Bitcoins and Fractals

tl;dr: Bitcoin is the first crypto-currency to receive mainstream attention, and it’s reception may decide the success or failure of this new form of financial interaction. I’ve been fascinated by Bitcoin for months, and have made reasonable profits trading with it, but I sought a means to develop a viable trading strategy. However, due to the novelty of the asset, most of the models I have access to at my level simply cannot  cope with it, so I had to search for something a little more rigorous, and I think I may have found my first step in Financial Multifractal Analysis.

Dragon curve (From Jurassic Park)

I’ve always fascinated by new technology, and Bitcoin is no exception. For the past few months, I’ve been trying to mine, trade, and learn as much as I can about this unique asset, in order to understand how it could change the way we deal with currency in the future. However, unlike my usual process of rambling on and on about my thoughts on the subject and distracting you with barely relevant gifs, this time I’ve actually got something semi-formal to show you; a psuedo-paper I’ve been working on for the past few weeks. It currently represents the most cohesive set of work I’ve put into this project so far. Before I start telling myself how I did it, you guys can get to it here.

Edit: With regards to how much attention this is getting, I’d just like to point out that at the time of this publication I was just an Undergraduate Finance student at UBC. This work is in no way backed by UBC or any other sort of professional institution, and is a purely personal application of theory that could not have been created without the assistance and direction of many generous professors  and actual experts. I am not an expert, I am just an enthusiast who wanted to make something cool.

I. Backstory.

There are already hundreds of articles claiming to know what the Bitcoin is and what it’ll become, so I won’t waste my breath repeating them there. It is my opinion that the most fascinating thing about the Bitcoin is that we do not know yet what it will become; for an explanation of why, I’d suggest you take a look at my paper. In this section, I will outline the route behind the creation of my paper, mainly for the sake of posterity. Again if you’d like to get to the actual report meat, please click here; I highly doubt any of this will be relevant to anyone but Future Me.  

Basically, this all started because I wasn’t playing too nice with my group during an informal 10-minute COMM 377 presentation, and was given the option by my professor Dr. Lazrak to present again with another group. I informed him that I wasn’t too keen on the idea of presenting again to such a small group of people, and asked if I could just do a project on my own instead to “prove I could work.” Thankfully, he said that was alright, and I now that I had a little more room, I immediately knew I was going to write my report on Bitcoin. However, I didn’t want to just write a typical report and just summarize a recent news story in order to feign a weak relationship to our course content; I wanted to build actual analysis that had something to add to the discussion, or at least my ability to trade.

However, when trying to analyze the asset using traditional financial models (such as CAPM, etc.) I hit roadblock after roadblock, because Bitcoin is unlike most traditional assets (And I’m kind of a n00b compared to the older smarter people who get paid to do this). It has no economy, no monetary policy, no centralized government, nor even any intrinsic value. It is driven by pure supply and demand, and is regulated at the entry and exit points of the system (currency exchanges), and is otherwise self-contained, decentralized, and anonymous. So, how do you analyze a currency that is not a currency? The answer eluded me for quite some time, until one of my late night Netflix binges, when I accidentally, I stumbled upon this Ted Talk by Benoit Mandelbrot:

Sierpinski Triangle as a Fractal.

Dr. Mandelbrot was widely recognized as a ‘rockstar’ of a mathematician, and his work was credited by people such as Arthur C. Clarke (Who wrote the novel that become the movie 2001 – A Space Odyssey) as “one of the most astounding discoveries in the history of mathematics.” To unfairly summarize, Dr. Mandelbrot had discovered a means to generate incredibly complex patterns from the repetition of relatively simple rules, and he called these patterns ‘Fractals’. A fractal is a pattern that is self-similar at various scales, and is generated by re-iterating a generally simple algorithm over time. While I will not go into the raw mathematics simply because I am not familiar enough with them to do them justice, perhaps the gifs interspersed throughout this post will be helpful in illustrating the concept.

‘Fractal Tree’

In any case, Dr. Mandelbrot’s discovery provided a framework to develop a whole host of new complex patterns that seemed bizarrely adept at corresponding to natural phenomenon. While much of the world is indeed random, his work suggested that there is an ‘order’ to this randomness, and, once cultivated, it is possible to describe the patterns that define everything from how a tree grows to how a shoreline develops. I’ve always been intrigued by fractals, but I did not realize their usefulness until I watched that Ted Talk, and discovered that, surprisingly, they had their roots in financial analysis, which is where Dr. Mandelbrot started. Fascinated, I wondered if this field could provide the tools I was looking for to analyze this unique asset, and I set out to teach myself Fractal Analysis.

Unfortunately, that is much easier said than done; Multifractal analysis represents one of the most cutting-edge fields of research in continuous-time finance theory, and its capabilities are not only extremely complex, but practically reserved for the more mathematically gifted. Naturally, I struck yet more hurdles, and after spending more than a reasonable amount sucking at it, I realized that perhaps I might need some help. In my search for resources, I ran into this little block of code, and then this textbook, written by a current professor at UBC, Dr. Adlai Fisher. Curious, I e-mailed him asking if I could meet with him about this personal quest of mine, and he informed that he was on sabbatical at the HEC, but was capable of putting me in touch with his PhD student, Mr. Charles Martineau. Mr. Martineau defies everything I thought I knew about the Finance department at UBC. Not only was he candid, direct, and comfortably informal, he had a capacity for creativity and curiousity that the cynic in me never expected to find at Sauder, and I consider meeting him one of the pivotal moments of this project. I could go on and on about how patient and helpful he was with how much of a novice I am and how much I had to learn, but there simply isn’t enough room to express how thankful I am. Suffice it to say, he is probably the  the main reason I have anything worth showing you at all.

In any case, Mr. Martineau informed me that while developing a forecasting strategy for Bitcoin is easier said than done, perhaps the first logical step might be to determine if Bitcoin data did indeed show properties of fractal behavior. He explained to me the tools and questions I needed to ask to determine this, and armed me with a page of notes and a Matlab script I knew literally nothing about to determine if Bitcoin data showed fractal behavior (Spoiler: It does.) I will reproduce my paper below to illustrate what I’ve done and discovered in my work. As a psuedo-academic paper, it represents one of the more…formal pieces I’ve done so far, but also one of the most personally rewarding, even though it isn’t really the trading strategy I was looking for. While I am still not much closer to developing a viable means of projecting Bitcoin’s behavior, perhaps my work will provide some useful data to other people curious about Bitcoin, and allow me to advance my work in the future. It’s definitely nothing I would submit to a journal or anything, but it was a fun exercise and I learnt a lot doing it, so feel free to point out any corrections or ask me any questions you might have. Enjoy:

II. Paper. 

Final1_2 Final1_22 Final1_23 Final1_24 Final1_25 Final1_26 Final1_27 Final1_28 Final1_29 Final1_210 Final1_211 Final1_212 Final1_213 Final1_214 Final1_215 Final1_216 Final1_217 Final1_218 Here’s a downloadable pdf. (Sorry about the blurry formulas, next time I’m going to just leave it in LaTex, and not export to png.)


4 thoughts on “Bitcoins and Fractals

  1. Pingback: Virtual Mining Bitcoin News » 6 Things Bitcoiners Are Talking About That You've Never Heard Of

  2. Pingback: Re: Wall Observer BTC/USD – Bitcoin price movement tracking & discussion |

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