Pawan IyerDec 28, 2017MCR3UE Creative Project Introduction: The financial market is a global powerhouse with countless values of currency being traded every minute. Mathematics is very crucial to finance and more specifically to Bitcoin and the stock market as we will come to know through my report. The study of mathematical finance, also known as quantitative finance, is a field of applied mathematics, concerned with financial markets. About Bitcoin: Bitcoin is a virtual currency. This means that it only exists digitally, it does not have physical notes or coins, and it can be used to buy things on the internet. One of the major reasons that people might want to use Bitcoin is that it exists internationally and isn’t controlled by any one government or company. This can be very useful as lots of businesses now operate online, and trade in multiple countries. Companies and individuals want to avoid paying transaction fees and taxes when they swap between currencies. With a digital currency like Bitcoin they don’t need to pay any of these charges. Bitcoin also ensures complete anonymity in its transactions, so one purchasing party cannot disclose personal details about the other party. One big drawback to Bitcoin is that currently there aren’t many retailers which will accept it as a currency although the United States has recently welcomed a growing number of businesses which accept Bitcoins. Another drawback is that digital currencies are a new technology, and as a result the price of Bitcoins fluctuates rather dramatically. We can investigate and prove the use of mathematics in Bitcoin trading and the stock market trends associated with it, by using specific mathematical subjects. These include:Elliptical CurvesDoubling TimeSinusoidal WavesElliptical CurvesHow does Bitcoin work?In order to work Bitcoin needs to overcome several major hurdles. Firstly it must make sure that transactions involving Bitcoins are secure. Bitcoin does this using something called ‘elliptic curve cryptography’ to ensure the security of transactions between owners of Bitcoins. Elliptic curve cryptography is a type of public key cryptography, relying on mathematics to ensure that a transaction can be secure. Elliptic curve cryptography is based on the difficulty of solving number problems involving elliptic curves. On a simple level, these can be regarded as curves given by equations of the form where and are constants. Below are some examples. In each case the graph shows all the points with coordinates , where and satisfy an equation of the form shown above.The elliptic curves corresponding to whole number values of a between -2 and 1 and whole number values of values of b between -1 and 2. Only the curve for a = b = 0 doesn’t qualify as an elliptic curve because it has a singular point. For the sake of accuracy we need to say a couple of words about the constants and For an equation of the form given above to qualify as an elliptic curve, we need that This ensures that the curve has no singular points. Informally, it means that the curve is nice and smooth everywhere and doesn’t contain any sharp points or cusps. In the examples above the constants and were chosen to be whole numbers between and and and respectively. But in general they can also take on other values. (For uses in cryptography and are required to come from special sets of numbers called finite fields). Given an elliptic curve, we can define the addition of two points on it as in the following example.Let’s consider the curve and the two points and which both lie on the curve. We now want to find an answer for which we would also like to lie on the elliptic curve. If we add them as we might vectors we get – but unfortunately this is not on the curve. So we define the addition through the following geometric steps.We join up the points and with a straight line. This line generally intersects the curve in one more place, We then reflect the point in the -axis.We declare this new point, to be the sum of and So In our example this means that Trying another example, (shown below), with and we have and . Therefore We also need a definition for the sum when to understand what we mean by In this case we take the tangent to the curve at the point , and then as before find the intersection of this tangent line and the curve, before reflecting the point. This is probably easier to understand with another graph:Here we use the elliptic curve and consider the point We have drawn the tangent to the curve at which intersects the curve at a second point Reflecting in the – axis gives Therefore we say or It’s now possible to define what we mean by for any point on the curve and any natural number and so on.The only situation in which our definition of doesn’t work is when is the reflection of in the -axis (which in the case where means that itself lies on the -axis). In this case, the line we use to define the sum is vertical and there isn’t a third point at which it meets the curve.We can get around this problem by adding an extra point to the usual plane, called the point at infinity and denoted by To make the addition work for our exceptional situation we simply define For any point we also define — so with our new notion of addition, the point plays the same role as the number in ordinary addition.It turns out that, given two points and on an elliptic curve, finding a number such that (if it exists) can take an enormous amount of computing power, especially when is large. Elliptic curve cryptography exploits this fact: the points and can be used as a public key, and the number as the private key. Anyone can encrypt a message using the publicly available public key (we won’t go into the details of the encryption method here), but only the person (or computer) in possession of the private key, the number can decrypt them.Conclusion: Currently the digital currency Bitcoin uses elliptic curve cryptography, and it is likely that its use will become more widespread as more and more data is digitalised. However, it’s worth noting that as yet no-one has proved that it has to be difficult to crack elliptic curves – there may be a novel approach which is able to solve the problem in a much shorter time. Indeed many mathematicians and computer scientists are working in this field. Government digital spy agencies like the NSA and GCHQ are also very interested in such encryption techniques. If there was a method of solving this problem quickly then overnight large amounts of encrypted data would be accessible – and for example Bitcoin currency exchange would no longer be secure. It also recently transpired that the NSA has built “backdoor” entries into some elliptic curve cryptography algorithms which have allowed them to access data that the people sending it thought was secure. Mathematics is at the heart of this new digital arms race.Doubling TimeWhat are the observed and predicted trends in the price of Bitcoin?The value of a Bitcoin is determined by the laws of supply and demand – there are a limited number of Bitcoins in circulation, and therefore their price is decided by how popular they are, and how many people want to trade in Bitcoins at that time. Analysts use the concept of doubling time to forecast the trend for a stock or currency. Bitcoin’s price could hit $100,000 per coin if it continues to follow one of tech’s “golden rules” — Moore’s law. The rule, which was devised in 1965 by Intel cofounder Gordon Moore, describes the exponential improvements of digital technology. Moore’s law specifically applies to the number of transistors on a circuit but can be applied to any digital technology. Any technology that is growing exponentially (i.e., ‘following Moore’s law’) has a doubling time. Typically, however, the rule applies to a technology’s computing power or capabilities. Bitcoin was the first time a technology’s price began following Moore’s law. What is the doubling time for the price of Bitcoin?We can determine the doubling time of Bitcoin since its inception in 2010 with the following logarithmic operation:Let’s assume P1 and P2, t = Time in months (All currency values are calculated in USD)July 17 2010: $0.05 (P1)Dec 31, 2017: $13,981.64 (P1)Time Interval = 2724 days (? 89 months)Given: P2 = P1 * 289/tP2/P1 = 289/tlog(P2/P1) = 89/t log2t = 89 log 2 / log (P2/P1)t ? 4.9 monthsHence we can see that the doubling time given the parameters is nearly every 4.9 months. This is an exceptionally high rate of growth for a currency. So this poses the question, what does this mean for someone in possession of one Bitcoin today? What is the forecasted of Bitcoin value in 5 years?Based on the doubling time of 4.9 months and the recent surge in Bitcoin valuation we see the following:P in 5 years?P5 = $13,981.64*25*(12/4.9)P5 = 65677296.83 By calculating the doubling time of Bitcoin, and forecasting the value just 5 years in the future, we can see the value of 1 Bitcoin capping nearly $65.6 million USD by the year 2023. Despite this being a hypothetical situation with no external factors affecting the price, it is still an immense growth for such a young currency. Conclusion: In the stock market and with many currency exchange rates, the concept of doubling time is used extensively. It requires a knowledge of logarithms as well as how to apply the doubling time formula. Doubling time not only allows analysts to see the rate at which a certain currency or stock is increasing, but also forecast the future values when making the decision to invest. In the case of Bitcoin and other cryptocurrencies this is crucial because of their high volatility. It is very unstable and depends entirely on concept of supply and demand with no regulations. To invest wisely and make spike or drop predictions one must be able to see trends that exist within them. Overall we can see that mathematics is once again at the core of the the stock and Bitcoin world. Sinusoidal Waves Do stock market value changes follow a pattern, can they be interpreted using sinusoidal waves? In mathematics, the Fourier Series is a method in general time signal analysis that is used to model data using a sum of sine and cosine waveforms. More formally, it decomposes any periodic function into the sum of a set of simple oscillating functions, namely sines and cosines. The number of sinusoidal terms used to create the series is defined by the order of the Fourier Series. For example, a Second-Order Fourier Series, contains two sine and two cosine terms. The more sinusoids that are used, the more accurate the dataset that is given will be modeled, for value interpolation and extrapolation. The Fourier Series has been applied to a variety of stocks and conclusions have been with mixed results. It was observed that in general, functions with lower volatilities were best modeled by a linear trend combined with a sinusoidal component, while higher-volatility stocks were more accurately modeled. It can be further observed that for very low volatility stocks, the sinusoidal component often over or under-predicted the actual peaks and dips of the function, respectively. Based on these observations, a new model was created using normalized volatility as a metric for interpolating past values and forecasting future prices of a given stock. Generally, the amplitudes and frequencies of these sinusoidal terms are found by converting the series into its Fourier Transform representation in the frequency domain. In the frequency spectrum, only the highest magnitude terms are kept, while the others are discarded. The number of terms to be kept is defined by the order of the series. This yields a signal with much less noise, since only a few fundamental sinusoids remain, while still following the general shape of the original signal. Conclusion: Though the price of a stock is often considered to be a seemingly random, with countless factors determining its future outcomes, it is seen we can use sinusoidal functions to provide positive outcomes for even the most inexperienced of investors. It has not only proven to be profitable for short term investments, but it also provides wealth of knowledge about different metrics for describing stocks. This acquired stock market intelligence will greatly help in making long-term retirement investment decisions. ConclusionThrough this report, we see there is a close relationship between established mathematical concepts and models, and the seemingly random, irrational world of cryptocurrencies and stocks. By further studying topics such as elliptical curves we can their significance in building a secure system for currencies such as Bitcoin. We also see applications of concepts learned in class such as doubling time having direct implications when forecasting stock and cryptocurrency price values. Finally when examining the patterns stock market trends follow, we see a direct use of sinusoidal functions in conjunction with new concepts such as the Fourier Series. This ultimately demonstrates the strong relationship that mathematics shares with the world of finance and how knowledge in these concepts will truly benefit any future investors. With that I would like to leave you with three equations any wise investors will benefit from knowing in their future endeavours. Equation 1S&P 500 dividend yield + about 4.5% = the expected long-term return on stocks This formula, known as the Gordon equation, assumes stocks get their ultimate value from being able to one day return earnings to investors. (That’s true whether or not a company currently pays a dividend or reinvests in the business.) Equation 2A 1.5% expense ratio = more than 40% of your money after 40 yearsMutual fund and adviser expenses seem so tiny— just 1% or so. But over many years expenses add up, or more mathematically precisely, they multiply up. For instance put $100,000 into a fund with a 1.5% expense ratio, assume a 6% underlying return, and you’ll get about $560,000 after 40 years. With the same pre-expense return in a very low-cost index fund charging 0.1%, you’d have $990,000.Equation 3Net income / shareholder equity = return on equityReturn on equity is a classic measure of a company’s ability to put shareholders’ money to good use. (Equity is roughly the cash investors put into the business, plus retained earnings.)Happy Investing!CitationsCoindesk. “Bitcoin Price Index – Real-Time Bitcoin Price Charts.” CoinDesk, www.coindesk.com/price/”Explaining The Math Behind Bitcoin.” CCN, 18 Oct. 2014, www.ccn.com/explaining-the-math-behind-bitcoin/.”How to Use Math to Gain Success in Stock Trading.” Finance – Zacks, finance.zacks.com/use-math-gain-success-stock-trading-11693.html.Mathematics in the Financial Markets, nrich.maths.org/1426.http://people.math.umass.edu/~gunnells/talks/crypt.pdfElliptic Cryptography, plus.maths.org/content/elliptic-cryptography.”3 Simple Math Equations All Investors Should Know | Money.” Time, Time, time.com/money/3735703/investing-basic-math-equations/.https://web.wpi.edu/Pubs/E-project/Available/E-project-030415-150803/unrestricted/GrottonVeilleuxIQP.pdf Image: Bitcoin price since inception, note the spike since 2017.