due to issues with downloading the data, there was a significant amount of data cleansing undertaken. The main issue being the date format being corrupted.

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2.3 Bitcoin Pricing Data

Here, the Bitcoin Pricing data will be examined, with a particular focus on historical volatil-ity.

2.3.1 Historical Price Data

To gain an idea of the overall pricing progression, the price is graphed over the entire data period. This will be graphed on a log10 scale.

Figure 1: Log Price over entire period

The decline towards the end of 2017 is evident. Furthermore, there were other sharp price rise and falls during the Bitcoin’s lifetime. Such as going from being essentially worthless in the beginning of 2013, to rising to over a $1000 by the end of the year and falling to $300 a year later. This suggests that during the Bitcoin’s life, ‘bubbles’ have started, burst and appeared again later. A small selection of other pricing graphs can be found in the Appendix. Next, the volatility will be examined.

2.3.2 Historical Volatility

Volatility is a statistical measure of the dispersion of returns for a given security. However, there are a number of competing methods of calculating volatility. The general idea is to look at prices at regular intervals during a certain period in which you believe that volatility should be reasonably constant, and to determine the standard deviation of the returns (in this case using log returns). There are many ways one can calculate volatility. The sampling interval can change oneminute intervals vs hourly intervals. The window length can differ -one-hour vs one day. The window style can change - overlapping vs non-overlapping. Usually, the overlapping percentage is 50% but it can differ. There can be different observation time periods - one can examine volatility over a month or over the entirety of the asset’s lifetime.

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For the most part, the observational period here will be over the entirety of the dataset. There are a number of different reporting time scales - daily, 30 day or annualized volatility. In this thesis, all volatility will be reported in terms of annualized volatility. How the volatility is scaled depends on the intervals, scaling by the square root of the interval length being used. Using one-minute intervals, the volatility must be scaled by √525600 as there are 365 ∗ 24 ∗ 60 = 525600 minutes in a year.

In this section, 5 distinct methods of calculating volatility will be briefly discussed. The main idea being to give graphical representation to the various volatility calculations. The volatility measures are created in R using basic functions. They are stored as vectors and then graphed against time. The below formula is used to calculate the various volatility measures.

V = Type of Volatility Vmi= sd(log( Pk∗m Pk∗m−i , Pk∗m−i Pk∗m−2i . . .Pk∗m−(m+i) Pk∗m−m )) Vmi= Vmi∗ sqrt(t) for K = 1, 2, 3 . . . k ∗ m < 3273377

Where m is the window length in minutes i.e. one day = 1440, where i is the interval length in minutes and t is the scaling value which depends on the sampling interval length.

The first volatility measure examined has a sampling interval of one-minute and non-overlapping window length of one-hour. Note on the y-axis 1=100%. Therefore, 75 on the y-axis actually corresponds the volatility of 7500%! This is the same for all volatility graphs.

Figure 2: Annualized Volatility, one-minute non-overlapping time intervals, one-hour window length

What is immediately clear from the above graph are the extremely high peaks. It is of interest to examine using overlapping window styles, namely 50% .3 Volatility is measured twice as

3

The formula provided must be edited slightly for overlapping window style. This can be found in the Appendix.

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frequently as there will actually be a calculation of volatility every half an hour. This is the only time (in this thesis) that overlapping intervals is used. This in many ways can be considered more accurate. The hope would be that maybe the volatility will have smoothed out using this method. However, when graphed it is essentially indistinguishable from the non-overlapping method. Another method is to examine a window length longer than an hour, specifically daily. In the graph below the window length is 1 day while the sampling interval and observation period remain the same (one-minute and entire period). The high

Figure 3: Annualized Volatility with one-minute time intervals and one day window length peaks, although still ludicrously large, are significantly smaller. Clearly this has a smoothing out effect in comparison to the volatility calculated every hour. This is especially highlighted in the below graphs which have a smaller observation period of 30 days.

(a) Annualized Volatility with one-minute intervals and one-hour window length

(b) Annualized Volatility with one-minute time in-tervals and one day window length

Figure 4: Annualized Volatility over 30 days

In all of the methods discussed so far, the sampling interval has been one-minute. This is partially due to the frequency of the data - if there is pricing information every minute why not use it? However, it is of interest to examine a larger sampling interval, namely an hour.

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Figure 5: Annualized Volatility with one-hour time intervals and one day window length

Note, volatility is scaled by a different value to arrive at annualized volatility (√8760). By using a larger sampling interval, there should be a further smoothing effect. The final method uses a 10-minute sampling interval, a window length of one hour and observation period over the entire period.

Figure 6: Annualized Volatility with 10-minute time intervals and one-hour window length

The main purpose of displaying all these techniques is the highlight that ‘volatility’ is not defined in certain terms. For example, using the above method the position of the high peaks have changed in comparison to Figure 2. This is due to using the 10-minute time intervals. Albeit, that in general, larger time intervals should have a smoothing out effect. In this case, it actually ‘increased’ volatility during this one hour, due to the nature of how the price of the Bitcoin progressed in this hour. It must be noted that the volatility in Figure 2 for the same hour is not exactly low - it is 4200%.

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There are numerous ways to calculate volatility. However, what is abundantly clear is that no matter the method used, the Bitcoin exhibits high volatility with extraordinarily high peaks. These high peaks could suggest that the volatility is not constant. This would have major implications for simulation and option pricing. To investigate this the volatility of the volatility is graphed. The volatility method used in Figure 2 will be used for this purpose -sampling interval one-minute, non-overlapping and window length of one-hour. For simplicity it is just denoted as Vol. The results are stored in a vector.

VolVolk= sd(Volk∗24+1, Volk∗24+2, . . . Volk∗24+24) for K = 0, 1, 2 . . . 2232

This produces the following graph:

Figure 7: Volatility of volatility

From the above graph it is clear that volatility of the Bitcoin is not constant. If the Bitcoin followed Geometric Brownian motion, the volatility should be relatively constant. This graph clearly indicates it is not. Furthermore, in Black-Scholes option valuation volatility is assumed to be constant. Therefore, when simulating the data, Geometric Brownian motion will not be used. Instead, stochastic volatility will have to be introduced. This will be dealt with in depth in a later section.

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2.4 Bitcoin Option Data

In this section the Bitcoin option data will be examined, with an emphasis on the implied volatility.

2.4.1 Implied Volatility

Implied volatilities are the volatilities implied by the option prices observed in the market [16, p. 341]. In other words, the expected magnitude of the future price changes of the underlying security as implied by the options. It is an estimate of the constant volatility used for Black-Scholes valuation during the lifetime of the option. Implied volatility is forward looking - it is an expectation of future volatility.

To begin with, how the implied volatility progresses with time will be examined. In the graphs below the implied volatility is compared between the put and call options during the same period. Note both graphs include all maturities and all strikes across all the available trading dates - the implied volatility for all trades during the time period is graphed.

Figure 8: Implied volatility of Call and Put options

The implied volatility of a European call option should be the same as a European put option when they have the same strike price and time to maturity [16, p. 431]. From above, it seems they follow roughly the same path. However, not much can be gained without examining the individual maturities and strikes. Below, the implied volatility of the call options is compared with the actual volatility (sampling interval of one-minute, a window length of one-hour and non-overlapping intervals) of the Bitcoin price process.

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Figure 9: Volatility and Implied Volatility

It can be seen that implied volatility is more stable than the actual volatility of the price process and also generally higher. Furthermore, it appears fairly well defined relative to the actual volatility observed.

2.4.2 The Volatility Smile

It is imperative to examine how the implied volatility relates to the strike price. According to the Black-Scholes model, keeping maturity constant, the implied volatility should be the same regardless of the strike price. Graphing the implied volatility against the various strikes should result in a flat line. However, in reality this is not the case. Graphing the implied volatility against the various strikes can result in a somewhat U-shaped curve, referred to as the Volatility Smile. Two other variants are the Reverse Skew (Volatility Smirk) and Forward Skew [1]. The volatility smile highlights the failure of the Black-Scholes model to capture the reality that volatility is often not constant. The presence of ‘smiles’ and ‘skews’ is observed in option data with all manner of underlying securities such as stocks or currency markets. A potential explanation for the presence of smiles and skews is the concept of nonconstant or stochastic volatility [4]. Therefore, the presence of smiles or skews in the data would be in line with the finding that the Bitcoin does not follow Geometric Brownian motion.

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Figure 10: Volatility Smile taken from [1]

Interestingly, prior to the crash of 1987 there was no marked volatility smile for equity options. This prompts the thought, was the market structure more in line with what the Black-Scholes model predicts prior to 1987? Was the assumption of constant volatility more consistent with the real world? Since then, in general the skew pattern referred to as a volatility smirk has been present for equity options [16]. This occurs when the implied volatility for options with lower strikes is higher than options with higher strikes. The final variant known as the forward skew occurs when the implied volatility is lower for options with lower strikes than options with higher strikes. This pattern is common in the commodities market.

(a) Volatility Smirk (b) Forward Skew

Figure 11: Skew Patterns taken from [1]

The presence of skew patterns in the Bitcoin must be investigated. Before discussing the results, a quick overview of how the analysis is conducted in R. A function is created that, keeping a certain maturity date constant, graphs the implied volatility against the strike

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prices for each trading date. The function creates a vast number of graphs - one for each trading day. Therefore, only a small number of the graphs created will be included in this report. This function was applied to both the put and call options individually and as a whole. To begin with, the results for the analysis of the call option data will be discussed. Peculiarly, often there seems to be no evident smile or skew of any form in the call data. In some cases, the graphs could be interpreted as having somewhat of forward skew. The below two examples contain the most data points (the most calls traded on both dates) for a call option with maturity 30-03-2018.

(a) Issue date 06-02-2018, Bitcoin High and Low:

[$7755,$5947]

(b) Issue date 07-12-2017, Bitcoin High and Low:

[$17020,$13666]

Figure 12: Implied volatility vs Strike for Calls with Maturity 30-03-18

In Figure 12 (a), the larger strikes tend to have higher implied volatility just like a forward skew. In Figure 12 (b) there seems to be no discernible relationship. However, what is telling is the level of variation for implied volatility present for the same strike. This is striking. What must be accounted for is the level of variation the price of Bitcoin on this trading day. Examining call options with strike price $20000, the price of a single call varied from $1507 to $3919. As the price of Bitcoin rises, so too would the price of such a call. The implied

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volatility increases with price - which is true for all options. This poses a problem later when attempting to simulate the volatility smiles found in the data. As a somewhat crude approximation, the price of the underlying asset will be taken to be the average of the daily high and low prices. Call options with a maturity date of 29-06-2018 are examined. A 4 day snapshot can be found in the Appendix. Again, a relation somewhat resembling a forward skew is evident. However, nothing definitive. And again, implied volatility can actually vary quite significantly for the same strike and maturity during a single trading day.

Now, to examine the put option data. Interestingly, it appears put option data seems to follow a pattern to a much greater extent than the call data. There seems to be multiple examples of trading days that display a volatility smirk (but not all). This is demonstrated in 4 day snapshot that can be found in the Appendix. Below is the most popular trade date for put options with maturity 29/06/18.

Figure 13: Issue date 23/05/2018, Bitcoin High and Low: [$7966,$7503]

One issue with the put option data is the lack of trades. This makes it more difficult to get a clear picture of the relation between strike price and implied volatility. One could conclude hastily that the forward skew is present in the call option data only and the volatility smirk is only present in the put option data. However, the range of strike prices for puts and calls must be addressed. On investigation the put options tends to have much lower strikes overall in comparison to the call options. This is highlighted in the below histogram:

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Figure 14: Histogram of strikes for call and put data

The dark shade of green is where the put and call options overlap. Furthermore, when taking the next most popular trade date for put options as an example, there is a noticeable somewhat U-shape resembling a volatility smile.

Figure 15: Issue date 29/03/2018, Bitcoin High and Low: [$7934,$6826]

Note the larger very much ITM strike prices present on this trade date for put options. To further the analysis, both put and call options are graphed together. Intriguingly, this often results in a resemblance of a volatility smile - it is somewhat U-shaped. Included in this report are examples of trading dates with at least 300 trades and ratio of at least 1:2 put options to call options. Below are two examples of trade dates that meet this criterion:

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Figure 16: Issue date 29-05-18, Bitcoin High and Low: [$7499,$7093]

However, it must be noted that not all trade dates that meet this criterion follow the pattern to such an extent:

Figure 17: Issue date 10-04-2018, Bitcoin High and Low: [$6856,$6701]

Graphs that meet this criterion can be found in the Appendix. Overall, a somewhat U shape is prevalent. This is an interesting and telling result. On a whole, the lower ITM strikes are simply not prevalent in the call option data. While the converse is true for the put option data. However, the Bitcoin, when traded over a sufficient number of ITM and OTM strikes often does exhibit a U-shape that resembles a volatility smile. These graphs will play an important role later when it comes to evaluating simulation results from the models proposed for the Bitcoin.

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3 The No Arbitrage Assumption

One of the main ideas behind option valuation is the principle of no arbitrage. The dictionary definition of arbitrage is “the nearly simultaneous purchase and sale of securities or foreign exchange in different markets in order to profit from price discrepancies” [27]. In other words, one can make a riskless profit. Potential arbitrage opportunities arising in the Bitcoin option market will be analysed in this section. The variation in implied volatility, holding maturity and strike constant, seen with in a single day has been the catalyst to this investigation. The assumption that there is no arbitrage is a common assumption in financial mathematics. An efficient market should rule out arbitrage. In that case, two securities that provide the same future cash flow and have the same level of risk should sell for the same price [28]. When examining arbitrage, the following are often assumed true for at least for some market participants:

1. There are no transaction costs

2. All trading profits are subject to the same tax rate 3. Borrowing and lending are possible at the risk free rate

However, the validity of these assumptions in reality is often questionable. In addition, there are a number of different arbitrage arguments that explore the relationships between option prices and the underlying security. The most important of these relationships is the put-call parity, which is a relationship between the price of a call option, the price of a put option and the underlying security [16]. When this parity breaks down arbitrage opportunities arise. First, the put-call parity, focusing on when it breaks down, will be discussed in a theoretical sense. The strategies to exploit arbitrage opportunities will be examined. Then, the reality and obstacles to exploiting such arbitrage opportunities will be discussed. Finally, there will be a discussion of the analysis of this parity using the put and call option data and the Bitcoin pricing data. 4

3.1 The put-call parity

Essentially, the put-call parity states that the value of a portfolio containing a European call option plus a zero-coupon bond with a payoff equal the to the strike price with the same maturity equals the value of a portfolio that contains European put option (neither option paying dividends) plus one unit of the underlying asset, with the both options having the same underlying asset, same strike and maturity.

put_t+ Xt = callt+ ˇKt (1)

Where put_tand calltequal the put and call values at time t, Xtin this our case being the value of one Bitcoin and ˇKt being the discounted strike price [16]. The strike price is discounted

4

The options studied as part of this thesis are European style contracts and not American - the put call parity only holds for European style contracts.

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by the discount factor at time t for maturity T. Under the principle of no arbitrage, this should always hold. This can be shown to be true with a quick proof. However, it is when this famous parity does not hold to be true that is of interest to us. That is, when one side is overvalued compared to the other. When this occurs it is quite simple (in theory) to exploit an arbitrage opportunity. There are two ways the put-call parity can break down. By examining both cases and detailing how to exploit the arbitrage opportunities in both cases, the put-call parity will be explicitly proven. Furthermore, for now all previous assumptions can still be assumed to be true.

3.1.1 Breakdown/Arbitrage Strategy (1) To begin with, the case where

put_t+ Xt > callt+ Ke−r(T −t)

will be examined. The arbitrage opportunity here can be exploited with the following strategy: taking a short position in the put and the underlying asset and a long position in the call. This strategy generates positive cash flows up front (at time t):

Xt+ putt− callt

Supposing that the sum of the cash flows were invested at the risk free rate at time T the amount grows to:

Xt∗ er(T −t)+ putt∗ er(T −t)− callt∗ er(T −t)

Now, the value of our portfolio at time t = T must be examined. If X > K at time T:

put_T = 0 callT = XT − K

Which will be the equivalent of purchasing one unit of the underlying asset for the strike price K at time T. Similarly, if X < K at time T:

put_T = K − XT callT = 0

Which is again equivalent of purchasing one unit of the underlying asset for the strike price K at time T (as we have taken a short position in the put). If X = K, neither option will be exercised. However, in this case one can still simply purchase one unit of the underlying asset for the strike price K. In any case, this can be used to close out the original short position. Therefore, at time T one has a net profit of

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Which is greater than zero as:

Xt∗ er(T −t)+ putt∗ er(T −t)> callt∗ er(T −t)− K ⇔ put_t+ Xt > callt+ Ke−r(T −t)

Which is known to be true (in this case). Therefore, a riskless profit has been achieved. It has been shown how to create an arbitrage in one instance of the breakdown of the put call parity. Now to examine the other scenario.

3.1.2 Breakdown/Arbitrage Strategy (2)

In this section the scenario where the inequality sign is reversed will be examined: put_t+ Xt < callt+ Ke−rT

To exploit the arbitrage opportunities in this scenario one would need to take a short position in the call and a long position in the put and the underlying asset. This would require initial investment of

Xt+ putt− callt

which when financed at the risk free rate would require a repayment of Xt∗ er(T −t)+ putt∗ er(T −t)− callt∗ er(T −t)

at time T. This is simply the converse of the last scenario, either XT > K, XT < K or XT = K, in any case it is the equivalent of selling one unit of the underlying asset for the strike price K. Net profit at time T is

K + callt∗ er(T −t)− Xt∗ er(T −t)− putt∗ er(T −t) Which is greater than zero as

K + callt∗ er(T −t)> Xt∗ er(T −t)+ putt∗ er(T −t)⇔ put_t+ Xt < callt+ Ke−r(T −t)

Which is known to be true (in this case). Thus, that the put-call parity should hold under no arbitrage has been explicitly proven. It has been shown, through a step by step guide, that arbitrage opportunities arise (at least in theory) in any case of its break down. However, it is of interest to move beyond theoretical study of arbitrage and examine the reality facing potential arbitrageurs.

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3.2 Exploiting Arbitrage Opportunities in Reality

In this section, potential arbitrage opportunities will be put to the test. Namely, when the two previously discussed breakdowns occur. Breakdown/Arbitrage Strategy (1) will be used to compare a sufficiently wealthy investor and an investor who needs to incorporate loans into their investment strategy. Breakdown/Arbitrage Strategy (2) will be used to compare an investor who wishes to realise any profits in dollars and an investor who wishes to realise their profits in Bitcoin. Additional barriers and risks associated with exploiting arbitrage will be discussed. First, the validity of the assumptions will be addressed.

3.2.1 Validity of Assumptions 1. There are no transaction costs.

In reality transaction cost exists. It is stated the assumption of no transaction costs need only to apply to large institutional investors [16]. In more traditional markets this could be true. However, transactions costs here involve purchasing/selling Bitcoin and taking long/short position in put and call options on the Bitcoin. For at least the purchasing and selling of options on the exchange platform Deribit (one of the few places where this is possible) there is no visible channel for a reduction in transaction fees due to the size of the investor. It cannot be assumed in reality that there are no transaction costs.

2. All trading profits are subject to the same tax rate.

Examining the taxation for investment profit and losses is beyond the scope of this thesis. However, this is a reasonable assumption and does not hold much credence in the investiga-tion.

3. Borrowing and lending are possible at the risk free rate.

Lending or making investments at the risk free rate is possible through the purchase of treasury bonds. However, the potential for investors to borrow at the risk free rate is dubious at best. Even Black stated that this was unrealistic [29]. Furthermore, borrowing related to the Bitcoin, an especially risky asset, will most definitely not be subject to the risk free rate, if it is granted at all. Therefore, it cannot be assumed that borrowing at the risk free rate is a valid assumption. In addition, going forward r will be taken to be 0.

3.2.2 Explicit Costs and Barriers

There are a number of explicit costs and barriers associated with carrying out the necessary trades. All of the below are assumed known to the investor. They are as follows:

1. Bid Ask spread 2. Transaction Costs 3. Exchange costs 4. Withdrawal costs

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5. Margins 6. Mistrade Rule

The bid ask spread must be taken into account. This is the difference between the price you pay for immediate purchase of an option (ask price) and the price you receive for immediate sale of an option (bid price). The bid price is lower than the ask price. The issue is that the data examined for this thesis the direction of the trades is not known. That is, it is not known if the reported price of past transactions is the bid price or ask price. Therefore, the extent of the difference in costs depending the actual direction of the trades must be addressed in some degree. This is done by examining a snapshot of the trades available on Deribit - as the bid and ask prices will be known for this trade data for various strike prices. It is useful for us to reformulate the put-call parity as follows:

callt− putt = Xt− ˇKt

To attempt to account for the effect of the bid ask spread, the difference between callask−putbid (the “best” case scenario) and callbid− putask (the “worst” case scenario) will be examined. This will give an idea of the range of costs that can be associated with the bid ask spread for the past trades examined as part of thesis. On the trade data examined, a single Bitcoin was valued at $7400 and 4 different time to maturities were examined (T=0.07, T=0.15, T=0.40 and T=0.64). The extent of the difference increased conclusively along with time to maturity and tended to increase as moneyness _KS decreased. The largest and smallest difference for each time to maturity can be seen below:

T=0.07 0.019BTC ($143.360) 0.0285BTC ($212.04) T=0.15 0.0195BTC ($145.08) 0.037BTC ($275.28) T=0.40 0.051BTC ($379.44) 0.0795BTC ($591.48) T=0.64 0.1195BTC ($889.08) 0.3035BTC ($2258.04)

Therefore, it can be seen that bid ask spread can be a rather large cost associated with execut-ing the trades. This will play an important role when analysexecut-ing the past transactions. For every option both the buyer and seller must pay transaction costs of 0.0004BTC per option. The fees cannot surpass 20% of the option cost. These fees are charged initially at the point of sale. If the option is exercised the buyer must pay further costs of 0.0002BTC on delivery [30].

All funds in Deribit must be uploaded in Bitcoin. There is no option to purchase BTC for fiat currency on Deribit. Therefore, to execute the necessary trades an investor must buy/sell Bitcoin on a different platform. The buying costs for buying Bitcoin for most major currency seems to be 1.5%. The exchange rate of 1.5% also applies to the sale of Bitcoin [31]. It will be assumed investors have a dollar account. It will be seen later that this is the large extra cost associated with executing arbitrage in this market.

The initial margin is the amount needed in the investor’s account to take a short position in an option. Margin maintenance is the amount of funds needed to be maintained during the lifetime of the option. Not necessarily a direct cost, these funds are needed to execute the trades. Therefore, they serve as a barrier to any investor without sufficiently deep pockets.

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They are calculated using the mark price, which is essentially the average between the bid and ask price. They are calculated as follows: 5

Initial margin (BTC) : 0.1 + Mark Price of the option Maintenance margin (BTC) : 0.08 + Mark Price of the option

[30]

Furthermore, the Deribit platform exchange has a mistrade rule. Full explanation in the platform’s own words can be found in the Appendix. Following the trades being executed the price can be updated on the platform. The prices can be updated to within 0.05BTC of the “theoretical price” which seems to be the mark price in most cases - the explanation is quite convoluted. Therefore, there could be a substantial change to the price equivalent to hundreds of dollars. This could leave the investor exposed to unwanted risk as it would not serve to protect Bitcoin bought or sold elsewhere. This rule attempts to protect people against mispricing, hoping to reduce risk for investors. However, for arbitrageurs, this is clearly a rather large additional risk. It could be postulated that the whole reason for this vague ruling’s existence is to act as a deterrent for traders wishing to exploit situations where it appears one side is undervalued. That is, maybe its purpose is to act as a deterrent for arbitrageurs.

In addition, there are costs associated with withdrawing funds from Deribit. According to Deribit they depend “on the current state of the Bitcoin network” [30]. This withdrawal cost will be denoted as Wt. Although the actual amount is unknown, it is an unavoidable cost incurred when one wishes to liquidate their funds stored on Deribit.

When examining the explicit known costs there already seems to be an advantage for suffi-ciently wealthy investors. This suggests there is an advantage for large scale investor over an individual investor. It will be seen that implicit costs and barriers are of a much greater hindrance to individual investors, especially those without sufficiently deep pockets.

3.2.3 Additional Barriers and Risks

There are additional barriers and risks to consider: 1. Monitoring price changes

2. Volatility of price changes 3. Loan processing

4. The type of investor

For one, someone (or some computer program) would have to continuously monitor prices. This would take up considerable time and effort. People do not (usually) stumble across

5

The actual calculations are bit more complicated and potentially depend on the out of the moneyness of the options.

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arbitrage opportunities. They spend considerable time and effort attempting to identify and exploit such opportunities. In large institutions, there are positions dedicated entirely to exploiting arbitrage in more traditional markets. However, for a single, independent investor this would be a major barrier.

However, even constant surveillance of the prices may not be enough. As seen the Bitcoin exhibits extremely volatile price changes. When examining the minute by minute price during the period from 29/11/17 to 27/03/18 (when the call options data begins and our pricing data ends) the minute high and low prices differ by more than $100 almost 3% of the time. As a percentage of the value of the Bitcoin, this difference of $100 or more ranged from 0.5% to 3.7%. To exploit any arbitrage opportunity, the trades should ideally be executed simultaneously. Again, this would affect a single investor to a much higher degree than large scale investors.

The potential to exploit arbitrage opportunities that incorporate loans into the strategy are hindered greatly (if not completely eradicated) due to borrowing costs. However, even if one could borrow at the risk free rate, the processing time for any loan would be far too long. The sufficient funds, in Bitcoin, will need to be previously loaded onto the Deribit platform and an exchange platform for immediate use. This is a major barrier to any investor who needs to incorporate a loan into their investment strategy.

Who would actually be interested in such an arbitrage opportunity? The nature of Bitcoin and the ever looming possibility of immense gains may seem to diminish the attractiveness of such an arbitrage. If we once again envision a theoretical world, this eliminates a vast number of investors who would be interested in such an opportunity. Potential investors can have three possible opinions about the future price of the Bitcoin (or any underlying asset):

• The price will go up. • The price will go down. • The price will stay the same.

It must be conceded that technically an arbitrageur should not be influenced by their expec-tations of future price. The profit is riskless and it does not matter if the price goes up or down. However, when examining the motive of individual traders the idea that their opinion of future prices should not matter becomes a little vaguer. Taking for example someone who already owns a Bitcoin. If they believe the price will go down, surely they would think it would make more financial sense to just sell the Bitcoin they own, rather still having Bitcoin in their portfolio in 3 months for a relatively small profit. In addition, optimistic investors would not want to tie up their funds for 3 months for a relatively small profit. On the other hand, if an investor really did think the Bitcoin price would stay level then this would be at-tractive. However, as seen throughout the thesis the Bitcoin exhibits overwhelming volatility. How many investors really believe the Bitcoin price will stay level in any future period? Implementing arbitrage on a larger scale would be more attractive. That is depending on the strategy; shorting/purchasing multiple Bitcoins and shorting/purchasing multiple put and call options. This reduces the overall effect of implicit costs such as monitoring price changes. This again suggests that any potential arbitrageurs in this market would be large scale investors.

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It can be seen that when accounting for implicit costs, exploiting the arbitrage opportunities seems much more attractive to large scale investors with sufficiently deep pockets.

3.2.4 Example from Deribit

Figure 18: Example from Data

The above is a snapshot of real trades that were available on the Deribit platform. There is no arbitrage opportunity here. However, by formulating the necessary trades needed to exploit an arbitrage opportunity, a better understanding of the actual costs involved can be gained. The case where the strike = $7500 will be used. Using Breakdown/Arbitrage Strategy (1) a sufficiently wealthy investor and an investor who needs incorporate loans into their investment strategy will be compared. Using Breakdown/Arbitrage Strategy (2) investors who want to realise any potential profits in dollar vs Bitcoin will be compared.

3.2.5 Breakdown/Arbitrage Strategy (1)

Let’s refresh, for someone to be in the position to construct this arbitrage they must able to:

• Short a Bitcoin • Short a Put option

• Take a long position in a Call option

If any of the above were not possible, then arbitrage essentially would not be possible. In short, executing all of the above is both possible and feasible. There are many costs associated with these transactions. Investors who are sufficiently wealthy and those who need to incorporate loans into their investment strategy will be examined separately. First an example from the concrete case:

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Figure 19: Trades executed for Strategy 1 in concrete case

There are a few things to note. For one, the trades are given in dollar amount - so that the significance of each cost can be understood to a greater extent. It can be seen that two different arbitrary values have been taken for the value of the Bitcoin at time T. This is not to suggest that an investor would have any idea of the value of the underlying asset at time T. It is to demonstrate that, when taking costs into account, there is now a difference in the profitability of an arbitrage depending on the value of the Bitcoin. Furthermore, it can be seen that even without costs, these trades would lead to a loss of close to $100. In addition, the strike price here is $7500. Going forward it will be assumed that r=0. Now to address both type of investors.

(1a) Sufficiently wealthy investor

The investor here has sufficiently deep pockets and sufficient Bitcoin in their portfolio and sufficient Bitcoin loaded onto the Deribit platform to execute the trades. To make this initial trade one would need to have a Bitcoin in their portfolio, sufficient Bitcoin to purchase the call options and sufficient Bitcoin for the initial margin for shorting the Bitcoin. Exchange costs would of course apply to the purchase of this sum of Bitcoin. However, the value of Bitcoin could have been much higher or lower than the current price, which would of course change the exchange cost amount. These exchange costs will be ignored, as they concern costs incurred prior to time t. Furthermore, the mistrade rule poses a great risk to this arbitrage. However, it essentially cannot be quantified and will have to be ignored. Going forward, Xt will denote the value of the Bitcoin at time t. The costs at time t depend on the value of the Bitcoin. They are mainly due to the exchange costs related to selling the Bitcoin.

The costs occurring at maturity date T must be addressed. There is now a difference in costs depending on whether the put option or the call option is exercised. If the put option is exercised, the arbitrage is actually more profitable. For one, the exchange costs associated with purchasing the Bitcoin would now be lower. On the other hand, if in fact the call option would to be exercised the costs would be higher. For one, there is the delivery cost of 0.0002BTC. Although this is not a fixed dollar amount it can be simply hedged by putting

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aside 0.0002BTC at time t, in the event that the call option is exercised. Furthermore, WT may apply to liquidating funds stored on Deribit. The profit from the call option may be needed to purchase a Bitcoin. Again, the exchange costs associated with purchasing the Bitcoin would be higher. All of the costs here are future projections and depend on the value of the Bitcoin at time T. Therefore, they cannot be quantified in analysis of the data as they involve future expectation, which would not be known to an investor. It is evident that an investor is no longer completely indifferent about the value of Bitcoin at time T. They would in fact be better off - ignoring their other investments - if the price of the Bitcoin dropped. This is in contrast with an arbitrageur in a traditional sense who should have no opinion on the price progression of the asset. In the concrete example, the difference in cost depending of the value of a Bitcoin at time T is clear. In this case there is a difference of almost $50. It is assumed for simplicity in this case that the withdrawal costs are the same as the exchange costs - 1.5%. Below is a formulation of the costs in general when executing the trades.

Figure 20: Trades executed for Strategy 1 in general

It is clearly of interest for an investor to avoid these exchange costs. They are clearly the largest explicit costs associated with this investment strategy. However, they are essentially unavoidable when dealing with Bitcoin to some extent. They will be a factor when a party wants/needs to increase the level of Bitcoin in their portfolio. And when they want/need to decrease the level of Bitcoin in their portfolio. Analysing this as a single trade there is potential to avoid these exchange costs by incorporating a future contract into the investment strategy. The Deribit platform also offers the possibility to buy or sell cash settled future contacts. However, the analysis of the future data will not be addressed in this thesis. (1b) Investor who needs to borrow

As it can be seen there are multiple extra costs associated with executing arbitrage. This means that essentially exploiting arbitrage opportunities is infeasible for those who need to borrow. For one, borrowing costs would presumably be too large. To execute the required trades need at a minimum need to borrow in dollar amount 1.015 ∗ (Xt+ Calla+ 0.1Xt+ P utm+ 0.0008Xt) which in the concrete case examined amounted to close to $10000. There are multiple factors that affect the rate of a loan, which are far beyond the scope of this thesis. However, even assuming a rather generous 10% rate for a short term 3 month loan the interest accrued would still be close to $250 in this case. At any case it would seem to be quite unlikely that any conventional bank would grant a loan for such purposes to a sole

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investor. There are platforms that offer margin trading as a way to short a Bitcoin without having the funds, but similar costs are associated with this. Even ignoring borrowing costs, the length of time needed to execute such a loan would be too long. Another approach would be to borrow sufficient funds in the hope of uncovering an arbitrage opportunity. However, that is simply risky behaviour and far from the idea behind riskless investment. Therefore, it can be concluded that this arbitrage opportunity is unattainable for those who do not have sufficient funds/Bitcoin to begin with. Intuitively, in any case one would assume that trading strategies like this would not be applicable to those with out sufficiently deep pockets.

3.2.6 Breakdown/Arbitrage Strategy (2) To exploit this arbitrage strategy one must be able to

• Purchase a Bitcoin

• Take a long position in a Put option • Short a Call option

By the same arguments as the previous breakdown, the arbitrage would not be open to an investor who needed to incorporate loans into their investment strategy. What will be compared here is an investor who wants to realize their funds in dollars and someone who is willing to keep their funds in Bitcoin. The second type of investor may avoid the enormously costly exchange costs associated with the trades. First the concrete example:

Figure 21: Trades executed for Strategy 2 in concrete cases

It can be seen that although not profitable, that these trades result in less of loss. (2a) Profit in Dollars

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To avoid repetition this investor will not be examined in great detail as the argument presented are very similar to the previous section. In the above case it is assumed that the investor buys a Bitcoin at time t and resells this Bitcoin at time T and wishes to realise any profits in dollars. As it was seen when the previous breakdown occurs, investors are no longer entirely indifferent about the value of the Bitcoin at T. In this case the costs once again differ. It cannot be conclusively stated which cost is higher as it would depend the value of XT. In the example the withdrawal costs are again taken to be the same as the exchange costs, but the costs at T are dependent on the actual withdrawal costs. Below are the costs associated with the trades in general:

Figure 22: Trades executed for Strategy 2 in general

However, what is of interest, is to examine an investor who wishes to keep any profit in Bitcoin. This type of investor will avoid the excessive exchange costs.

(2b) Investor willing to keep funds as Bitcoin

The investor here is assumed to be sufficiently wealthy in Bitcoin and has no issue realising any profits in Bitcoin and keeping them as such. This investor would not have to purchase a Bitcoin a time t - avoiding the exchange costs at the initial leg of the trades. At time T riskless profit could be achieved - when valued in dollars. If the call option would be exercised a single Bitcoin will be worth more. However, they would have to sell a portion of this Bitcoin to cover the losses on the call. They would have less Bitcoin but the value of their portfolio in dollars will not be effected. On the other hand, if the put option is exercised, they would have more Bitcoin but at a much lower value. Again, when valued in dollars this will not affect the value of their portfolio. This is demonstrated here taking for example 1 BTC = $5000 at time t and either $10000 or $1000 at time T (shorting an ATM call option and long position in an ATM put option):

% 0.5BTC($5000) 1BTC($5000)

& 5BTC($5000)

Modelling the bitcoin