2504.18982v1

# On Bitcoin Price Prediction On Bitcoin Price Prediction Grégory Bournassenko gregory.bournassenko@etu.u-paris.fr Université Paris Cité In recent years, cryptocurrencies have attracted growing attention from both private investors and institutions. Among them, Bitcoin stands out for its impressive volatility and widespread influence. This paper explores the predictability of Bitcoin’s price movements, drawing a parallel with traditional financial markets. We examine whether the cryptocurrency market operates under the efficient market hypothesis (EMH) or if inefficiencies still allow opportunities for arbitrage. Our methodology combines theoretical reviews, empirical analyses, machine learning approaches, and time series modeling to assess the extent to which Bitcoin’s price can be predicted. We find that while, in general, the Bitcoin market tends toward efficiency, specific conditions, including information asymmetries and behavioral anomalies, occasionally create exploitable inefficiencies. However, these opportunities remain difficult to systematically identify and leverage. Our findings have implications for both investors and policymakers, particularly regarding the regulation of cryptocurrency brokers and derivatives markets. Contents 1. 1 Introduction 1. 2 The Cryptocurrency Market is Efficient 1. 2.1 Eugene Fama and the Notion of No Arbitrage Opportunities 1. 2.1.1 Efficient Market Hypothesis Adaptation to Cryptocurrencies 1. 2.1.2 Random Walk and Martingale 1. 2.1.3 Cryptocurrencies and Fundamental Value 1. 2.2 From Louis Bachelier to Contemporary Models 1. 2.2.1 Modeling of Traditional Finance 1. 2.2.2 Modeling Crypto-Finance 1. 2.3 Time Series Studies and Analyses 1. 2.3.1 Fundamental Analysis 1. 2.3.2 Chartist / Technical Analysis 1. 2.3.3 Machine Learning 1. 3 The Cryptocurrency Market is Inefficient 1. 3.1 Robert Shiller and the Notion of an Inefficient Market in Terms of Arbitrage 1. 3.1.1 Volatility and Expected Dividends 1. 3.1.2 Behavioral Finance and Market Anomalies 1. 3.1.3 Speculative Bubbles 1. 3.2 Informational Inefficiency 1. 3.2.1 Market Manipulation 1. 3.2.2 Pump & Dump 1. 3.2.3 Natural Language Processing 1. 3.3 Operational Inefficiency 1. 3.3.1 At the Macroscopic Scale 1. 3.3.2 At the Mesoscopic Scale 1. 3.3.3 At the Microscopic Scale 1. 4 Conclusion 1. A isRandomBetter( $\Omega,n,k$ ) 1. B isSMABetter( $\Omega,n,r$ ) 1. C getHoldReturn(asset) 1. D getSMAReturn(asset, n, r) 1. E getRandomReturn(asset) 1. F getRandomPerc( $\Omega$ ) 1. G getAverageAccuracy( $\Omega,n$ ) 1. H NLP Trading Bot List of Figures 1. 1 Introduction 1. 3 $\blacktriangle 9,000\%$ BTC/USD [2014-2022] 1. 2.1 Eugene Fama and the Notion of No Arbitrage Opportunities 1. 6 $\blacktriangle 14,000\%$ DOGE/USD [01/2021-05/2021] 1. 2.1.1 Efficient Market Hypothesis Adaptation to Cryptocurrencies 1. 9 $\blacktriangle 150\%$ BTC/USD [13/06/2017-01/09/2017] 1. 10 $\blacktriangledown 15\%$ BTC/USD [16/01/2018-17/01/2018] 1. 11 $\blacktriangledown 22\%$ BTC/USD [14/04/2021-25/04/2021] 1. 12 Correlation between BTC/USD, GOLD/USD, and S&P500 1. 13 S&P500 over the period available with BTC/USD 1. 14 GOLD/USD over the period available with BTC/USD 1. 2.1.3 Cryptocurrencies and Fundamental Value 1. 2.2.1 Modeling of Traditional Finance 1. 2.2.1 Modeling of Traditional Finance 1. 21 RSI Signals for BTC/USD 1. 22 SAR Signals for BTC/USD 1. 2.3.3 Machine Learning 1. 2.3.3 Machine Learning 1. 27 Results of the ARIMA model 1. 28 Evolution of the S&P500 and dividends 1. 29 Results from the NLP trading bot List of Tables 1. 1 Results of isRandomBetter( $\Omega,n,k$ ) 1. 2 Results of isSMABetter( $\Omega,n,r$ ) 1 Introduction The price of Bitcoin has lost almost 50% of its value since last November, almost as much as Orpea’s stock value after its scandal. In Orpea’s case, the correlation is clear with the scandal, but for Bitcoin, such irrational volatility is rather usual. <details> <summary>extracted/6391907/images/btc-new.png Details</summary>  ### Visual Description ## Candlestick Chart: Stock Price Movement (Oct 2021 - Apr 2022) ### Overview The image displays a candlestick chart tracking stock price fluctuations over eight months. The chart shows significant volatility, with a peak in late 2021 followed by a sharp decline and partial recovery. Green candlesticks represent upward price movement, while red candlesticks indicate downward trends. ### Components/Axes - **X-axis (Horizontal)**: Timeframe labeled with months (Oct 2021 to Apr 2022), divided into approximate weekly intervals. - **Y-axis (Vertical)**: Price scale ranging from 30,000 to 70,000, marked in 5,000 increments. - **Legend**: Located in the top-right corner, using green (#228B22) for price increases and red (#B22222) for decreases. - **Gridlines**: Subtle gray lines at 5,000-unit intervals for reference. ### Detailed Analysis 1. **Initial Uptrend (Oct-Nov 2021)**: - Prices rose from ~48,000 to ~68,000. - Notable peaks: ~65,000 (Oct 15), ~67,000 (Oct 29), ~68,000 (Nov 12). - Sharp correction in late November to ~60,000. 2. **December 2021 Crash**: - Steep decline from ~68,000 to ~45,000. - Largest single drop: ~13,000 between Nov 26 and Dec 3. - Bottomed at ~45,000 by Dec 24. 3. **January 2022 Recovery**: - Gradual climb to ~50,000 by Jan 15. - Volatile mid-January: ~48,000 to ~52,000. - Sharp drop to ~42,000 by Jan 29. 4. **February 2022**: - Recovery to ~48,000 by Feb 12. - Sharp decline to ~35,000 by Feb 26. - Final rebound to ~40,000 by Mar 5. 5. **March-April 2022**: - Sideways movement between ~38,000 and ~45,000. - Notable volatility: ~42,000 to ~46,000 in late March. - Ended at ~39,000 by Apr 30. ### Key Observations - **Peak-to-Trough Decline**: ~23,000 drop from Nov 12 peak to Dec 24 low. - **Recovery Phase**: ~13,000 gain from Dec 24 to Mar 5. - **Volatility Clusters**: - High volatility in late November/early December. - Secondary volatility in late January/February. - **Price Range**: ~35,000 (lowest) to ~68,000 (highest). ### Interpretation The chart reveals a classic bear market pattern with: 1. **Bullish Phase (Oct-Nov 2021)**: Strong upward momentum followed by profit-taking. 2. **Bearish Phase (Dec 2021-Feb 2022)**: Sharp correction and prolonged consolidation. 3. **Recovery Attempt (Mar-Apr 2022)**: Limited rebound with persistent selling pressure. The data suggests external factors (e.g., macroeconomic news, sector-specific events) likely triggered the December crash. The subsequent recovery indicates resilience but fails to reclaim pre-crash highs, implying structural weakness or ongoing market uncertainty. The April 2022 price (~39,000) represents a 20% drawdown from the November peak, signaling potential long-term bearish sentiment. </details> Figure 1: $\blacktriangledown 50\%$ BTC/USD [11/2021-02/2022] <details> <summary>extracted/6391907/images/orpea-new.png Details</summary>  ### Visual Description ## Candlestick Chart: Stock Price Movement (Dec 2021 - Apr 2022) ### Overview The image displays a candlestick chart tracking stock price movements over six months (December 2021 to April 2022). The chart shows daily price fluctuations with green candlesticks indicating upward movement (closing price > opening price) and red candlesticks indicating downward movement (closing price < opening price). The y-axis represents price in increments of 20, while the x-axis marks time in monthly intervals. ### Components/Axes - **Y-Axis (Price):** Labeled "Price" with gridlines at 20-unit intervals (0, 20, 40, 60, 80, 100, 120). No explicit units (e.g., USD) are provided. - **X-Axis (Time):** Labeled with months (Dec 2021, Jan 2022, Feb 2022, Mar 2022, Apr 2022). Days are not explicitly labeled but inferred from candlestick spacing. - **Legend:** Located on the right side of the chart. Green = upward movement (closing price > opening price), Red = downward movement (closing price < opening price). ### Detailed Analysis 1. **December 2021:** - Price starts near 90, fluctuates between 80–90, and closes at ~85. - Green candlesticks dominate early December, followed by red candlesticks toward month-end. 2. **January 2022:** - Price opens at ~85, peaks at ~90, then drops sharply to ~80. - Mixed green/red candlesticks, with a notable red candlestick near month-end. 3. **February 2022:** - Sharp decline from ~80 to ~60, followed by a partial recovery to ~50. - Large red candlestick in mid-February (price drop from ~80 to ~60). - Green candlesticks appear in late February, but price remains below 60. 4. **March 2022:** - Price stabilizes between ~35–40, with minor fluctuations. - Alternating green/red candlesticks, ending at ~35. 5. **April 2022:** - Price rises to ~40, then declines to ~30. - Green candlesticks in early April, followed by red candlesticks toward month-end. ### Key Observations - **Overall Trend:** Bearish trajectory from December 2021 (starting ~90) to April 2022 (ending ~30), a ~60-point decline. - **Volatility Peaks:** - Sharp drop in February 2022 (~80 → ~60). - Significant rebound in late February (~60 → ~50). - **Price Range:** - High: ~90 (Dec 2021). - Low: ~30 (Apr 2022). - **Candlestick Patterns:** - Long red candlesticks in February indicate panic selling. - Smaller green/red candlesticks in March/April suggest consolidation. ### Interpretation The chart demonstrates a prolonged bear market with a 67% price decline over six months. The February 2022 crash (80 → 60) suggests a critical event (e.g., earnings report, macroeconomic shock) triggered a sell-off. The subsequent recovery in late February and March/April volatility may reflect short-term trading activity or sector-specific news. The lack of sustained recovery below the 50 mark indicates weak buyer confidence. Investors might interpret this as a high-risk asset with declining fundamentals or external market pressures. The absence of volume data limits analysis of institutional participation. </details> Figure 2: $\blacktriangledown 50\%$ ORP [01/2022-03/2022] The notion of prediction is vague, especially regarding price prediction: isn’t price itself the result of agents’ predictions about the value of an asset? Are we therefore predicting a prediction? For simplicity, we will use the term prediction as defined by American economist Alfred Cowles in his paper [Cowles 3rd, 1933], particularly in the second part, where he analyzes the reliability of "forecasters" on stock market volatility. Bitcoin, for its part, is a decentralized cryptocurrency, created in 2008, based on a "proof of work" mining protocol and a robust transaction system as explained by Satoshi Nakamoto [Nakamoto, 2008]. <details> <summary>extracted/6391907/images/btc-new2.png Details</summary>  ### Visual Description ## Line Chart: Dual Time Series Analysis (2015–2022) ### Overview The image depicts a line chart with two overlapping data series (red and green lines) plotted against a grid background. The chart spans 8 years (2015–2022) on the x-axis and a value range of 0–70,000 on the y-axis. Both lines exhibit distinct trends, with significant divergence in 2021–2022. ### Components/Axes - **X-Axis**: Labeled "Year" with annual ticks (2015–2022). No gridlines extend beyond the axis. - **Y-Axis**: Labeled "Value" with increments of 10,000 (0, 10k, 20k, ..., 70k). Gridlines span the full chart. - **Legend**: Located in the bottom-right corner. Red line = "Line A"; Green line = "Line B". - **Grid**: Light gray horizontal and vertical lines at 10k intervals. ### Detailed Analysis - **Line A (Red)**: - **2015–2017**: Remains near 0 (≈0–500). - **2018**: Sharp rise to ~15,000, followed by volatility (10k–18k). - **2019–2020**: Stabilizes between 5k–10k, with minor fluctuations. - **2021**: Rapid ascent to ~60k, peaking at ~70k in 2022. - **Line B (Green)**: - **2015–2017**: Mirrors Line A’s flatline (≈0–500). - **2018**: Peaks at ~18,000, then declines to ~8k by 2019. - **2020**: Dips to ~4k, then rises sharply to ~65k in 2022. - **2021–2022**: Shows sharper volatility than Line A, with a peak of ~65k in 2022. ### Key Observations 1. **2015–2017**: Both lines are nearly flat, suggesting minimal activity or baseline values. 2. **2018–2019**: Both lines experience volatility, with Line B peaking higher than Line A. 3. **2020**: Line B dips below Line A, indicating a temporary divergence. 4. **2021–2022**: Both lines surge, but Line B grows faster, reaching 65k vs. Line A’s 70k. The gap narrows in 2022. ### Interpretation The chart likely represents economic, financial, or operational metrics (e.g., revenue, stock prices, or production output). The 2021–2022 surge aligns with post-pandemic recovery trends, suggesting external shocks (e.g., COVID-19, policy changes) drove the divergence. Line B’s sharper rise in 2022 may indicate sector-specific growth or delayed effects of interventions. The 2020 dip in Line B could reflect sector-specific vulnerabilities (e.g., travel, retail). The data underscores the importance of contextual factors in interpreting time-series trends. </details> Figure 3: $\blacktriangle 9,000\%$ BTC/USD [2014-2022] As shown above, Bitcoin has progressively gained success: initially used for anonymous transactions on illegal markets, it became a speculative tool for individuals, and eventually attracted institutional interest, despite limited daily usage [Baur et al., 2015]. Notably, Bitcoin’s underlying technology, Blockchain, was actually invented by researchers Haber and Stornetta [Haber and Stornetta, 1990], not Nakamoto, although Nakamoto was the first to apply it at large scale. The literature on cryptocurrency prediction remains relatively poor, given the recent emergence of the technology. Virtually no academic papers referenced cryptocurrencies before 2008. Instead, much research focuses on machine learning techniques for cryptocurrency prediction. However, similarities with financial markets exist (closer to forex than stocks due to the monetary nature of cryptocurrencies), a domain extensively studied since the early 1900s. From Louis Bachelier’s Gaussian model [Bachelier, 1900] to Mathieu Rosenbaum’s rough Heston model [Gatheral et al., 2018], and Gordon-Shapiro’s valuation model [Gordon and Shapiro, 1956], numerous theories have been proposed. Yet, debates persist regarding market behavior. According to Eugene Fama [Fama, 1970], a rational market cannot be systematically beaten. Louis Bachelier [Bachelier, 1900] states, "The determination of these activities depends on an infinite number of factors: therefore, a precise mathematical forecast is absolutely impossible." Nevertheless, Keynes [Keynes, 1937] compared the market to a beauty contest: predicting what the majority will find beautiful, not objective beauty itself. This idea echoes momentum strategies and aligns with Charles Dow’s technical analysis [Brown et al., 1998]. Alternatively, Warren Buffett promotes stock-picking and value investing, diverging from Markowitz’s modern portfolio theory [Steinbach, 2001]. However, Buffett’s method, focusing on selecting promising assets, differs from our study, where the asset (Bitcoin) is preselected. Burton Malkiel [Malkiel, 2003] famously claimed that "a blindfolded monkey throwing darts at a newspaper’s financial pages could perform as well as professional investors," although empirical studies [Pernagallo and Torrisi, 2020] challenge this assertion. To explore random versus selected portfolios, we define a Python function isRandomBetter( $\Omega,n,k$ ) (code in Appendix A). Results: | 1 | 141 | 998 | 10 | 10 | 20% | False | | --- | --- | --- | --- | --- | --- | --- | | 2 | 141 | 998 | 10 | 20 | 30% | False | | 3 | 141 | 998 | 20 | 10 | 40% | False | | 4 | 141 | 998 | 20 | 20 | 30% | False | | 5 | 141 | 998 | 20 | 30 | 60% | True | | 6 | 141 | 998 | 30 | 20 | 57% | True | | 7 | 141 | 998 | 30 | 30 | 47% | False | | 8 | 141 | 998 | 30 | 10 | 27% | False | | 9 | 141 | 998 | 10 | 30 | 20% | False | | 10 | 141 | 998 | 40 | 5 | 25% | False | Table 1: Results of isRandomBetter( $\Omega,n,k$ ) Choosing a random crypto portfolio in 2021 was not optimal. We will investigate whether Bitcoin price predictability depends on market efficiency. Given the cryptocurrency market’s heterogeneity, various scenarios (competitive markets, manipulated markets, rational/irrational agents) are expected. We will show that, by default, the crypto market tends to be efficient, although inefficiencies sometimes appear, albeit difficult to exploit systematically. We will address prediction methods under efficient market conditions, focusing on time series analysis and machine learning algorithms. We will also study prediction under inefficiency contexts, emphasizing empirical observations and stylized facts. Let’s first examine the case when the market is efficient. 2 The Cryptocurrency Market is Efficient We first assume an efficient market. We will explain the concept’s origins, assumptions, verify some of them, discuss model evolutions, and their implications for cryptocurrencies. We will also analyze this through machine learning and quantitative techniques, reflecting critically on the results. 2.1 Eugene Fama and the Notion of No Arbitrage Opportunities We start with Fama’s [Fama, 1970] definition of efficient markets, comparing the US stock market and cryptocurrencies. Fama’s idea implies no arbitrage opportunities. However, as we will see later, arbitrage is relatively common in crypto markets (price differences between brokers). <details> <summary>extracted/6391907/images/arb1.png Details</summary>  ### Visual Description ## Screenshot: KoinKnight Website Interface ### Overview The image depicts the homepage of KoinKnight, a cryptocurrency arbitrage platform. The layout combines a clean white background with a central graphic illustration and text-based content. Key elements include a navigation menu, promotional text, call-to-action buttons, and a footer. ### Components/Axes 1. **Header**: - **Logo**: "KoinKnight" with a stylized "K" icon (blue and dark blue). - **Navigation Menu** (top-right): - Pricing - API Services - Crypto Analytics - Refer & Earn - Language dropdown (default: English) - **Action Links** (top-right): - "Sign in" (blue text) - "Sign up" (blue button) 2. **Main Content**: - **Headline**: "Your personal assistance for cryptocurrency arbitrage" (text split into two lines: "Your personal assistance" in blue, "for cryptocurrency arbitrage" in black). - **Subheading**: "Find the best trade and arbitrage opportunities using KoinKnight's powerful algorithm and real-time data exploration tools." - **Buttons**: - "Try for free" (white button with blue text) - "View Demo" (blue button with white text) - **Login Prompt**: "Already using KoinKnight? Log in" (underlined "Log in" text). 3. **Graphic**: - A stylized illustration of a laptop displaying multiple charts/tables, a calculator, and a yellow coin icon on a green background. 4. **Footer**: - Copyright notice: "© 2023 KoinKnight. All rights reserved." ### Detailed Analysis - **Textual Content**: - All text is in English. - No numerical data, charts, or tables are present. - Buttons and links use consistent color coding (blue for primary actions, black for secondary text). - **Spatial Grounding**: - Logo and navigation are positioned at the top-left and top-right, respectively. - Main content is centered, with the graphic illustration to the right. - Footer spans the bottom of the page. ### Key Observations - The design emphasizes simplicity and clarity, targeting users interested in cryptocurrency arbitrage. - The graphic reinforces the platform's focus on data analysis and financial tools. - No interactive elements (e.g., dropdowns, modals) are visible in the static image. ### Interpretation KoinKnight positions itself as a user-friendly tool for cryptocurrency traders, leveraging algorithmic analysis and real-time data. The absence of complex visualizations in the screenshot suggests the platform prioritizes accessibility over advanced technical interfaces. The "Refer & Earn" feature hints at a growth strategy focused on user acquisition through incentives. The green color scheme and coin icon align with financial technology branding conventions. </details> Figure 4: KoinKnight <details> <summary>extracted/6391907/images/arb2.png Details</summary>  ### Visual Description ## Screenshot: ArbiTool Website Interface ### Overview The image depicts the homepage of a website named "ArbiTool," which focuses on cryptocurrency arbitrage. The design uses a vibrant purple background with abstract geometric shapes. Key elements include a navigation menu, promotional text, interactive buttons, and an illustrative diagram of digital tools and collaboration. ### Components/Axes - **Header**: - Logo: "AT ArbiTool Professional Arbitrage" (top-left). - Navigation Menu (horizontal, top-center): - HOME - ABOUT ARBITOOL (dropdown) - TUTORIAL - PRICING - ARBITRAGE COURSE - JOIN OUR COMMUNITY (dropdown) - FAQ'S - CONTACT - Language Selector: UK flag icon (top-right). - Action Buttons (top-right): - LOGIN (white text on purple). - SIGN UP FREE (gradient pink-to-orange button). - **Main Content**: - Headline (left-center, large white text): - "Did you know that the rate of the same cryptocurrency may vary by up to 50% on two different exchanges?" - Subheading (smaller white text): - "Our tool will show you where and when to buy LOW and sell HIGH." - Call-to-Action Buttons (below headline): - "TELL ME MORE!" (outlined in yellow, white text). - "TEST IT FOR FREE" (gradient pink-to-orange, white text). - Token Trading Section (bottom-center, dark blue gradient): - Text: "Trade our token on:" - Logo: "altilly" (white background with a dark blue spiral icon). - **Footer**: - Live Chat Button (bottom-right, dark gray box): - Text: "We are here! Live chat now." - Feedback Button (below live chat, red box): - Text: "Laissez un message" (French, translates to "Leave a message"). - **Illustration**: - Right-side diagram showing: - Laptop displaying Bitcoin (₿) and ArbiTool interface. - Server rack with green light. - Tablet with shield icon (cybersecurity). - Desktop monitor showing graphs. - Group of people in a meeting (symbolizing collaboration). ### Detailed Analysis - **Textual Content**: - All text is in English except for the French phrase "Laissez un message" in the footer. - Key phrases emphasize cryptocurrency arbitrage opportunities ("buy LOW and sell HIGH") and the tool's utility. - Buttons use contrasting colors (yellow, pink-orange) to draw attention to actions. - **Visual Elements**: - The illustration uses flat design with minimalistic icons (e.g., shield, server rack). - Color scheme: Purple background with orange, pink, and blue accents for buttons and icons. ### Key Observations - The website prioritizes user engagement through bold headlines and actionable buttons. - The illustration reinforces the theme of digital collaboration and arbitrage opportunities. - The mention of "50% variation" highlights the core value proposition of the tool. ### Interpretation The website positions itself as a solution for cryptocurrency arbitrage, targeting users interested in maximizing profits by exploiting price discrepancies across exchanges. The use of a live chat and feedback button suggests a focus on customer support and user interaction. The partnership with Altilly (token trading platform) indicates integration with broader financial services. The French text implies multilingual support or a target audience in French-speaking regions. </details> Figure 5: ArbiTool At a discretionary level, however, arbitrage opportunities are rarely exploitable due to transfer fees and liquidity issues. 2.1.1 Efficient Market Hypothesis Adaptation to Cryptocurrencies Fama [Fama, 1970] outlined several conditions for market efficiency and its three forms. Let’s check them for crypto markets. First, agents should be rational. In crypto, this is unlikely. For example, Dogecoin rose by 14,000% mainly due to memes and social media [Chohan, 2021]: <details> <summary>extracted/6391907/images/doge-new.png Details</summary>  ### Visual Description ## Line Chart: Metric Value Over Time ### Overview The chart displays two time-series data sets (red and green lines) tracking a metric value from July 2020 to April 2022. The y-axis represents "Metric Value" (0–0.7), and the x-axis shows dates. A vertical inset box highlights activity in April 2021. The red line ("Series A") shows extreme volatility, while the green line ("Series B") exhibits gradual growth followed by decline. ### Components/Axes - **Y-Axis**: Labeled "Metric Value" with increments of 0.1 from 0.0 to 0.7. - **X-Axis**: Dates from July 2020 to April 2022, with monthly labels (Jul 2020, Oct 2020, Jan 2021, Apr 2021, Jul 2021, Oct 2021, Jan 2022, Apr 2022). - **Legend**: Located on the right, outside the chart. Red = "Series A", Green = "Series B". - **Inset Box**: A zoomed-in view of April 2021 activity, positioned above the main chart. ### Detailed Analysis - **Series A (Red Line)**: - **July 2020–January 2021**: Flat at ~0.01. - **April 2021**: Spikes to ~0.7 (peak), then declines to ~0.2 by July 2021. - **July 2021–April 2022**: Fluctuates between ~0.15–0.3, with a notable dip to ~0.1 in January 2022. - **Series B (Green Line)**: - **July 2020–January 2021**: Flat at ~0.01. - **April 2021**: Rises to ~0.4, then declines to ~0.2 by July 2021. - **July 2021–April 2022**: Gradual decline to ~0.12, with minor fluctuations. ### Key Observations 1. **April 2021 Anomaly**: Both series experience a sharp increase in April 2021, with Series A reaching ~0.7 and Series B ~0.4. This suggests a shared external driver (e.g., market event, policy change). 2. **Post-April 2021 Decline**: Both series show sustained declines after April 2021, though Series A remains more volatile. 3. **Inset Focus**: The April 2021 inset emphasizes the magnitude of the spike, with Series A’s red line showing erratic oscillations (e.g., ~0.65–0.75 range) and Series B’s green line peaking at ~0.45. ### Interpretation - **Causal Relationship**: The synchronized spike in April 2021 implies a common cause (e.g., economic stimulus, regulatory shift). The divergent post-spike trajectories suggest differing sensitivities to subsequent market conditions. - **Volatility vs. Stability**: Series A’s extreme fluctuations (e.g., ~0.7 → ~0.15 in 12 months) contrast with Series B’s smoother decline, indicating differing risk profiles or sector exposures. - **Uncertainty**: Exact values are approximate due to overlapping data points and lack of gridlines for precise interpolation. For example, Series A’s April 2021 peak is estimated at ~0.7 but could range between 0.65–0.75 based on the inset’s granularity. ### Spatial Grounding - **Legend**: Positioned on the right, outside the chart area. - **Inset Box**: Located above the main chart, centered over the April 2021 timeframe. - **Axes**: Y-axis on the left, X-axis at the bottom. ### Trend Verification - **Series A**: Slopes upward sharply in April 2021, then declines with oscillations. - **Series B**: Gradual upward slope until April 2021, followed by a steady decline. ### Component Isolation 1. **Main Chart**: Dominates the image, showing long-term trends. 2. **Inset Box**: Focuses on April 2021 activity, providing granular detail. 3. **Legend**: Separated from the chart to avoid visual clutter. ### Content Details - **Data Points**: - Series A: Peaks at ~0.7 (April 2021), troughs at ~0.01 (July 2020–January 2021). - Series B: Peaks at ~0.4 (April 2021), troughs at ~0.01 (July 2020–January 2021). - **Axis Ranges**: - Y-axis: 0.0–0.7 (increments of 0.1). - X-axis: July 2020–April 2022 (monthly intervals). ### Notable Outliers - **April 2021 Spike**: Both series deviate sharply from prior flat trends, suggesting an exogenous event. - **Series A’s Volatility**: Post-April 2021 oscillations (e.g., ~0.3 → ~0.15 in 6 months) indicate instability. ### Final Notes The chart highlights a critical inflection point in April 2021, with both series reacting similarly to an unspecified catalyst. The subsequent divergence in behavior underscores the need for further analysis of external factors (e.g., policy changes, market shocks) to explain the differing trajectories. </details> Figure 6: $\blacktriangle 14,000\%$ DOGE/USD [01/2021-05/2021] Individuals should not influence the market. Elon Musk, however, can shift prices with a single tweet: <details> <summary>extracted/6391907/images/musk1.png Details</summary>  ### Visual Description ## Screenshot: Twitter Post by Elon Musk ### Overview This image is a screenshot of a Twitter post by Elon Musk (@elonmusk), featuring a meme with text and an image of a couple sitting apart on a couch. The post includes hashtags, emojis, and engagement metrics. ### Components/Axes - **Text Content**: - Tweet text: - "Her: I know I said it would be over between us if you quoted another Linkin Park song but I've found someone else." - "Him: So in the end it didn't even matter?" - Hashtags: `#Bitcoin` (blue text), `#BTC` (yellow Bitcoin emoji), `#BrokenHeart` (red broken heart emoji). - Image watermark: "made with mematic" (bottom-left corner). - Timestamp: "3:07 AM · 4 juin 2021 · Twitter for iPhone" (translated to "June 4, 2021"). - **Image**: - A meme image of a man and woman sitting on a gray couch, facing away from each other. - The woman (left) wears a white top and black leggings; the man (right) wears a striped shirt and gray pants. - Background: Neutral room with white walls and a window. - **Engagement Metrics**: - Retweets: 21.1k - Tweets with citations: 9,986 - Likes: 210.1k ### Detailed Analysis - **Textual Elements**: - The tweet references a meme format where a couple’s breakup is humorously tied to a Linkin Park song lyric. - The hashtags `#Bitcoin` and `#BTC` suggest a connection to cryptocurrency, while `#BrokenHeart` implies emotional context. - The French phrase "j'aime" (translated as "I love") appears in the engagement metrics, likely part of Twitter’s interface. - **Image Details**: - The meme’s dialogue mirrors the lyrics of Linkin Park’s "In the End," creating a meta-humor about relationships and music. - The couple’s body language (crossed arms, physical distance) reinforces the theme of emotional detachment. ### Key Observations - The tweet’s engagement metrics are exceptionally high, indicating viral spread. - The juxtaposition of a breakup meme with Bitcoin-related hashtags may reflect Elon Musk’s known interest in cryptocurrency and his tendency to blend humor with financial topics. - The use of "j'aime" in the metrics suggests the post was viewed in a French-speaking region or by users with localized settings. ### Interpretation The post leverages meme culture to comment on relationships, possibly as a metaphor for the volatility of Bitcoin or Musk’s own public persona. The high engagement underscores the intersection of pop culture, technology, and finance in social media discourse. The meme’s reference to Linkin Park’s lyrics adds a layer of irony, as the song’s themes of futility and regret align with the breakup narrative. The inclusion of Bitcoin hashtags may hint at Musk’s influence on crypto markets or his playful approach to branding. </details> Figure 7: Negative tweet on 04/06/2021 <details> <summary>extracted/6391907/images/btc-new3.png Details</summary>  ### Visual Description ## Candlestick Chart: Price Movement Analysis (May 26 - June 17, 2021) ### Overview The image displays a candlestick chart tracking price movements over a 22-day period (May 26 to June 17, 2021). The chart uses green and red candlesticks to represent upward and downward price trends, respectively. The y-axis shows price values in thousands (25k–45k), while the x-axis marks specific dates. The chart exhibits significant volatility, with alternating periods of price increases and declines. --- ### Components/Axes - **X-Axis (Horizontal)**: - Labeled with dates: May 26, May 29, June 1, June 4, June 7, June 10, June 13, June 16, June 17. - Dates are spaced approximately every 3–4 days. - **Y-Axis (Vertical)**: - Labeled "Price" with grid lines at 25k, 30k, 35k, 40k, and 45k. - Values are approximate, with no explicit numerical labels on the axis. - **Legend**: - Located on the right side of the chart. - Green: Upward price trend (closing price > opening price). - Red: Downward price trend (closing price < opening price). --- ### Detailed Analysis #### Candlestick Structure - Each candlestick represents a trading day, with: - **Body**: The range between the opening and closing prices. - **Wicks**: The high (top) and low (bottom) prices of the day. - **Key Observations**: - **May 26–May 29**: Initial downward trend (red candlesticks), with prices dropping from ~40k to ~35k. - **May 30–June 1**: Slight recovery (green candlesticks), stabilizing around 35k–37k. - **June 4–June 7**: Sharp decline (red candlesticks), reaching a low of ~32k. - **June 8–June 13**: Volatile recovery (mixed green/red), peaking at ~40k on June 14. - **June 14–June 17**: Sustained upward trend (green candlesticks), closing near 38k–40k. #### Notable Data Points - **Highest Price**: ~42k (June 14, green candlestick). - **Lowest Price**: ~32k (June 8, red candlestick). - **Most Volatile Day**: June 7–8, with a ~10k drop. - **Longest Green Candlestick**: June 14, indicating strong buying pressure. - **Longest Red Candlestick**: June 7, reflecting significant selling pressure. --- ### Key Observations 1. **Volatility**: The chart shows frequent price swings, with no clear sustained trend until late June. 2. **Recovery Phase**: After mid-June, prices rebounded sharply, suggesting a shift in market sentiment. 3. **Uncertainty**: The alternating green/red candlesticks indicate indecision among traders during the middle period (June 4–13). 4. **Closing Trend**: The final days (June 14–17) show a consistent upward trend, closing near the 40k mark. --- ### Interpretation The chart reflects a market characterized by **short-term volatility** and **long-term recovery**. The initial decline (May 26–29) may indicate negative sentiment or external shocks, while the subsequent rebound (June 8–17) suggests renewed investor confidence. The sharp drop on June 7–8 could signal a market correction or panic selling, followed by a recovery phase. The final upward trend (June 14–17) implies a potential breakout or consolidation phase, with prices stabilizing near the 40k mark. **Critical Insight**: The chart highlights the importance of monitoring both short-term fluctuations and long-term trends. The recovery after mid-June may indicate underlying strength in the asset, but the volatility underscores the need for caution in decision-making. </details> Figure 8: Observed correlation: $\blacktriangledown 15\%$ BTC/USD [04/06/2021-08/06/2021] No information asymmetry should exist. Yet, insider knowledge (e.g., hacks) creates advantages [Biais et al., 2020]. Information should be free. For crypto, public data is widely available, though high-frequency trading data is costly [Grossman and Stiglitz, 1976]. Taxes should be low. Given international diversity, this varies. Regarding efficiency forms: Strong form: all public and private info is priced. However, events like Binance’s launch in 2017 or the Bitconnect scandal in 2018 show that insiders could have benefited: <details> <summary>extracted/6391907/images/btc-new4.png Details</summary>  ### Visual Description ## Candlestick Chart: Price Movement Over Time (June 11 - October 1, 2017) ### Overview The image displays a candlestick chart tracking price fluctuations over 126 trading days. The chart shows significant volatility, with prices oscillating between ~1,800 and ~5,800 units. Green candlesticks indicate upward price movement (closing higher than opening), while red candlesticks show downward movement. The chart contains 126 candlesticks, representing daily price action. ### Components/Axes - **X-axis**: Dates from June 11, 2017 (left) to October 1, 2017 (right), with labels every 14 days (Jun 11, Jun 25, Jul 9, Jul 23, Aug 6, Aug 20, Sep 3, Sep 17, Oct 1) - **Y-axis**: Price scale from 1,000 to 7,000 in 1,000-unit increments - **Legend**: Located at bottom-right corner - Green: Upward price movement (closing > opening) - Red: Downward price movement (closing < opening) - **Grid**: Light gray grid lines with darker axes ### Detailed Analysis 1. **Initial Phase (June 11 - July 15, 2017)**: - Price starts at ~2,800 - Gradual decline to ~2,200 by July 9 - Sharp drop to ~1,800 on July 15 (lowest point) - Recovery to ~2,400 by July 23 2. **Mid-Phase (July 23 - August 20, 2017)**: - Steady climb from ~2,400 to ~4,200 - Notable volatility: 10%+ daily swings observed - Peak at ~4,500 on August 20 3. **Late Phase (August 20 - October 1, 2017)**: - Sharp correction from ~4,500 to ~3,000 by September 15 - Recovery phase: - 30% gain from September 15 to September 29 (~3,000 → ~4,000) - Final surge to ~5,800 on October 1 (highest point) - Ends with 3 consecutive green candlesticks ### Key Observations - **Volatility Clusters**: - Highest volatility between July 15-23 (15%+ daily moves) - Sharpest single-day drop: July 15 (-22%) - Strongest single-day gain: October 1 (+12%) - **Pattern Recognition**: - "M" shaped pattern in August-September (rally-correction-rally) - Bullish engulfing pattern observed in final 3 days - **Volume Implication**: - Larger candlesticks correlate with increased price movement magnitude - Red candlesticks dominate during correction phases ### Interpretation The chart demonstrates a classic market cycle with: 1. **Accumulation Phase** (June-July): Institutional buying during price dips 2. **Markup Phase** (August): Momentum-driven rally to new highs 3. **Distribution Phase** (September): Profit-taking and consolidation 4. **Final Accumulation** (October 1): Late-stage buying pressure The 100%+ total return from June 11 (~2,800) to October 1 (~5,800) suggests strong bullish momentum despite mid-term volatility. The chart's "W" pattern (double bottom) around September 15 indicates potential for institutional accumulation. The absence of bearish reversals in the final days suggests positive market sentiment heading into October. **Note**: No explicit volume data is visible in this chart. All price interpretations assume standard candlestick conventions (open/close/high/low). The chart lacks annotations for fundamental events that may have influenced price action. </details> Figure 9: $\blacktriangle 150\%$ BTC/USD [13/06/2017-01/09/2017] <details> <summary>extracted/6391907/images/btc-new5.png Details</summary>  ### Visual Description ## Candlestick Chart: Price Movement Analysis (Dec 2017–Mar 2018) ### Overview The image displays a candlestick chart tracking price fluctuations over time, with green candlesticks representing upward price movements and red candlesticks indicating downward trends. The chart spans from December 2017 to March 2018, with price values ranging from ~4,000 to ~22,000 units (likely currency or stock index points). --- ### Components/Axes - **X-Axis (Horizontal)**: - Labeled with dates: - December 10, 2017 (leftmost) - December 24, 2017 (midpoint) - January 7, 2018 (right of midpoint) - January 21, 2018 (center-right) - February 4, 2018 (far right) - Tick marks at ~1-week intervals. - **Y-Axis (Vertical)**: - Labeled with price values in increments of 2,000: - 4,000 (bottom) - 6,000 - 8,000 - 10,000 - 12,000 - 14,000 - 16,000 - 18,000 - 20,000 - 22,000 (top). - **Legend**: - Located on the right side of the chart. - Green: Upward price movement (closing price > opening price). - Red: Downward price movement (closing price < opening price). --- ### Detailed Analysis 1. **Price Trends**: - **December 2017**: - Initial upward trend from ~11,000 to ~19,500 (peak ~Dec 15). - Sharp decline to ~13,000 by December 24. - **January 2018**: - Volatile period with a high of ~17,500 (Jan 7) and a low of ~9,500 (Jan 15). - Gradual recovery to ~12,000 by January 21. - **February–March 2018**: - Sideways consolidation between ~8,000 and ~12,000. - Final upward movement to ~11,500 by March 4. 2. **Key Data Points**: - **Highest Peak**: ~19,500 (Dec 15, 2017). - **Lowest Trough**: ~9,500 (Jan 15, 2018). - **Final Value**: ~11,500 (Mar 4, 2018). 3. **Candlestick Patterns**: - Large green candlesticks dominate early December, indicating strong buying pressure. - Red candlesticks with long lower shadows appear in mid-January, suggesting panic selling. - Smaller green/red candlesticks in February–March reflect indecision and consolidation. --- ### Key Observations - **Bearish Phase (Dec 15–Jan 15)**: - A ~50% drop from the December peak to the January trough. - Sharpest decline occurs between December 24 (~13,000) and January 15 (~9,500). - **Bullish Reversal (Jan 15–Mar 4)**: - Recovery of ~20% from the January low to the March value. - No single candlestick exceeds ~2,000 in size during this period, indicating muted momentum. - **Volatility Clustering**: - High volatility in December and January, followed by reduced activity in February–March. --- ### Interpretation The chart suggests a classic **bearish-to-bullish reversal** pattern. The December–January decline may reflect external factors (e.g., market corrections, economic news), while the February–March recovery hints at renewed investor confidence. The lack of strong upward momentum in March raises questions about sustainability. - **Notable Anomalies**: - The January 7 candlestick (~17,500) stands out as a false peak, followed by a steep drop. - The March 4 closing value (~11,500) is 25% below the December high, suggesting unresolved bearish sentiment. - **Implications**: - The chart could represent a stock, cryptocurrency, or commodity index. - The absence of volume data limits conclusions about liquidity or institutional activity. - The pattern aligns with a "head and shoulders" formation, though incomplete without additional data. --- **Note**: All values are approximate, derived from visual estimation of candlestick heights and axis labels. Uncertainty arises from the lack of gridlines and precise numerical annotations. </details> Figure 10: $\blacktriangledown 15\%$ BTC/USD [16/01/2018-17/01/2018] Semi-strong form: all public info is priced. The crypto market reacts quickly to news, as seen with Coinbase’s NASDAQ listing: <details> <summary>extracted/6391907/images/btc-new6.png Details</summary>  ### Visual Description ## Candlestick Chart: Price Movement Analysis (Mar 14 - May 23, 2021) ### Overview The image displays a candlestick chart tracking price movements over time, with dates on the x-axis (March 14 to May 23, 2021) and price values on the y-axis (50k to 65k). Green candlesticks represent upward price movements, while red candlesticks indicate declines. The chart shows significant volatility with multiple peaks, troughs, and trend reversals. ### Components/Axes - **X-Axis**: Dates (Mar 14, Mar 15, ..., May 23) spaced at daily intervals. - **Y-Axis**: Price values labeled in increments of 5k (50k, 55k, 60k, 65k). - **Legend**: Implied by candlestick colors: - **Green**: Price increased (close > open). - **Red**: Price decreased (close < open). - **Gridlines**: Horizontal lines at 5k intervals for reference. ### Detailed Analysis 1. **Early March (Mar 14-18)**: - Price starts at ~52k, rises to ~61k by Mar 15 (green candles). - Sharp decline to ~54k by Mar 18 (red candles). 2. **Mid-March (Mar 19-28)**: - Volatile range between ~54k and ~59k. - Notable green candle on Mar 23 (price closes at ~58k). 3. **Early April (Apr 11-15)**: - Sharp upward trend to ~64k (green candles). - Followed by a steep drop to ~55k by Apr 16 (red candles). 4. **Mid-April (Apr 16-25)**: - Price oscillates between ~50k and ~57k. - Significant red candle on Apr 25 (price closes at ~50k). 5. **Early May (May 9-15)**: - Price rises to ~58k (green candles) before dropping to ~52k (red candles). 6. **Late May (May 16-23)**: - Steep decline from ~52k to ~45k (red candles). - Final candlestick on May 23 closes at ~44k. ### Key Observations - **Peaks**: Highest price (~64k) on Apr 14; lowest price (~44k) on May 23. - **Volatility**: Multiple sharp reversals (e.g., Apr 14-15, Apr 25-26). - **Downtrend**: Sustained decline from Apr 16 to May 23, with no recovery above 58k. - **Outliers**: - Apr 14: Sharp spike to 64k (potential catalyst event). - May 15: Abrupt drop from 58k to 52k (possible market correction). ### Interpretation The chart suggests a bearish trend over the observed period, with price failing to sustain gains after initial rallies. The prolonged decline from Apr 16 to May 23 indicates strong selling pressure or negative market sentiment. The Apr 14 spike may reflect a temporary positive catalyst (e.g., earnings report, news event), but the subsequent drop suggests profit-taking or loss of confidence. The lack of recovery above 58k after May 9 implies a loss of momentum in upward movement. This pattern could signal a bear market phase or sector-specific downturn, warranting further analysis of external factors (e.g., economic data, company-specific news). </details> Figure 11: $\blacktriangledown 22\%$ BTC/USD [14/04/2021-25/04/2021] The day before its IPO, BTC/USD increased by almost 7%, before losing more than 20% ten days later. The weak form assumes that all historical price information is already reflected in the current price. This form challenges technical analysis, which specializes precisely in analyzing past returns. These analyses are widely shared on social media, due to their ease of implementation, and attract a (too?) proselytizing community. The idea is to use indicators mainly based on past fluctuations to make future predictions. Among the usual indicators (according to the TA-Lib library, considered a reference) are: RSI (Relative Strength Index), SMA (Simple Moving Average), BBANDS (Bollinger Bands). Let us check, for example, whether a "mean-reversion" strategy would be more effective than a simple "hold" (buy-sell only once) and more effective than a random strategy by backtesting these strategies on 2021. If not, we could conjecture that, over the entire year of 2021, it was useless to use a "mean-reversion" strategy (which assumes that when the current price is too "far" from the moving average (SMA), the price will return to its "mean")). This may also give us an indication about the market efficiency form. We will base our analysis on a set $\Omega$ of crypto-assets. For each element in $\Omega$ , we will test three strategies: mean-reversion, hold, and random. We assume short-selling is allowed. Let $P_{t}$ be the price at time $t$ , $M_{t}(n)$ the moving average at time $t$ with a window of $n$ days, $\omega_{i}$ the $i^{th}$ element of $\Omega$ , and $r∈[0,100]$ a percentage around $M_{t}(n)$ indicating the threshold at which we open/close a position. The mean-reversion strategy will be constructed as follows: if $P_{t}>M_{t}(n)+(\frac{M_{t}(n)× r}{100})$ , then sell $\omega_{i}$ at price $P_{t}$ ; if $P_{t}<M_{t}(n)-(\frac{M_{t}(n)× r}{100})$ , then buy $\omega_{i}$ at price $P_{t}$ , with $t$ ranging from [01/01/2021, 31/12/2021]. The hold strategy will be constructed as follows: if $t=01/01/2021$ , then buy $\omega_{i}$ at price $P_{t}$ ; if $t=31/12/2021$ , then sell $\omega_{i}$ at price $P_{t}$ . The random strategy will be constructed as follows: generate a signal $S∈[\text{buy, sell, hold}]$ with $P(S=\text{buy})=P(S=\text{sell})=P(S=\text{hold})=\frac{1}{3}$ . For each $\omega_{i}$ and for each $t$ , if $S=\text{"buy"}$ we buy $\omega_{i}$ at price $P_{t}$ , if $S=\text{"sell"}$ we sell $\omega_{i}$ at price $P_{t}$ , if $S=\text{"hold"}$ we do nothing. Thus, we create a Python function isSMABetter( $\Omega,n,r$ ) that takes as parameters $\Omega$ (the set of crypto-assets), $r$ (the percentage for the SMA thresholds), and $n$ (the window size in days for the SMA), and returns True if the average SMA returns of $\omega_{i}$ are greater than the average returns of the hold strategy and (strictly) the random strategy in at least 50% of the cases, and False otherwise. We only consider daily returns. Indeed, how could we backtest a strategy that only opens positions? We thus place ourselves in a short-term trading scale for each trade, which is consistent with the chartist approach (otherwise, we would prefer a passive investment strategy that requires almost no analysis). The results of isSMABetter( $\Omega,n,r$ ), whose code is in Appendix B, are as follows: | 116 | 1179 | -484 | -4 | 50 | 20 | 0.00 | False | | --- | --- | --- | --- | --- | --- | --- | --- | Table 2: Results of isSMABetter( $\Omega,n,r$ ) It appears that in 2021, among the 116 crypto-assets tested, it was more optimal to have a passive strategy or, at worst, a random strategy, rather than using the moving average in an attempt to generate profits with a day-trading approach (speculation aiming to make a profit within the same day of a market order execution), since the average return obtained with the SMA strategy was the lowest among the three (-484%), and strictly no crypto-asset (0%) showed any interest in being traded with an SMA strategy. We can conjecture that the cryptocurrency market efficiency form is at least weak, and possibly semi-strong, depending on the crypto-assets and periods, but hardly strong. 2.1.2 Random Walk and Martingale In almost all the literature ([Lardic and Mignon, 2006], [Jovanovic, 2009] …), a random walk is modeled by two elements: the previous observation and white noise. The literature explains that a price can be modeled as: $P_{t+1}=P_{t}+\varepsilon_{t+1}$ , with $\varepsilon=\{\varepsilon_{t},t∈ N\}$ being white noise. This implies that the best (and only) way to predict the price of an asset is by using its current price. We will perform a Dickey-Fuller test [Dickey and Fuller, 1979] on each element of a set of assets $\Omega$ with a significance level of $\alpha=5\%$ . We define a Python function getRandomPerc( $\Omega$ ) that takes as input a set of crypto-assets $\Omega$ and returns the percentage of assets in that set that appear to follow a random walk, that is, for which we do not reject the null hypothesis "the time series is non-stationary". The result of getRandomPerc( $\Omega$ ), whose code is provided in Appendix F, returns 69 %. It seems that more than half of the cryptocurrencies follow a random walk. There is often confusion between efficiency and random walk. Indeed, when reading the Wikipedia page on the efficient market hypothesis, one might think that an efficient market necessarily implies prices following a random walk. However, this is false. The market is not necessarily inefficient if prices do not follow a random walk because, as [Lardic and Mignon, 2006] states, "It suffices, for example, that the hypothesis of risk neutrality is not satisfied, or that individuals’ utility functions are not separable and additive [LeRoy, 1982], meaning that it is impossible to separate consumption and investment decisions." Many studies show that cryptocurrencies (most studies focus on Bitcoin) do not follow a random walk ([Palamalai et al., 2021], [Aggarwal, 2019] …). However, these studies mainly rely on the very restrictive assumption of autocorrelation, and conclude that the Bitcoin market is not efficient. Samuelson [Samuelson, 2016] already addressed this problem in his time and proposed a modification to the random walk hypothesis: the martingale model. This model is less restrictive than the random walk model because it imposes no condition on the autocorrelation of residuals. Very similar to the previous model, a price process $P_{t}$ follows a martingale if: $E[P_{t+1}|I_{t}]=P_{t}$ , where $P_{t}$ is the price at time $t$ and $I_{t}$ is the information set at time $t$ . Thus, under the martingale model, the current price is the sole (and best) predictor of the next price, even if there are successive dependencies in returns. As previously noted, an analysis of most cryptocurrencies (the most widely used) shows that the returns of more than half of the assets seem to follow a random walk. With the martingale model, one might be tempted to assert that the crypto market is efficient. However, many studies have investigated the relationship between Bitcoin and the martingale model ([Zargar and Kumar, 2019], [Nadarajah and Chu, 2017] …) and conclude that the Bitcoin market is not efficient, mainly due to endogenous factors of an emerging and immature market, and the absence of traders relying on fundamental value. It is difficult to extend this conclusion to the entire cryptocurrency market. However, we know that a study showing market inefficiency between 2012 and 2015 is not highly relevant for 2022, as much has happened since then (especially for Bitcoin). Thus, we highlight the application of Lo’s adaptive market hypothesis [Lo, 2004] to Bitcoin through a study [Khuntia and Pattanayak, 2018], which explains that efficiency improves over time. This study particularly well summarizes the evolution of crypto market returns: episodes of efficiency and inefficiency, creating opportunities for arbitrage and above-average returns, but an impossibility to predict these opportunities systematically or mathematically. 2.1.3 Cryptocurrencies and Fundamental Value As explained by [Delcey et al., 2017], there are two definitions of an efficient market. Fama’s definition implies that the randomness of a price is explained by the fact that prices converge toward the fundamental value. Samuelson’s definition implies that unpredictable price variations are simply the result of competition among investors, regardless of fundamental value. This raises the following question: What is a fundamental value for a cryptocurrency? According to [Biais et al., 2020], the fundamental value of Bitcoin (and by extension most other cryptocurrencies, as they hardly differ in their characteristics) lies in its stream of net transactional benefits, which depend on its future prices. These transactional benefits may, for instance, represent the ability to exchange money in an unstable economic and financial system (such as in Venezuela or Zimbabwe), or when exchanges are blocked or heavily taxed. To determine the net value, [Biais et al., 2020] consider various costs: limited convertibility, transaction fees from brokers, mining costs, and crash risk. They thus provide a definition of Bitcoin’s fundamental value (and technically of other cryptocurrencies) and answer the question of whether a cryptocurrency can have a fundamental value. Obviously, this value differs depending on the cryptocurrency. For instance, if there is a strong demand for privacy in transactions, Monero (XMR) would dominate in volume, since it uses a private blockchain by default (making transactions untraceable, unlike Bitcoin where the blockchain is public and all transactions are identifiable). However, the very idea that Bitcoin has a fundamental value is debated both in the media and academic literature. According to [Yermack, 2013], cryptocurrencies have no fundamental value because, if they did, there would be no incentive to mine cryptocurrency. According to [Hanley, 2013], Bitcoin’s value merely floats relative to other currencies as a market estimate without any fundamental value to support it. [Woo et al., 2013] suggests Bitcoin may have a certain fair value because of its features similar to fiat currencies (means of exchange and store of value), but without any other underlying basis. [Hayes, 2015] links the importance of Bitcoin’s mining network to the dependency of altcoin holders on Bitcoin, given that most altcoins must be exchanged into Bitcoin before being converted into fiat currency for real-world use. Furthermore, [Garcia et al., 2014] highlights the importance of mining production costs in the fundamental value of cryptocurrencies, as it provides a kind of “floor value”. Cryptocurrencies are often criticized for being "backed by nothing", a misconception regarding the role of money in an economy. For example, according to the U.S. Federal Reserve, “ Federal Reserve notes are not redeemable in gold, silver, or any other commodity. Federal Reserve notes have not been redeemable in gold since January 30, 1934, when the Congress amended Section 16 of the Federal Reserve Act to read: "The said [Federal Reserve] notes shall be obligations of the United States….They shall be redeemed in lawful money on demand at the Treasury Department of the United States, in the city of Washington, District of Columbia, or at any Federal Reserve bank." ” Beyond the purely economic definition of value (utility and scarcity), for which Bitcoin qualifies (its utility lying in being an alternative to the centralized financial system, and its scarcity from the 21 million unit limit and diminishing accessibility over time), there is also a subjective characteristic to this value. We highlight two relevant elements: network value and safe-haven value. According to Metcalfe’s law [Metcalfe, 1995], although nuanced [Odlyzko and Tilly, 2005], the value of a network is proportional to the square of the number of its users: a single fax machine is useless, but the value of each fax increases with the total number of machines in the network. One could thus infer a similar characteristic for cryptocurrencies. According to [Baur and McDermott, 2010], a safe-haven asset can be defined as one that is negatively correlated with equities during crises. Gold is often a reference point. Let us verify this. We cannot directly compare superimposed charts due to vastly different magnitudes: <details> <summary>extracted/6391907/images/cor1.png Details</summary>  ### Visual Description ## Line Chart: Cryptocurrency and Financial Market Performance (2015-2022) ### Overview The chart displays three financial assets' price movements over eight years: gold (GOLD/USD), Bitcoin (BTC/SD), and the S&P500 index. The y-axis represents price in USD, while the x-axis spans 2015-2022. Three distinct lines show divergent volatility patterns. ### Components/Axes - **X-axis**: Time (2015-2022), marked annually - **Y-axis**: Price (USD), scaled 0-70,000 - **Legend**: Right-aligned, color-coded: - Blue: GOLD/USD - Red: BTC/SD - Green: S&P500 - **Grid**: Light gray lines with 10,000 USD increments ### Detailed Analysis 1. **GOLD/USD (Blue Line)** - Flat line between 1,200-1,300 USD throughout 2015-2022 - Minor fluctuations (±50 USD) with no significant trend - Spatial grounding: Bottom-most line, consistently near y=1,250 2. **BTC/SD (Red Line)** - 2015-2016: Near-zero value (~100 USD) - 2017: Sharp rise to ~20,000 USD, followed by 70% correction - 2018: Volatile consolidation between 3,000-10,000 USD - 2019-2020: Gradual increase to ~10,000 USD - 2021: Parabolic surge to ~65,000 USD (peak), then 38% correction - 2022: Recovery to ~40,000 USD with continued volatility - Spatial grounding: Dominates upper chart space post-2017 3. **S&P500 (Green Line)** - 2015-2016: Stable ~2,000-2,200 USD - 2017-2018: Gradual rise to ~2,800 USD - 2019-2020: Sharp increase to ~4,000 USD during pandemic - 2021: Peaks at ~4,500 USD, then declines to ~3,800 USD - Spatial grounding: Middle line, shows steady upward trend until 2021 ### Key Observations - BTC/SD exhibits extreme volatility (100x increase from 2015-2021) - S&P500 shows 100% total return over 8 years with accelerated growth post-2020 - GOLD/USD remains remarkably stable despite macroeconomic changes - BTC/SD's 2021 peak (~65,000 USD) approaches but doesn't reach y-axis maximum - S&P500's 2021 peak correlates with BTC/SD's surge, suggesting market correlation ### Interpretation The chart reveals divergent risk profiles: 1. **BTC/SD** demonstrates speculative asset characteristics with 100x+ volatility 2. **S&P500** reflects traditional market growth with pandemic-era acceleration 3. **GOLD/USD** maintains safe-haven status with minimal volatility 4. 2021 correlation between crypto and equities suggests macroeconomic drivers 5. BTC/SD's 2022 correction (38% from peak) indicates renewed volatility despite prior gains The data suggests BTC/SD as a high-risk/high-reward asset, S&P500 as a growth benchmark, and gold as a stability anchor. The 2021 synchronized rally across all assets may indicate broader market optimism, while BTC/SD's 2022 correction highlights crypto-specific risks. </details> Figure 12: Correlation between BTC/USD, GOLD/USD, and S&P500 Thus, we will separately analyze the correlation between S&P500 crashes and BTC/USD prices: <details> <summary>extracted/6391907/images/sp.png Details</summary>  ### Visual Description ## Line Graph: Value Over Time (2015–2022) ### Overview The image depicts a line graph illustrating a time series trend of a metric labeled "Value" from 2015 to 2022. The graph shows fluctuations with notable increases, decreases, and volatility over the period. ### Components/Axes - **X-Axis (Horizontal)**: Labeled "Year," with markers at 2015, 2016, 2017, 2018, 2019, 2020, 2021, and 2022. - **Y-Axis (Vertical)**: Labeled "Value," with increments of 500, ranging from 2000 to 4500. - **Legend**: Located in the top-right corner, indicating the blue line represents "Value Over Time." - **Gridlines**: Subtle gridlines divide the chart into 500-unit intervals for reference. ### Detailed Analysis - **2015–2016**: The line begins at approximately **2000** in 2015, fluctuates slightly, and dips to ~1900 in 2016. - **2017–2018**: A steady upward trend occurs, reaching ~2800 by 2018. - **2019**: A sharp decline to ~2400 in mid-2019, followed by recovery to ~2800 by year-end. - **2020**: A significant drop to ~2300 in early 2020, followed by a rebound to ~3200 by late 2020. - **2021**: Consistent growth, peaking at ~4700 in 2022. - **2022**: The line stabilizes around ~4500–4700, with minor fluctuations. ### Key Observations 1. **Initial Stability (2015–2016)**: Minimal change, hovering near 2000. 2. **Gradual Growth (2017–2018)**: Steady increase to ~2800. 3. **Volatility (2019–2020)**: Sharp dips and recoveries, suggesting external shocks (e.g., economic/pandemic impacts). 4. **Exponential Rise (2021–2022)**: Rapid growth from ~3200 to ~4700, indicating a potential structural shift. ### Interpretation The data suggests a metric that remained stable until 2017, then grew gradually until disrupted by external factors in 2019–2020. The post-2020 surge implies a recovery or acceleration driven by policy changes, technological adoption, or market dynamics. The 2020 dip aligns with global events like the COVID-19 pandemic, while the 2021–2022 rebound may reflect stimulus measures or sector-specific growth. **Uncertainties**: - Exact values are approximate due to lack of gridline precision. - The 2020 dip could represent a temporary anomaly or a systemic shift. - The 2022 plateau may indicate saturation or ongoing growth depending on external factors. </details> Figure 13: S&P500 over the period available with BTC/USD We notice graphical correlations during several crash periods: - Early 2018 - Late 2019 - Early 2020 - Early 2022 These correlations are weaker, or even negative, with gold: <details> <summary>extracted/6391907/images/gold.png Details</summary>  ### Visual Description ## Line Graph: Unlabeled Time Series Data (2015–2022) ### Overview The image depicts a line graph tracking a numerical metric over time, spanning from 2015 to 2022. The y-axis ranges from 1000 to 2000, with the line fluctuating between these bounds. A notable upward trend emerges after 2020, culminating in a peak near 2000 by 2022. ### Components/Axes - **X-Axis (Horizontal)**: Labeled with years from 2015 to 2022, marked at 1-year intervals. - **Y-Axis (Vertical)**: Labeled with numerical values from 1000 to 2000, marked at 200-unit intervals. - **Line**: A single blue line represents the data series. No legend is present to clarify the metric’s meaning. - **Grid**: Light gray gridlines align with axis ticks for reference. ### Detailed Analysis 1. **2015–2016**: - The line begins at ~1200 in 2015, fluctuating slightly between 1150–1250. - A sharp dip occurs in mid-2016, reaching ~1050, followed by a recovery to ~1150 by year-end. 2. **2017–2018**: - Volatility increases, with peaks near 1300 in 2017 and ~1250 in 2018. - A minor decline to ~1150 occurs in late 2018. 3. **2019–2020**: - Stabilizes around 1200–1250 until early 2020. - A steep rise begins in mid-2020, surging to ~1600 by late 2020. 4. **2021–2022**: - Continued growth, peaking at ~2000 in 2021. - A minor correction occurs in 2022, stabilizing near 1900–1950. ### Key Observations - **Pre-2020 Stability**: The metric remains relatively flat (~1150–1300) from 2015–2019. - **Post-2020 Surge**: A 60% increase occurs between 2020 and 2021, followed by a plateau. - **Volatility**: Notable fluctuations in 2016 and 2017 suggest external shocks or cyclical patterns. ### Interpretation The data suggests a significant external event (e.g., economic, technological, or global crisis) triggered the post-2020 surge. The pre-2020 stability implies a baseline equilibrium disrupted by an unforeseen factor. The lack of a legend or contextual labels limits definitive conclusions, but the sharp rise aligns with real-world events like the COVID-19 pandemic (2020), which often correlates with metrics such as digital adoption, healthcare costs, or remote work trends. The 2022 plateau may indicate market saturation or stabilization post-crisis. **Note**: The image contains no textual labels, units, or contextual clues beyond the axes. All interpretations are speculative and require corroboration with external data sources. </details> Figure 14: GOLD/USD over the period available with BTC/USD Let us graphically check the correlation of daily returns: <details> <summary>extracted/6391907/images/cor-btc-sp.png Details</summary>  ### Visual Description ## Line Chart: BTC/SD vs S&P500 Performance (2015–2022) ### Overview The chart compares the performance of two financial instruments—BTC/SD (blue line) and S&P500 (red line)—over an 8-year period (2015–2022). The y-axis represents normalized values ranging from -0.4 to 0.2, while the x-axis spans years. Both lines exhibit volatility, but with distinct patterns. The legend is positioned in the top-right corner, with blue denoting BTC/SD and red denoting S&P500. ### Components/Axes - **X-axis (Years)**: Labeled with annual increments from 2015 to 2022. No explicit title, but contextually represents time. - **Y-axis (Normalized Value)**: Ranges from -0.4 to 0.2 in 0.1 increments. No explicit title, but implied to measure performance deviation. - **Legend**: Top-right corner. Blue = BTC/SD, Red = S&P500. - **Gridlines**: Faint horizontal and vertical lines for reference. ### Detailed Analysis - **BTC/SD (Blue Line)**: - **Trend**: Highly volatile, with sharp peaks and troughs. Notable spikes in 2017 (~0.15), 2020 (~-0.4), and 2021 (~0.18). Persistent oscillations around zero. - **Key Data Points**: - 2015: ~0.05 - 2016: ~0.02 - 2017: ~0.15 (peak) - 2018: ~-0.1 - 2019: ~0.03 - 2020: ~-0.4 (trough) - 2021: ~0.18 (peak) - 2022: ~0.01 - **Uncertainty**: Approximate values derived from visual estimation; exact values may vary by ±0.02 due to chart resolution. - **S&P500 (Red Line)**: - **Trend**: Relatively stable, fluctuating narrowly around zero. Minor peaks in 2017 (~0.02) and 2021 (~0.01). No extreme deviations. - **Key Data Points**: - 2015: ~0.0 - 2016: ~0.0 - 2017: ~0.02 - 2018: ~0.0 - 2019: ~0.0 - 2020: ~0.0 - 2021: ~0.01 - 2022: ~0.0 - **Uncertainty**: Values stable within ±0.01 due to minimal variation. ### Key Observations 1. **Volatility Disparity**: BTC/SD exhibits ~10x greater volatility than S&P500, with extreme swings (e.g., -0.4 in 2020 vs. 0.18 in 2021). 2. **Correlation**: Both lines show synchronized movements in 2017 and 2021, suggesting potential market-wide influences (e.g., crypto booms/busts). 3. **Anomalies**: - BTC/SD’s 2020 trough (-0.4) aligns with the global pandemic crash. - S&P500 remains flat during the same period, indicating resilience. ### Interpretation - **BTC/SD Dynamics**: The extreme volatility reflects cryptocurrency market instability, likely driven by speculative trading, regulatory shifts, and macroeconomic factors. The 2020 crash and 2021 rebound highlight sensitivity to external shocks. - **S&P500 Stability**: The red line’s consistency underscores the index’s role as a benchmark for traditional equity markets, less affected by crypto-specific risks. - **Implications**: The divergence in 2020–2021 suggests crypto markets operate independently of traditional indices during crises, while overlapping peaks (2017, 2021) indicate shared exposure to speculative trends. *Note: All values are approximate, derived from visual inspection of the chart. No textual annotations or embedded data tables are present.* </details> Figure 15: Correlation between BTC/USD and S&P500 (daily returns) <details> <summary>extracted/6391907/images/cor-gold-sp.png Details</summary>  ### Visual Description ## Line Chart: GOLD/USD vs S&P500 Performance (2015–2022) ### Overview The chart displays two financial time series: **GOLD/USD** (blue line) and **S&P500** (red line), plotted against a timeline from 2015 to 2022. Both series oscillate around a baseline of 0, with values ranging between **-0.1 and +0.1** on the y-axis. The legend is positioned in the **top-right corner**, clearly associating colors with their respective data series. --- ### Components/Axes - **X-axis (Horizontal)**: Represents years from **2015 to 2022**, with gridlines marking each year. - **Y-axis (Vertical)**: Labeled from **-0.1 to +0.1**, with no explicit unit provided. The axis is centered at 0, suggesting percentage changes or normalized values. - **Legend**: Located in the **top-right**, with: - **Blue**: GOLD/USD (gold price relative to USD) - **Red**: S&P500 (stock market index) --- ### Detailed Analysis 1. **GOLD/USD (Blue Line)**: - Exhibits **moderate volatility**, with fluctuations typically within **±0.05** of the baseline. - Notable peaks occur in **2016** and **2020**, reaching ~+0.04 and ~+0.03, respectively. - A sharp dip to **-0.03** in **2020** aligns with the S&P500's volatility but is less extreme. 2. **S&P500 (Red Line)**: - Shows **higher volatility**, with frequent spikes and troughs exceeding **±0.05**. - A dramatic **plunge to -0.12** in **2020** (likely reflecting the 2020 market crash) is followed by a rapid rebound to **+0.08** by mid-2020. - Post-2020, the line stabilizes but remains more erratic than GOLD/USD. 3. **Correlation**: - The two series **do not consistently align**; for example, GOLD/USD peaks in 2020 while S&P500 bottoms, suggesting **inverse behavior during crises**. - Both lines remain within the **-0.1 to +0.1** range, indicating normalized or percentage-based metrics. --- ### Key Observations - **2020 Anomaly**: The S&P500 experiences a **120% drop** (from ~+0.05 to -0.12) in early 2020, followed by a **130% recovery** by mid-2020. GOLD/USD shows a smaller but synchronized dip (-0.03) during the same period. - **Stability vs. Volatility**: GOLD/USD demonstrates **relative stability** compared to the S&P500, which exhibits **extreme swings**. - **Normalization**: The y-axis range (-0.1 to +0.1) suggests the data may represent **percentage changes** or **z-scores** rather than absolute values. --- ### Interpretation - **Market Dynamics**: The S&P500's volatility reflects broader market risks (e.g., 2020 pandemic crash), while GOLD/USD's stability aligns with gold's historical role as a **safe-haven asset**. - **Divergence in 2020**: The inverse relationship during the 2020 crash highlights how investors may shift from equities (S&P500) to gold during uncertainty, though the chart does not explicitly show this causality. - **Normalization Limitation**: Without explicit units, the y-axis values are ambiguous. A value of +0.1 could represent a 10% increase or a z-score of 0.1, significantly altering interpretation. --- ### Conclusion The chart illustrates contrasting behaviors between gold and the S&P500 over seven years, emphasizing gold's stability and the stock market's susceptibility to sharp corrections. The 2020 divergence underscores the importance of diversification, though further context (e.g., absolute prices, economic events) is needed for precise analysis. </details> Figure 16: Correlation between GOLD/USD and S&P500 (daily returns) A numerical analysis of the correlation of daily returns over the entire period shows 16% for Bitcoin with the S&P500 and 5% for gold with the S&P500. Bitcoin does not appear to be a better safe-haven asset than gold, which is confirmed by other studies ([Smales, 2019], [Bouri et al., 2017]). 2.2 From Louis Bachelier to Contemporary Models Eugène Fama is not the inventor of the idea of a random market. We can trace it back to 1863, when Jules Regnault [Regnault, 1863] proposed a model of randomly volatile markets. Then, in 1900, Bachelier [Bachelier, 1900] formalized it. It was only from the 1930s that the random aspect of the market began to be considered, notably in the United States with the emergence of econometrics, and then, from the 1960s, financial economics in the United States started to connect the model to economic theory, giving rise to the theory of informational efficiency of financial markets. However, this theory, although constituting the foundation of the random walk model, would never achieve unanimous acceptance. In this subsection, we will present the theoretical models that have explained the variations of financial assets since 1900, from Louis Bachelier’s theory of speculation to the present day. 2.2.1 Modeling of Traditional Finance It is important to understand that the cryptocurrency market is not disconnected from traditional financial markets in its creation. - The Louis Bachelier Model Bachelier is a pioneer of modern finance in the sense that he was the first to use Brownian motion in modeling stock prices, five years before [Einstein, 1956]. From his model, the Wiener process [Wiener, 1976] would later be formalized. The model simply explains that the stock market follows a Gaussian distribution. Of course, such a model today would not be considered rigorous, but for its time, it was already remarkably close to a correct model. Indeed, Brownian motion applied to stock price fluctuations is based on questionable assumptions: Markov chain (memoryless process), stationarity (constant mean and standard deviation), and normal distribution. We can clearly see, for example, for the four largest cryptocurrencies, that the distribution of daily returns is not really Gaussian: <details> <summary>extracted/6391907/images/dist-btc.png Details</summary>  ### Visual Description ## Histogram with Normal Distribution Overlay: Value Frequency Distribution ### Overview The image displays a histogram overlaid with a normal distribution curve. The histogram shows the frequency distribution of data points across a range of values, while the curve represents the theoretical normal distribution that fits the data. The x-axis spans from -0.3 to 0.2, and the y-axis measures frequency up to 0.004. ### Components/Axes - **X-axis (Value)**: Labeled "Value," with tick marks at -0.3, -0.25, -0.2, -0.15, -0.1, 0, 0.05, 0.1, and 0.15. The scale increments by 0.05. - **Y-axis (Frequency)**: Labeled "Frequency," with tick marks at 0.0005, 0.001, 0.0015, 0.002, 0.0025, 0.003, 0.0035, and 0.004. The scale increments by 0.0005. - **Legend**: Located at the bottom-right corner, with two entries: - **Normal Distribution Curve**: Black line. - **Histogram Bars**: Gray bars. ### Detailed Analysis 1. **Histogram Bars**: - The tallest bar is centered at **0**, with a frequency of approximately **0.0035**. - Bars decrease symmetrically as they move away from 0: - At **-0.1** and **0.1**, frequency ≈ **0.0025**. - At **-0.2** and **0.2**, frequency ≈ **0.0015**. - At **-0.3** and **0.15**, frequency ≈ **0.0005**. - Bars are evenly spaced at 0.05-unit intervals. 2. **Normal Distribution Curve**: - Peaks at **0** with a frequency of **0.0035**, matching the histogram's tallest bar. - The curve tapers off symmetrically toward the edges, closely following the histogram's shape. - The curve's inflection points (where concavity changes) occur near **-0.1** and **0.1**, aligning with the histogram's steepest declines. ### Key Observations - The data is **symmetric** around 0, consistent with a normal distribution. - The histogram bars and curve align closely, suggesting the data fits a normal distribution well. - The highest frequency occurs at **0**, with frequencies decreasing by ~0.001 per 0.05-unit step away from the center. - The histogram shows slight asymmetry in the tails (e.g., the left tail extends slightly further to -0.3 than the right to 0.2), but the curve smooths this out. ### Interpretation The data likely represents a process governed by a normal distribution, such as measurement errors, natural variations, or standardized test scores. The histogram and curve together confirm that: - **Most data points cluster near the mean (0)**, with fewer extreme values. - The distribution is approximately symmetric, though minor skewness may exist in the tails. - The normal distribution curve provides a good fit, enabling probabilistic interpretations (e.g., ~68% of data lies within ±0.1 of the mean). This visualization is critical for identifying outliers, validating assumptions of normality, and informing statistical analyses like confidence intervals or hypothesis testing. </details> Figure 17: Distribution of daily returns for BTC/USD <details> <summary>extracted/6391907/images/dist-eth.png Details</summary>  ### Visual Description ## Line Graph with Normal Distribution Curve and Histogram ### Overview The image depicts a line graph overlaid with a histogram, illustrating a dataset's distribution. A smooth, bell-shaped curve (normal distribution) is plotted alongside vertical bars representing frequency counts. The x-axis spans from -0.4 to 0.2, while the y-axis ranges from 0 to 0.005. The curve peaks at x = 0, and the bars are concentrated around this central value, decreasing symmetrically toward the edges. ### Components/Axes - **X-axis**: Labeled with values from -0.4 to 0.2, with tick marks at -0.4, -0.3, -0.2, -0.1, 0, 0.1, and 0.2. - **Y-axis**: Labeled with values from 0 to 0.005, with tick marks at 0.001, 0.002, 0.003, 0.004, and 0.005. - **Legend**: A single entry labeled "Normal Distribution" in black, positioned near the top-right corner of the graph. - **Data Elements**: - **Normal Distribution Curve**: A smooth, continuous line in black, peaking at x = 0. - **Histogram Bars**: Vertical black bars clustered around x = 0, with heights decreasing as distance from 0 increases. ### Detailed Analysis - **Normal Distribution Curve**: - The curve follows the classic bell shape, with its maximum at x = 0. - The curve's height at x = 0 is approximately 0.004, decreasing to near 0 at x = ±0.2. - The curve's symmetry suggests a mean of 0 and a standard deviation of ~0.1 (estimated from the spread of the bars). - **Histogram Bars**: - Bars are evenly spaced along the x-axis, with widths of ~0.05 units. - The tallest bars are centered at x = 0, with heights reaching ~0.003. - Bars decrease in height as they move away from 0, with the shortest bars at x = ±0.2 (height ~0.001). - The bars align closely with the normal distribution curve, indicating the data follows a normal distribution. ### Key Observations 1. **Symmetry**: The data is symmetrically distributed around x = 0, consistent with a normal distribution. 2. **Peak at 0**: The highest frequency of data points occurs at x = 0, with a gradual decline toward the edges. 3. **Scale**: The y-axis values are extremely small (0–0.005), suggesting the data may be standardized or the sample size is large. 4. **Alignment**: The histogram bars closely match the normal distribution curve, confirming the data's adherence to the theoretical model. ### Interpretation This graph demonstrates that the dataset follows a normal distribution, with most values concentrated near the mean (x = 0). The histogram bars and the normal distribution curve are in close agreement, indicating the data is well-modeled by a Gaussian distribution. The symmetry and central peak suggest no significant skewness or outliers. The small y-axis values imply the data may have been normalized or the sample size is large, resulting in low frequency counts per bin. This visualization is critical for understanding the data's central tendency, variability, and distributional properties, which are essential for statistical analysis and hypothesis testing. </details> Figure 18: Distribution of daily returns for ETH/USD <details> <summary>extracted/6391907/images/dist-ltc.png Details</summary>  ### Visual Description ## Line Graph with Histogram Overlay: Distribution of Data Points ### Overview The image depicts a line graph overlaid with a histogram, illustrating the distribution of data points. The graph features a bell-shaped curve (normal distribution) and vertical bars representing frequency counts. The x-axis spans from -0.4 to 0.6, while the y-axis ranges from 0 to 0.004. The histogram bars are concentrated near the center (x=0), with decreasing frequency as values move away from the mean. ### Components/Axes - **X-Axis**: Labeled with numerical values from -0.4 to 0.6 in increments of 0.2. No explicit label is visible in the image. - **Y-Axis**: Labeled "Frequency" (approximate, inferred from context). Values range from 0 to 0.004 in increments of 0.0005. - **Legend**: Located in the top-right corner. Contains two entries: - **Normal Distribution**: Represented by a smooth black curve. - **Histogram**: Represented by vertical black bars. ### Detailed Analysis - **Normal Distribution Curve**: - Peaks sharply at x=0 with a height of approximately 0.003. - Symmetrically declines on both sides, approaching zero at x=±0.4. - The curve follows the classic bell shape of a Gaussian distribution. - **Histogram**: - Bars are clustered tightly around x=0, with the tallest bars (height ~0.0025) centered at x=0. - Bar heights decrease symmetrically as x moves toward ±0.2, with minimal bars beyond ±0.2. - Total bar count is approximately 20–25, with most bars concentrated in the range -0.1 to 0.1. ### Key Observations 1. **Symmetry**: Both the histogram and curve are symmetric about x=0, indicating a balanced distribution. 2. **Concentration**: ~70% of data points lie within x=±0.1, as suggested by the histogram's peak and curve alignment. 3. **Tails**: The distribution has thin tails, with negligible frequency beyond x=±0.2. 4. **Alignment**: The histogram closely matches the normal distribution curve, confirming the data's adherence to a Gaussian model. ### Interpretation The graph demonstrates that the dataset follows a normal distribution, a common pattern in natural and experimental data. The histogram's alignment with the curve validates the assumption of normality, which is critical for statistical methods like hypothesis testing or confidence interval estimation. The absence of outliers or skewed bars suggests the data is reliable for parametric analyses. The tight clustering around the mean implies low variability, which could indicate controlled experimental conditions or a highly consistent process. The y-axis scale (up to 0.004) suggests the data represents probabilities or proportions rather than raw counts. </details> Figure 19: Distribution of daily returns for LTC/USD <details> <summary>extracted/6391907/images/dist-xrp.png Details</summary>  ### Visual Description ## Line Graph: Normal Distribution with Error Bars ### Overview The image depicts a line graph representing a normal distribution curve with superimposed error bars. The x-axis spans from -0.4 to 0.8, while the y-axis ranges from 0 to 0.005. The graph includes a smooth black line forming a bell-shaped curve and vertical error bars clustered around the center. The legend is positioned at the bottom-right but contains no visible text or labels. ### Components/Axes - **X-Axis**: Labeled with approximate values at intervals: -0.4, -0.2, 0, 0.2, 0.4, 0.6, 0.8. No explicit title is visible. - **Y-Axis**: Labeled with approximate values at intervals: 0.001, 0.002, 0.003, 0.004, 0.005. No explicit title is visible. - **Legend**: Located at the bottom-right corner but empty (no labels or color keys). - **Grid**: Light gray grid lines span the entire plot area. - **Main Line**: A smooth black curve forming a normal distribution, peaking at x=0 with a y-value of approximately 0.002. - **Error Bars**: Vertical black lines clustered around the center (x=0), with heights decreasing symmetrically toward the edges. Heights range from ~0.001 to ~0.004. ### Detailed Analysis - **Main Line**: The curve follows a Gaussian distribution, peaking at x=0 (y≈0.002) and tapering off to near-zero values at x=±0.4. The curve is symmetric, with no visible anomalies. - **Error Bars**: - Tallest bars (~0.004) are centered at x=0, decreasing in height as x moves away from 0. - Bars become sparser and shorter beyond x=±0.2, with minimal presence beyond x=±0.4. - Heights suggest variability in measurements, with standard deviations likely decreasing away from the mean. ### Key Observations 1. **Symmetry**: The distribution and error bars are symmetric about x=0. 2. **Peak Variability**: The highest error bars (0.004) align with the curve’s maximum, indicating maximum uncertainty at the mean. 3. **Tapered Distribution**: Both the curve and error bars diminish rapidly beyond x=±0.2, suggesting limited data or confidence in extreme values. 4. **Missing Legend Text**: The legend’s absence of labels implies either a generic plot or omitted contextual information. ### Interpretation The graph likely represents a dataset with a central tendency around x=0, where measurements cluster most densely. The error bars’ heights and distribution suggest: - **Highest Uncertainty at the Mean**: The tallest error bars at x=0 imply greater variability or measurement error near the central value. - **Confidence in Central Values**: The curve’s peak and dense error bars indicate strong agreement in measurements around x=0. - **Rapid Decline in Confidence**: The sharp drop in error bar heights and curve amplitude beyond x=±0.2 suggests limited data or lower confidence in extreme values. The absence of a legend title or axis labels reduces interpretability, but the visual structure strongly aligns with a normal distribution model. The error bars may represent standard deviations or confidence intervals, though their exact meaning remains unclear without textual clarification. </details> Figure 20: Distribution of daily returns for XRP/USD It might be more appropriate to refer to a Lévy law or an $\alpha$ -stable distribution. - The Gordon-Shapiro Model The Gordon-Shapiro model [Gordon and Shapiro, 1956] is very well-known in finance and provides a very simple formula to model the price of a stock: $$ P_{0}=\frac{D_{1}}{k-g} \tag{1} $$ where $P_{0}$ is the theoretical value of the stock, $D_{1}$ the anticipated dividend for the first period, $k$ the expected return rate for the shareholder, and $g$ the growth rate of the gross earnings per share. The first thing to note is that this model is useless for the crypto market: there are no dividends. Therefore, this model can be dismissed, even though it is attractive. - Contemporary Models With the development of quantitative finance and derivative pricing, many models have emerged, one of the most famous being the Black & Scholes model [Black and Scholes, 2019]. However, as with the binomial model (Cox, Ross & Rubinstein model), the problem of constant volatility of the underlying assets appeared. Indeed, in the Black & Scholes formula: $$ C=S_{t}N(d_{1})-Ke^{-rt}N(d_{2}) \tag{2} $$ where: $$ d_{1}=\frac{\ln\frac{S_{t}}{K}+(r+\frac{\sigma^{2}}{2})t}{\sigma\sqrt{t}} \tag{3} $$ $$ d_{2}=d_{1}-\sigma\sqrt{t} \tag{4} $$ with: $C$ the price of the call option $P$ the stock price $K$ the strike price $r$ the risk-free interest rate $t$ the time in years to maturity $N$ a normal distribution $\sigma$ the volatility of the underlying asset We notice that volatility is considered constant. This led to the development of stochastic volatility models, treating the volatility of the underlying as a random process. As explained, for instance, by [Mantegna and Stanley, 1999], the price of an asset can be characterized by a standard geometric Brownian motion: $$ dS_{t}=\mu S_{t}dt+\sigma S_{t}dW_{t} \tag{5} $$ with: $\mu$ the drift (often negligible) $\sigma$ constant volatility $dW_{t}\hookrightarrow N(0,1)$ an increment of Brownian motion then replacing $\sigma$ by a process $\nu_{t}$ . This is indeed how the Heston model [Heston, 1993] is built, one of the most well-known stochastic volatility models. Its formulas are: $$ dS_{t}=rS_{t}dt+\sqrt{V_{t}}S_{t}dW_{1t} \tag{6} $$ with $V_{t}$ the instantaneous variance: $$ dV_{t}=\kappa(\theta-V_{t})dt+\sigma\sqrt{V_{t}}dW_{2t} \tag{7} $$ where: $S_{t}$ the asset price at time $t$ $r$ the risk-free interest rate $\sqrt{V_{t}}$ the volatility (standard deviation) of the price $\sigma$ the volatility of the volatility (i.e., of $\sqrt{V_{t}}$ ) $\theta$ the long-term variance $\kappa$ the reversion rate to $\theta$ $dt$ an infinitesimally small time increment $W_{1t}$ the Brownian motion for the asset price $W_{2t}$ the Brownian motion for the variance of the asset price with the property that, for Brownian motions, $W_{0}=0$ , the $W_{t}$ are independent, and $W_{t}$ is continuous in $t$ . This model seems well suited for modeling the price of cryptocurrencies. Indeed, [Kachnowski, 2020] explains that an adaptation of the Heston model to Bitcoin improves the accuracy of predictions over time windows ranging from 7 days to 2 months. However, as shown by [Gatheral et al., 2018], the log-volatility is not actually a classic Brownian motion but rather a fractional Brownian motion, as in the Fractional Stochastic Volatility Model by [Comte et al., 2012], but with a Hurst exponent of 0.1 (and not 0.5 as in [Comte et al., 2012], who did not take into account the rough aspect of volatility). 2.2.2 Modeling Crypto-Finance - Quantitative Theory of (Crypto)Currency As we know [Fisher, 2006], $$ MV=PY \tag{8} $$ where: $M$ is the money supply $V$ is the velocity of money $P$ is the price level $Y$ is the output of the economy Let’s adapt this model to cryptocurrencies. For $M$ , it is simple: it is constant at 21 million. However, we can already anticipate that $M$ tends towards 0. Indeed, 21 million is the maximum number of Bitcoins that can be mined. Once mined, Bitcoins can disappear for several reasons: lost passwords, hacking, computer errors, etc. For $V$ , it is more complicated. We would need to differentiate between economically meaningful transactions and meaningless ones. And this is very difficult, even though all transactions are listed on the Blockchain, the reasons behind them are not. Thus, we cannot distinguish "real" transactions from "fake" ones. For $P$ , it refers to the goods and services that can be purchased with Bitcoin. In November 2020, the Venezuelan branch of Pizza Hut accepted Bitcoin. On that day, you could buy around 1,800 pizzas (worth approximately 10 USD each) with one Bitcoin. Today, you could buy around 4,000 pizzas with one Bitcoin. Thus, $P$ has been continuously falling for BTC/USD. For $Y$ , it represents the amount of goods and services available for purchase and sale. We can admit that very few goods and services are currently bought and sold with cryptocurrencies. Thus, over time, cryptocurrencies are expected to depreciate. Indeed, we know that the number of Bitcoins in circulation initially increases (then will decrease), which should induce inflation. However, the opposite is observed. If $Y$ is exogenous to Bitcoin (goods and services offered are not really dependent on Bitcoin’s price), and $M$ is constant, then $V$ will influence $P$ . In this case, two scenarios arise: if Bitcoin (same reasoning for other cryptocurrencies) is merely a means of exchange without any fundamental value, $V$ will increase, as it will become just another payment option for households. If Bitcoin is rather seen as a store of value, with a fundamental value, then households will invest and hold their Bitcoins, causing $V$ to decrease, which will raise $P$ . Studies, including [Pernice et al., 2020], show a link between price and velocity in cryptocurrencies. - Other Models of the Crypto Market [Cretarola and Figà-Talamanca, 2018] propose modeling the crypto market by the interest it generates. They explore the link between Bitcoin’s price behavior and investor attention in the network. They conclude that the attention index impacts Bitcoin’s price through dependence of the drift and diffusion coefficients and potential correlation between the sources of randomness represented by Brownian motions. [Hou et al., 2020] propose a model for pricing crypto options, SVCJ (stochastic volatility with a correlated jump), similar to [Pascucci and Palomba,], and compare it with the cojump model of [Bandi and Reno, 2016]. It is very likely that the future of cryptocurrency price modeling will develop towards derivative products. 2.3 Time Series Studies and Analyses We are still considering the case where the crypto market is efficient. Thus, it is impossible to predict its price movements, regardless of the methods employed. However, these methods are still widely used by both retail and professional investors. Therefore, we will examine these methods to understand whether they can be effective in prediction. Nevertheless, we will see that it is sometimes difficult to answer this question with a simple yes or no. 2.3.1 Fundamental Analysis To perform fundamental analysis on a company, there are a number of well-established methods (financial ratios, EBITDA, cash flows, etc.). For cryptocurrencies, there are not really established methods. We have therefore chosen 5 themes. We cannot develop a full analysis due to the lack of data, but these indicators can, in our opinion, allow a good fundamental analysis: - Supply Measures: - Is the number of coins fixed in advance? - How many coins have been mined, and how many remain? - What is the inflation rate? - What is the coin-to-flow ratio? - What is the granularity of the coins? - Value Measures: - What is the current price? - What is the current gross market capitalization? - What is the current net market capitalization (excluding lost coins)? - What is the interest rate (borrowing cryptocurrencies)? - What are the yearly high and low points? - What are the returns by day, week, month, year, and overall? - Network Activity Measures: - How many active addresses are there? - How many new addresses are there? - How many transactions are there? - What is the average transaction size? - Broker Activity Measures: - What is the total traded volume? - On how many brokers is the cryptocurrency listed? - What is the broker flow? - In which geographical areas do the flows occur? - Mining Measures: - What is the consensus mechanism (Proof of Work, Proof of Stake, etc.)? - What is the governance of the mining network? - How long does it take to mine a block? - How are miners rewarded? - What are the median fees? - What is the hash rate? 2.3.2 Chartist / Technical Analysis Technical analysis aims to predict a price using future prices, and more precisely through repetitive patterns or technical indicators. Beyond the SMA tested previously, let’s simply perform a graphical analysis. Let’s take the RSI on BTC/USD: <details> <summary>x1.png Details</summary>  ### Visual Description ## Candlestick Chart: Bitcoin/Dollar (1W) TradingView ### Overview The image displays a Bitcoin/Dollar (BTC/USD) 1-week candlestick chart from COINBASE TradingView. The chart shows price movements over time with green (upward) and red (downward) candlesticks, accompanied by two moving average lines (purple and yellow) and a price axis on the right. A horizontal dotted line at ~$38,652.55 is visible, likely indicating a key support/resistance level. --- ### Components/Axes 1. **X-Axis (Time)**: - Labeled "1W" (1 Week) in the top-left corner. - No explicit date markers, but candlesticks represent sequential time intervals. 2. **Y-Axis (Price)**: - Right-axis labeled "USD" with values ranging from $0.00 to $70,000.00 in increments of $10,000. - Left-axis shows price markers at $10,000, $20,000, ..., $70,000. 3. **Legend**: - Located in the bottom-right corner. - **Purple line**: 200-period Simple Moving Average (SMA). - **Yellow line**: 50-period Simple Moving Average (SMA). 4. **Additional Elements**: - Horizontal dotted line at ~$38,652.55 (bottom of the chart). - Timestamp in the bottom-left corner: "TV" (likely TradingView branding). --- ### Detailed Analysis 1. **Candlestick Trends**: - **Early Period (Left Side)**: - Price remains flat near $37,607.87 (red candlestick at the far left). - Gradual upward movement to ~$39,695.03 (green candlesticks). - **Mid-Period**: - Volatile price action with alternating green/red candlesticks. - Price peaks near $40,000, then declines to ~$38,652.55. - **Late Period (Right Side)**: - Sharp upward surge to ~$65,000 (green candlesticks dominate). - Subsequent correction to ~$50,000 (red candlesticks). 2. **Moving Averages**: - **Purple (200 SMA)**: - Starts near $37,607.87, trends upward to ~$40,000, then declines to ~$38,652.55. - **Yellow (50 SMA)**: - Begins at ~$37,607.87, rises to ~$40,000, dips to ~$38,652.55, then surges to ~$60,000 before correcting. 3. **Key Data Points**: - **High**: ~$65,000 (right side, green candlesticks). - **Low**: ~$37,607.87 (left side, red candlestick). - **Support Level**: Horizontal dotted line at ~$38,652.55. --- ### Key Observations 1. **SMA Crossover**: - The 50 SMA (yellow) crosses above the 200 SMA (purple) near $39,695.03, signaling a potential bullish trend. - Later, the 50 SMA crosses below the 200 SMA during the price drop to ~$38,652.55, indicating bearish momentum. 2. **Volatility**: - Sharp price swings observed between $38,652.55 and $65,000, with candlestick wicks extending beyond these levels. 3. **Support/Resistance**: - The horizontal dotted line at ~$38,652.55 acts as a critical support level, with price repeatedly testing it during the mid-period. --- ### Interpretation 1. **Trend Analysis**: - The initial flat price near $37,607.87 suggests consolidation. - The 50/200 SMA crossover near $39,695.03 marks a bullish breakout, followed by a correction to the support level. - The late-period surge to $65,000 reflects strong buying pressure, but the subsequent drop to ~$50,000 indicates profit-taking or market correction. 2. **Market Dynamics**: - The support level at ~$38,652.55 appears robust, as price bounces off it multiple times. - The divergence between the 50 SMA (yellow) and price action suggests potential for renewed upward momentum if the 50 SMA stabilizes above the 200 SMA. 3. **Anomalies**: - The extreme volatility between $38,652.55 and $65,000 may reflect external factors (e.g., regulatory news, macroeconomic events) not visible in the chart. --- ### Conclusion The chart illustrates Bitcoin's price volatility over a 1-week period, with key technical indicators (SMAs) and support levels providing insights into market sentiment. The 50/200 SMA crossover and repeated tests of the $38,652.55 support level highlight critical decision points for traders. The late-period surge to $65,000 underscores Bitcoin's susceptibility to rapid price swings, emphasizing the importance of risk management in cryptocurrency trading. </details> Figure 21: RSI Signals for BTC/USD This indicator tells us that when it is below 30, we should buy, and above 70, we should sell. It is clearly seen that the RSI is useless for a long-term vision: what is the use of selling at 10,000 in 2018 when one could simply buy, hold, and sell at 60,000 in 2022? Now, let’s take another very famous technical indicator: the SAR. <details> <summary>x2.png Details</summary>  ### Visual Description ## Line Chart: Bitcoin / Dollar (1W) - COINBASE TradingView ### Overview The chart displays Bitcoin's price fluctuations against the US Dollar over a 5-year period (July 2017–April 2023). It features two primary data series (green upward trends, red downward trends) and a blue dotted trendline. The current price (April 2023) is highlighted in green at $38,652.70. ### Components/Axes - **X-Axis (Time)**: - Labeled with monthly intervals (e.g., "Juill" = July, "Oct" = October). - Covers July 2017 to April 2023. - **Y-Axis (Price in USD)**: - Range: -$4,000 to $72,000. - Increment: $4,000 per major tick. - **Legend**: - Green: Upward price movement. - Red: Downward price movement. - **Title**: "Bitcoin / Dollar - 1W · COINBASE · TradingView" (top-left). - **Current Price Marker**: Green dot at $38,652.70 (bottom-right). ### Detailed Analysis 1. **Price Trends**: - **July 2017**: Starts at ~$38,500 (red marker). - **October 2017**: Sharp rise to ~$63,000 (green marker). - **April 2018**: Sharp decline to ~$3,800 (red marker). - **July 2018–April 2019**: Sideways consolidation (~$3,800–$10,000). - **May 2020**: Surge to ~$60,000 (green marker). - **July 2021**: Peaks at ~$72,000 (green marker). - **October 2021–April 2022**: Sharp drop to ~$16,000 (red marker). - **July 2022–April 2023**: Recovery to ~$38,652.70 (green marker). 2. **Trendline**: - Blue dotted line tracks overall price direction. - Shows cyclical volatility with alternating bull (green) and bear (red) markets. 3. **Key Data Points**: - **Highest Peak**: ~$72,000 (July 2021). - **Lowest Trough**: ~$3,800 (April 2018). - **Current Price**: $38,652.70 (April 2023). ### Key Observations - **Volatility**: Extreme price swings (e.g., 2017–2018 crash, 2021 peak). - **Cyclical Patterns**: Repeated bull/bear cycles with diminishing amplitude post-2021. - **Current Position**: Price near 2017 starting level but with higher volatility. ### Interpretation The chart reflects Bitcoin's speculative nature, driven by macroeconomic factors (e.g., inflation, regulatory news) and market sentiment. The 2021 peak aligns with institutional adoption and ETF approvals, while the 2022 crash correlates with interest rate hikes. The current price suggests cautious optimism but remains below the 2021 high, indicating unresolved market uncertainty. The blue trendline underscores the long-term upward trajectory despite short-term fluctuations. </details> Figure 22: SAR Signals for BTC/USD This indicator tells us that when the blue dots are below the candlesticks, we should buy, and when they are above, we should sell (first appearance of dots per sequence). The same reasoning applies: what is the point of selling in 2018? These indicators are ultimately only signals for day-trading, with the aim of making quick profits. However, statistics show that more than 70% of day-traders lose money. Yet, they all have access to all available indicators. Technical analysis would therefore seem useless both in the long term and in the short term, a priori. According to [Park and Irwin, 2007], the literature on the subject is inconclusive: some studies are positive, others negative, and others mixed. 2.3.3 Machine Learning Let’s now check the effectiveness of Machine Learning in predicting cryptocurrency prices. We will not test all algorithms, but only two. The first, Support Vector Machine classification, was introduced by [Cortes and Vapnik, 1995]. It consists of classifying "good" trades from "bad" trades. For this, we create a Python function getAverageAccuracy( $\Omega,n$ ), which takes as parameters $\Omega$ and $n$ the window for technical indicators and returns the average accuracy percentage of our model across all tested cryptocurrencies (over 100). The features considered are: price (OHLC), previous prices, previous returns, SAR, RSI, SMA, ADX, ATR, and 80% training dataset. The function, whose code is in Appendix G, returns 38%. This is low. Here are the confusion matrices for the 4 largest cryptocurrencies (read as "Perfect prediction on the top-left/bottom-right diagonal, inverse prediction on the bottom-left/top-right"): <details> <summary>extracted/6391907/images/btc-ml.png Details</summary>  ### Visual Description ## Heatmap: Unlabeled 3x3 Grid with Numerical Values and Color Coding ### Overview The image depicts a 3x3 grid where each cell contains a numerical value and is colored differently. There are no axis labels, legends, or explicit titles. The grid appears to represent categorical or comparative data, with color intensity potentially indicating magnitude or grouping. ### Components/Axes - **Grid Structure**: 3 rows × 3 columns. - **Cell Contents**: Each cell contains a single integer (3–13) centered in the cell. - **Color Coding**: Cells are colored in red, orange, purple, yellow, and dark blue. No legend is present to define color-to-value mapping. - **Absence of Labels**: No axis titles, row/column headers, or legends are visible. ### Detailed Analysis | Row \ Column | 1 | 2 | 3 | |--------------|---------|---------|---------| | **1** | 9 (red) | 10 (orange) | 4 (purple) | | **2** | 12 (orange) | 8 (purple) | 3 (dark blue) | | **3** | 13 (yellow) | 9 (red) | 3 (dark blue) | - **Values**: Numbers range from 3 (lowest) to 13 (highest). - **Color Distribution**: - **Red**: 9 (Row 1, Col 1), 9 (Row 3, Col 2). - **Orange**: 10 (Row 1, Col 2), 12 (Row 2, Col 1). - **Purple**: 4 (Row 1, Col 3), 8 (Row 2, Col 2). - **Yellow**: 13 (Row 3, Col 1). - **Dark Blue**: 3 (Row 2, Col 3), 3 (Row 3, Col 3). ### Key Observations 1. **Highest Value**: The cell with the maximum value (13) is located in the bottom-left corner (Row 3, Col 1), colored yellow. 2. **Lowest Values**: Two cells contain the minimum value (3), both in the bottom-right quadrant (Row 2, Col 3 and Row 3, Col 3), colored dark blue. 3. **Color-Value Relationship**: No clear gradient or pattern links colors to values (e.g., yellow is not consistently the highest, and dark blue is not exclusively the lowest). 4. **Clustering**: The top-left quadrant (Rows 1–2, Cols 1–2) contains higher values (8–12), while the bottom-right quadrant (Rows 2–3, Cols 2–3) has lower values (3–9). ### Interpretation - **Ambiguity in Color Coding**: Without a legend, the purpose of color differentiation remains unclear. Colors may represent categories (e.g., groups, statuses) or magnitudes, but this cannot be confirmed. - **Potential Patterns**: The highest value (13) and second-highest (12) are adjacent in the top-left quadrant, suggesting a possible regional or categorical dominance. The repeated low values (3) in the bottom-right quadrant might indicate a shared characteristic or constraint. - **Uncertainty**: The lack of axis labels, legends, or contextual text prevents definitive interpretation. The grid could represent anything from survey results to performance metrics, but the absence of metadata limits analysis. ### Conclusion This heatmap provides numerical data in a 3x3 grid with arbitrary color coding. While the values suggest variability, the absence of labels and legends makes it impossible to draw concrete conclusions. Further context is required to determine the grid’s purpose or the significance of its color scheme. </details> Figure 23: Confusion matrix for BTC/USD <details> <summary>extracted/6391907/images/eth-ml.png Details</summary>  ### Visual Description ## Heatmap: Unlabeled Numerical Distribution Grid ### Overview The image displays a 3x3 heatmap with numerical values embedded in colored cells. Colors transition from purple (low values) to yellow (high values), with no axis labels, legends, or contextual text present. The grid structure suggests a categorical comparison across rows and columns, though specific dimensions (e.g., time, categories) are undefined. ### Components/Axes - **Grid Structure**: 3 rows × 3 columns. - **Color Gradient**: - Purple → Orange → Yellow (increasing magnitude). - Dark blue appears for the lowest value (1). - **No Labels**: Rows, columns, or legend are unlabeled, limiting interpretability of categories. ### Detailed Analysis | Row \ Column | 1 | 2 | 3 | |--------------|-------|-------|-------| | **1** | 9 | 11 | 1 | | **2** | 12 | 17 | 3 | | **3** | 4 | 9 | 5 | - **Color Consistency**: - Highest value (17) is yellow (center cell). - Lowest value (1) is dark blue (top-right cell). - Intermediate values use orange/purple gradients. ### Key Observations 1. **Central Peak**: The middle cell (17) is the maximum value, surrounded by lower magnitudes. 2. **Lowest Value**: Top-right cell (1) is an outlier, significantly smaller than adjacent cells. 3. **Row Totals**: - Row 1: 21 (9 + 11 + 1) - Row 2: 32 (12 + 17 + 3) – **highest total**. - Row 3: 18 (4 + 9 + 5). 4. **Column Totals**: - Column 1: 25 (9 + 12 + 4) - Column 2: 37 (11 + 17 + 9) – **highest total**. - Column 3: 9 (1 + 3 + 5). ### Interpretation - **Data Distribution**: The heatmap reveals a **centralized peak** (17) with diminishing values toward the edges, suggesting a focal point of intensity. The top-right cell (1) is an anomalous outlier, potentially indicating an error or unique condition. - **Row/Column Dynamics**: - Row 2 dominates in magnitude, while Row 3 has the lowest aggregate. - Column 2 (middle) contains the two highest values (11, 17), reinforcing its central importance. - **Color Utility**: The gradient effectively visualizes magnitude, though the absence of a legend prevents precise quantification of color-to-value mapping. - **Contextual Limitations**: Without axis labels or legends, the grid’s purpose (e.g., time-series, categorical comparison) remains ambiguous. The data structure alone suggests a matrix of interrelated metrics, but external context is required for deeper analysis. </details> Figure 24: Confusion matrix for ETH/USD <details> <summary>extracted/6391907/images/ltc-ml.png Details</summary>  ### Visual Description ## Heatmap: Unlabeled 3x3 Grid with Numerical Values and Color Coding ### Overview The image depicts a 3x3 grid where each cell contains a numerical value and a distinct color. No axis labels, legends, or contextual text are present. The grid appears to represent a matrix of discrete data points, with colors potentially indicating categories or ranges, though no explicit mapping is provided. ### Components/Axes - **Grid Structure**: - 3 rows and 3 columns. - No axis titles, legends, or scale markers. - **Cell Contents**: - Each cell contains a single integer (e.g., 8, 10, 2) and a solid color (e.g., red, orange, blue). - Colors vary across cells but lack a defined legend or gradient scale. ### Detailed Analysis | Position | Value | Color | |----------------|-------|-----------| | Top-Left | 8 | Red | | Top-Center | 10 | Orange | | Top-Right | 2 | Blue | | Middle-Left | 12 | Orange | | Middle-Center | 13 | Yellow | | Middle-Right | 5 | Purple | | Bottom-Left | 8 | Red | | Bottom-Center | 8 | Red | | Bottom-Right | 5 | Purple | - **Numerical Range**: Values span from **2** (minimum) to **13** (maximum). - **Color Distribution**: - Red: 3 cells (values 8, 8, 8). - Orange: 2 cells (values 10, 12). - Blue: 1 cell (value 2). - Yellow: 1 cell (value 13). - Purple: 2 cells (values 5, 5). ### Key Observations 1. **Highest Value**: The cell at **Middle-Center** (13) is the only yellow cell, suggesting a potential outlier or unique category. 2. **Lowest Value**: The cell at **Top-Right** (2) is the only blue cell, possibly indicating a distinct group or threshold. 3. **Color Ambiguity**: Without a legend, the relationship between colors and values is unclear. For example: - Red cells contain mid-range values (8). - Orange cells contain higher values (10, 12). - Blue and yellow cells represent extremes (2, 13). 4. **Repetition**: The value **8** appears three times (all red), while **5** appears twice (both purple). ### Interpretation - **Data Structure**: The grid likely represents a categorical or grouped dataset, with colors serving as visual markers rather than a continuous scale. - **Trends**: - The central cell (13) is the highest value, surrounded by lower values (12, 5, 8). - The bottom row has two identical values (8, 8), suggesting possible grouping or repetition. - **Uncertainties**: - The purpose of color coding is undefined. - No contextual labels prevent interpretation of what the numbers represent (e.g., counts, scores, measurements). - **Notable Patterns**: - The top row shows a sharp drop from 10 to 2, while the middle row peaks at 13. - The bottom row has no variation in value (8, 8) but shares the same color (red). ### Conclusion This heatmap-like grid highlights variability in numerical values across a 3x3 matrix. However, the absence of axis labels, legends, or contextual text limits the ability to draw definitive conclusions. The colors may serve as categorical indicators, but their meaning remains speculative without additional information. The data suggests potential groupings or thresholds, but further clarification is required to validate interpretations. </details> Figure 25: Confusion matrix for LTC/USD <details> <summary>extracted/6391907/images/xrp-ml.png Details</summary>  ### Visual Description ## Heatmap: Value Distribution Across Rows and Columns ### Overview The image is a 3x3 heatmap with numerical values in colored cells. The rows are labeled with values (10, 10, 3) in the first column, and the columns are labeled with values (5, 8, 13) in the top row. Each cell contains a numerical value, and the color of the cell corresponds to the magnitude of the value, as defined by a legend. ### Components/Axes - **Rows**: Labeled with values **10**, **10**, and **3** (leftmost column). - **Columns**: Labeled with values **5**, **8**, and **13** (top row). - **Legend**: - **Dark purple**: Low values (3–5). - **Medium purple**: Medium values (6–8). - **Yellow**: High values (9–13). - **Color Gradient**: The heatmap transitions from dark purple (low) to yellow (high), with red appearing in the middle range (6–8). ### Detailed Analysis | Row \ Column | 5 | 8 | 13 | |--------------|-------|-------|-------| | **10** | 10 (red) | 5 (dark purple) | 9 (red) | | **10** | 10 (red) | 8 (medium purple) | 13 (yellow) | | **3** | 3 (dark purple) | 6 (medium purple) | 7 (medium purple) | - **Row 1 (10)**: - Column 5: **10** (red, medium value). - Column 8: **5** (dark purple, low value). - Column 13: **9** (red, medium value). - **Row 2 (10)**: - Column 5: **10** (red, medium value). - Column 8: **8** (medium purple, medium value). - Column 13: **13** (yellow, high value). - **Row 3 (3)**: - Column 5: **3** (dark purple, low value). - Column 8: **6** (medium purple, medium value). - Column 13: **7** (medium purple, medium value). ### Key Observations 1. **Highest Value**: **13** (yellow) in the bottom-right cell (Row 3, Column 13). 2. **Lowest Value**: **3** (dark purple) in the bottom-left cell (Row 3, Column 5). 3. **Medium Values**: Most cells (6–8) are medium purple, except for the red cells (9, 10). 4. **Color Discrepancy**: The legend specifies **medium purple** for values 6–8, but the cells with values **9** and **10** are red, which is not explicitly listed in the legend. This suggests a potential inconsistency in the color mapping. ### Interpretation - The heatmap illustrates a distribution of values across rows and columns, with the highest value (13) concentrated in the bottom-right corner and the lowest (3) in the bottom-left. - The medium values (6–8) dominate the central cells, while the red cells (9, 10) introduce ambiguity due to their color not aligning with the legend. This could indicate an error in the legend or an intentional design choice to highlight specific ranges. - The row labeled **3** (bottom row) shows a gradual increase in values from left to right (3 → 6 → 7), suggesting a trend of increasing magnitude. - The column labeled **13** (rightmost column) contains the highest values (9, 13, 7), indicating a potential correlation between column position and value magnitude. ### Conclusion The heatmap effectively visualizes value distributions, but the red cells (9, 10) deviate from the legend’s defined color ranges. This inconsistency may require clarification to ensure accurate interpretation. The data suggests a spatial pattern where higher values cluster in specific regions, though further analysis is needed to resolve the color-legend mismatch. </details> Figure 26: Confusion matrix for XRP/USD The second model is ARIMA, introduced by [Box et al., 2015], and it aims to predict future trends. The results of our model are as follows: <details> <summary>extracted/6391907/images/arima.png Details</summary>  ### Visual Description ## ARIMA Model Results: ARIMA(4, 1, 0) for D.Close ### Overview The image presents the results of an ARIMA(4, 1, 0) model applied to the dependent variable "D.Close" (likely a time series of closing prices). The analysis includes model parameters, statistical metrics, and diagnostic outputs. The data spans 291 observations, with a sample period from 05-03-2021 to 02-17-2022. ### Components/Axes - **Model Specification**: - Dependent Variable: `D.Close` - Model: `ARIMA(4, 1, 0)` - Method: `css-mle` (conditional sum of squares with maximum likelihood estimation) - Number of Observations: `291` - Log Likelihood: `-2586.480` - Standard Deviation of Innovations: `1753.258` - AIC: `5184.959` - BIC: `5206.999` - HQIC: `5193.789` - Date: `Mon, 02 May 2022` - Time: `03:16:01` - Sample Period: `05-03-2021` to `02-17-2022` - **Coefficients Table**: - **Columns**: `coef` (coefficient), `std err` (standard error), `z` (z-score), `P>|z|` (p-value), `[0.025` (lower 95% CI), `[0.975` (upper 95% CI) - **Rows**: - `const`: `-55.2311` (std err: `101.025`, z: `-0.547`, p: `0.585`) - `ar.L1.D.Close`: `-0.0541` (std err: `0.059`, z: `-0.923`, p: `0.357`) - `ar.L2.D.Close`: `-0.0278` (std err: `0.059`, z: `-0.470`, p: `0.638`) - `ar.L3.D.Close`: `-0.0297` (std err: `0.060`, z: `-0.499`, p: `0.618`) - `ar.L4.D.Close`: `0.0947` (std err: `0.060`, z: `1.592`, p: `0.113`) - **Roots Table**: - **Columns**: `Real`, `Imaginary`, `Modulus`, `Frequency` - **Rows**: - `AR.1`: Real: `-1.7241`, Imaginary: `-0.0000j`, Modulus: `1.7241`, Frequency: `-0.5000` - `AR.2`: Real: `0.0309`, Imaginary: `-1.7600j`, Modulus: `1.7603`, Frequency: `-0.2472` - `AR.3`: Real: `0.0309`, Imaginary: `+1.7600j`, Modulus: `1.7603`, Frequency: `0.2472` - `AR.4`: Real: `1.9763`, Imaginary: `-0.0000j`, Modulus: `1.9763`, Frequency: `0.0000` ### Detailed Analysis 1. **Model Parameters**: - The ARIMA(4,1,0) model includes 4 autoregressive terms (AR.1–AR.4), 1 differencing step (d=1), and no moving average terms (q=0). - The constant term (`const`) has a coefficient of `-55.2311`, but its p-value (`0.585`) suggests it is not statistically significant. 2. **Autoregressive Coefficients**: - **AR.1**: `-0.0541` (p=0.357) – non-significant. - **AR.2**: `-0.0278` (p=0.638) – non-significant. - **AR.3**: `-0.0297` (p=0.618) – non-significant. - **AR.4**: `0.0947` (p=0.113) – marginally significant (close to 0.05). 3. **Roots Analysis**: - **AR.1**: Root modulus = `1.7241` (outside the unit circle, indicating potential instability). - **AR.2/AR.3**: Complex conjugate roots with modulus `1.7603` (also outside the unit circle). - **AR.4**: Root modulus = `1.9763` (outside the unit circle). - All roots have frequencies of `-0.5000`, `-0.2472`, `0.2472`, and `0.0000`, respectively. 4. **Model Fit Metrics**: - AIC (`5184.959`), BIC (`5206.999`), and HQIC (`5193.789`) are relatively high, suggesting the model may not be optimal. - The standard deviation of innovations (`1753.258`) indicates the average error in the model’s predictions. ### Key Observations - **Non-Significant Coefficients**: Most AR terms (AR.1–AR.3) have p-values > 0.05, implying they do not significantly contribute to the model. - **Root Instability**: All roots have moduli > 1, which may indicate the model is not stable or the series is not properly differenced. - **Marginal Significance of AR.4**: The AR.4 coefficient (`0.0947`) is the only term with a p-value near 0.05, suggesting a weak but potentially relevant relationship. ### Interpretation The ARIMA(4,1,0) model for `D.Close` shows limited statistical significance in its autoregressive terms, with most coefficients failing to meet the 0.05 threshold. The roots’ moduli exceeding 1 suggest potential instability, which could undermine the model’s reliability. While the AR.4 term is marginally significant, the high AIC/BIC values indicate the model may not adequately capture the underlying data structure. The non-significant constant term further questions the model’s explanatory power. These results suggest the need for further model refinement, such as adjusting the order of differencing (d) or exploring alternative models (e.g., including moving average terms). The high standard deviation of innovations (`1753.258`) also highlights the model’s poor fit to the data. </details> Figure 27: Results of the ARIMA model 3 The Cryptocurrency Market is Inefficient In 1981, Robert Shiller [Shiller, 1980] showed a higher volatility than that predicted by the rational behavior of agents. Shiller concluded that no rationality could explain the observed volatility, which ultimately had no link with dividend expectations. Thus, if the market is inefficient, it is possible to achieve performances superior to the market. 3.1 Robert Shiller and the Notion of an Inefficient Market in Terms of Arbitrage This section deals with elements that prove that the Bitcoin market admits arbitrage opportunities. For example, we observe that the price of Bitcoin varies from one exchange to another. This is even more true for the altcoin market. Intuitively, we can imagine that the price will tend to move closer to the average price across exchanges. 3.1.1 Volatility and Expected Dividends In his book, [Shiller, 2015], Shiller shows the difference between stock price volatility and expected dividends: <details> <summary>extracted/6391907/images/shiller-plot.png Details</summary>  ### Visual Description ## Line Chart: Real S&P Stock Price Index, Earnings, and Dividends (1871 = 100) ### Overview The chart visualizes four economic metrics over time: 1. **Price** (red line) 2. **Earnings** (blue line) 3. **Dividends** (green line) 4. **Interest Rates** (black line) The x-axis spans 1870–2010, while the y-axis for Price/Earnings/Dividends ranges from 0 to 2,500 (base year 1871 = 100), and Interest Rates from 0% to 100%. The legend is positioned on the right, with labels matching line colors. The source is cited as `irrationalexuberance.com/shiller_downloads/ie_data.xls`. --- ### Components/Axes - **X-axis (Year)**: - Labels: 1870, 1890, 1910, 1930, 1950, 1970, 1990, 2010. - Scale: Linear progression from 1870 to 2010. - **Y-axis (Real S&P Stock Price Index, Earnings, and Dividends)**: - Labels: 0, 500, 1,000, 1,500, 2,000, 2,500. - Base year: 1871 = 100. - **Secondary Y-axis (Interest Rate %)**: - Labels: 0%, 20%, 40%, 60%, 80%, 100%. - **Legend**: - Position: Right side of the chart. - Labels: - Red = Price - Blue = Earnings - Green = Dividends - Black = Interest Rates --- ### Detailed Analysis #### Price (Red Line) - **Trend**: - Starts near 0 in 1870, fluctuates modestly until ~1990. - Sharp upward spike from ~1995 to 2000 (peak ~2,000). - Plummets to ~1,500 in 2002 (post-dot-com crash). - Recovers to ~2,000 by 2008, then drops sharply during the 2008 financial crisis. - **Key Data Points**: - 1995: ~1,000 - 2000: ~2,000 (peak) - 2002: ~1,500 - 2008: ~2,000 (pre-crash) - 2010: ~1,800 #### Earnings (Blue Line) - **Trend**: - Mirrors Price but with less volatility. - Gradual rise from ~100 in 1870 to ~800 in 2000. - Drops to ~500 in 2002, recovers to ~1,000 by 2008. - **Key Data Points**: - 1995: ~300 - 2000: ~800 - 2002: ~500 - 2008: ~1,000 #### Dividends (Green Line) - **Trend**: - Slow, steady growth from ~50 in 1870 to ~400 in 2008. - Sharp drop to ~200 in 2002, partial recovery to ~400 by 2008. - **Key Data Points**: - 1995: ~100 - 2000: ~300 - 2002: ~200 - 2008: ~400 #### Interest Rates (Black Line) - **Trend**: - Peaks at ~15% in 1980, then declines to ~5% by 2000. - Drops to ~2% in 2008, then rises slightly to ~4% by 2010. - **Key Data Points**: - 1980: ~15% - 2000: ~5% - 2008: ~2% - 2010: ~4% --- ### Key Observations 1. **Price Volatility**: - Price exhibits extreme volatility, with a 100%+ surge from 1995–2000 and a 25% drop during the 2008 crisis. 2. **Earnings-Dividend Correlation**: - Earnings and Dividends track Price but with dampened amplitude, suggesting corporate profitability lags behind market speculation. 3. **Interest Rate Inversion**: - Interest Rates peak in 1980 (15%) and decline inversely with Price until 2008, then rise again. 4. **2008 Crisis Impact**: - All metrics collapse in 2008, but Price recovers faster than Earnings/Dividends. --- ### Interpretation - **Economic Bubbles and Crashes**: - The 1995–2000 Price surge aligns with the dot-com bubble, while the 2008 crash reflects the global financial crisis. - **Corporate Profitability vs. Market Speculation**: - Earnings and Dividends grow steadily but remain subordinate to speculative Price movements, highlighting a disconnect between fundamentals and market sentiment. - **Monetary Policy Influence**: - Falling Interest Rates (post-1980) correlate with rising Price, suggesting accommodative monetary policy fueled asset inflation. - **Anomalies**: - The 2002 Price drop (post-dot-com) outpaces Earnings/Dividends, indicating market overreaction. - The 2008–2010 Interest Rate rebound contradicts typical post-crisis easing, possibly reflecting quantitative easing effects. The chart underscores the S&P 500’s transformation from a stable index (1870–1990) to a highly volatile, speculation-driven asset (post-1990), with earnings and dividends struggling to keep pace. </details> Figure 28: Evolution of the S&P500 and dividends According to him: "the price-to-earnings ratio is still (as of 2005) far from its historical average from the mid-20th century. Investors place too much trust in the market and overestimate the positive developments of their investments without sufficiently hedging against a market downturn." It is therefore difficult to determine whether the crypto market is inefficient based solely on this information, as cryptocurrencies do not pay dividends. 3.1.2 Behavioral Finance and Market Anomalies Shiller introduces the concept of behavioral finance. In the crypto market, we mainly think of herd behavior: investors buy simply because other investors are buying. This phenomenon is less visible in day-trading because time scales are too short to draw conclusions about the trend. Indeed, the 2017 bubble still took some time to form and partially burst. 3.1.3 Speculative Bubbles When it comes to cryptocurrencies, speculative bubbles are often mentioned. It is true that cryptocurrencies provide fertile ground for such phenomena, but this only matters for medium-term investors. A long-term investor will mainly seek to minimize diversifiable risk through cryptocurrencies, while a short-term trader will hope to enter the market before a hype event. Moreover, these hypes can sometimes be artificially created by one or several people, sometimes even behind fraudulent projects. Over time, as projects repeat, fraud risks decrease, and hypes also tend to diminish, making the crypto market increasingly efficient and reducing the possibility of bubbles. 3.2 Informational Inefficiency We will look at scenarios where information asymmetries allow an individual or a group to achieve superior returns to the market. In such situations, cryptocurrency prices do not reflect all available information. 3.2.1 Market Manipulation The most famous example is the public use of Twitter by Elon Musk, with each of his crypto-related tweets causing abrupt movements in the crypto market. By deduction, we can imagine similar scenarios involving other public figures, broker managers, intermediaries, etc. 3.2.2 Pump & Dump Pump & Dump was a strong practice during the early days of crypto hype. It consisted of gathering the largest possible group of users around a well-promoted cryptocurrency. The initiator of the movement would encourage the entire community to engage with the project for a single purpose: to artificially inflate the price of the cryptocurrency. Once the cryptocurrency reached a satisfactory price, the initiator—who had taken care to invest as much as possible when the crypto was worth nothing—would sell everything and exit the project. This type of phenomenon was also seen with ICOs. 3.2.3 Natural Language Processing Natural Language Processing (NLP) can be used to analyze market sentiment without manually reading content. For example, the bot, whose code is in Appendix H, returns the following results: <details> <summary>x3.png Details</summary>  ### Visual Description ## Transaction Log: Buy/Sell Activity with Profit Margins ### Overview The image displays a chronological list of 24 trading transactions, alternating between BUY and SELL orders. Each entry includes: - Transaction amount (BUY/Sell price) - Profit/loss percentage - Color-coded formatting (purple for numbers, blue for labels) ### Components/Axes - **Labels**: - "BUY" and "SELL" in uppercase (purple text) - "Profit = X%" (blue text) - **Formatting**: - Buy prices: Purple text, 8-10 digit format (e.g., `38088.01953125`) - Sell prices: Purple text, 8-10 digit format (e.g., `38038.94921875`) - Profit percentages: Blue text with ±0.001% precision ### Detailed Analysis 1. **Transaction Pattern**: - Alternating BUY/SELL sequence (12 BUY, 12 SELL) - Buy prices generally higher than sell prices (except 3 instances) - Profit percentages range from **-0.376%** to **+0.924%** 2. **Key Transactions**: - **Highest Profit**: - BUY: 35800.88671875 - SELL: 36131.80078125 - Profit: **+0.924%** - **Largest Loss**: - BUY: 36106.11328125 - SELL: 35970.46484375 - Profit: **-0.376%** 3. **Numerical Distribution**: - Buy prices: 35,119.19140625 to 38,840.0859375 - Sell prices: 35,562.3359375 to 38,811.48046875 - Profit range: -0.376% to +0.924% ### Key Observations - **Profit Variance**: - 6 transactions show positive returns (max +0.924%) - 8 transactions show negative returns (min -0.376%) - 2 transactions break the BUY/SELL pattern (36123.421875 and 36123.828125) - **Price Correlation**: - Higher buy prices correlate with larger absolute profit/loss values - Example: 38,840.0859375 BUY → 38,811.48046875 SELL (-0.074%) ### Interpretation This dataset represents a trading strategy with mixed outcomes: 1. **Risk-Return Profile**: - Small average profit margin (net -0.023% across all trades) - High volatility in individual trade outcomes - Suggests speculative rather than conservative trading approach 2. **Pattern Anomalies**: - Two consecutive BUY orders (36123.421875 and 36123.828125) break the alternating pattern - 38,840.0859375 BUY price matches 38,840.984375 BUY price within 0.001% tolerance 3. **Technical Implications**: - Decimal precision suggests high-frequency trading or algorithmic execution - Profit calculation methodology appears consistent (likely (Sell-Buy)/Buy * 100) - Data format indicates possible binary/float representation artifacts The log demonstrates a trading system with tight margin management but inconsistent profitability, potentially indicating either market volatility challenges or suboptimal execution timing. </details> Figure 29: Results from the NLP trading bot 3.3 Operational Inefficiency The price of Bitcoin can be predicted if one knows in advance the factors likely to influence the network as a whole, or a significant part of it. We will explore whether one or several elements hindering cryptocurrency exchanges can induce market movements. 3.3.1 At the Macroscopic Scale We can take the example of countries that ban cryptocurrencies. These bans have a notable effect on liquidity, or cases of massive adoption like in El Salvador or the Marshall Islands, or the future rise of CBDCs (such as the digital euro). Environmental concerns, which are becoming a major issue, also hinder liquidity, as cryptocurrencies require a significant amount of electricity resources. 3.3.2 At the Mesoscopic Scale Certain cryptocurrencies can have a negative impact on others. For example, Monero, with its private blockchain, can very well absorb all the demand for cryptocurrencies that also aim to respect user privacy. The same goes for issues related to transaction speed. 3.3.3 At the Microscopic Scale There are usage barriers to crypto-assets among households that strongly impact the markets, such as the prohibition of cryptocurrency usage for minors, broker restrictions regarding certain trading positions, security risks and broker compliance concerning suspicious activities, tainted bitcoins, and money laundering (KYC/AML requirements), the impact of taxation on crypto-related capital gains, and various hacks. It can be observed that these phenomena have much less impact on the markets than macro or even meso factors. However, households and discretionary traders still represent a large part of the crypto market landscape. 4 Conclusion In conclusion, by default, it is not possible to predict Bitcoin since it is an asset very similar in nature to others (notably the stock market), but, as with any market, there are moments when the market is inefficient, and thus it is possible to profit from these moments and predict Bitcoin prices accurately. Among the limitations, we focus only on the spot market, we do not consider the influence of other cryptocurrencies, and we are limited in our expertise in time series analysis. Among public policy recommendations, we agree with the view of [Brito et al., 2014] regarding the regulation of brokers and particularly of derivatives products, which are becoming increasingly significant in the crypto market. Appendix A isRandomBetter( $\Omega,n,k$ ) ⬇ 1 # The set Omega is a subset of all cryptocurrencies on the market (between 10,000 and 20,000) 2 Omega = [’1INCH-USD’, ’AAVE-USD’, ’ACH-USD’, ’ADA-USD’, ’AERGO-USD’, ’AGLD-USD’, 3 ’AIOZ-USD’, ’ALCX-USD’, ’ALGO-USD’, ’ALICE-USD’, ’AMP-USD’, ’ANKR-USD’, 4 ’APE-USD’, ’API3-USD’, ’ARPA-USD’, ’ASM-USD’, ’ATOM-USD’, ’AUCTION-USD’, 5 ’AVAX-USD’, ’AVT-USD’, ’AXS-USD’, ’BADGER-USD’, ’BAL-USD’, ’BAND-USD’, 6 ’BAT-USD’, ’BCH-USD’, ’BICO-USD’, ’BLZ-USD’, ’BNT-USD’, ’BOND-USD’, 7 ’BTC-USD’, ’BTRST-USD’, ’CHZ-USD’, ’CLV-USD’, ’COMP-USD’, 8 ’COTI-USD’, ’COVAL-USD’, ’CRO-USD’, ’CRPT-USD’, ’CRV-USD’, ’CTSI-USD’, 9 ’CTX-USD’, ’CVC-USD’, ’DAI-USD’, ’DASH-USD’, ’DDX-USD’, ’DESO-USD’, 10 ’DIA-USD’, ’DNT-USD’, ’DOGE-USD’, ’DOT-USD’, ’ENJ-USD’, ’ENS-USD’, 11 ’EOS-USD’, ’ERN-USD’, ’ETC-USD’, ’ETH-USD’, ’FARM-USD’, 12 ’FET-USD’, ’FIDA-USD’, ’FIL-USD’, ’FORTH-USD’, ’FOX-USD’, ’FX-USD’, 13 ’GALA-USD’, ’GFI-USD’, ’GLM-USD’, ’GNT-USD’, ’GODS-USD’, 14 ’GRT-USD’, ’GTC-USD’, ’GYEN-USD’, ’HIGH-USD’, ’ICP-USD’, ’IDEX-USD’, 15 ’IMX-USD’, ’INV-USD’, ’IOTX-USD’, ’JASMY-USD’, ’KEEP-USD’, ’KNC-USD’, 16 ’KRL-USD’, ’LCX-USD’, ’LINK-USD’, ’LOOM-USD’, ’LPT-USD’, ’LQTY-USD’, 17 ’LRC-USD’, ’LTC-USD’, ’MANA-USD’, ’MASK-USD’, ’MATIC-USD’, ’MCO2-USD’, 18 ’MDT-USD’, ’MINA-USD’, ’MIR-USD’, ’MKR-USD’, ’MLN-USD’, ’MPL-USD’, 19 ’MUSD-USD’, ’NCT-USD’, ’NKN-USD’, ’NMR-USD’, ’NU-USD’, ’OGN-USD’, 20 ’OMG-USD’, ’ORCA-USD’, ’ORN-USD’, ’OXT-USD’, ’PERP-USD’, 21 ’PLA-USD’, ’PLU-USD’, ’POLS-USD’, ’POLY-USD’, ’POWR-USD’, ’PRO-USD’, 22 ’QNT-USD’, ’QSP-USD’, ’QUICK-USD’, ’RAD-USD’, ’RAI-USD’, ’RARI-USD’, 23 ’RBN-USD’, ’REN-USD’, ’REP-USD’, ’REQ-USD’, ’RGT-USD’, ’RLC-USD’, 24 ’RLY-USD’, ’RNDR-USD’, ’SHIB-USD’, ’SHPING-USD’, ’SKL-USD’, ’SNT-USD’, 25 ’SNX-USD’, ’SOL-USD’, ’SPELL-USD’, ’STORJ-USD’, ’STX-USD’, ’SUKU-USD’, 26 ’SUPER-USD’, ’SUSHI-USD’, ’SYN-USD’, ’TBTC-USD’, ’TRAC-USD’, ’TRB-USD’, 27 ’TRIBE-USD’, ’TRU-USD’, ’UMA-USD’, ’UNFI-USD’, ’UNI-USD’, ’UPI-USD’, 28 ’USDC-USD’, ’USDT-USD’, ’UST-USD’, ’VGX-USD’, ’WBTC-USD’, 29 ’WCFG-USD’, ’WLUNA-USD’, ’XLM-USD’, ’XRP-USD’, ’XTZ-USD’, ’XYO-USD’, 30 ’YFI-USD’, ’YFII-USD’, ’ZEC-USD’, ’ZEN-USD’, ’ZRX-USD’] ⬇ 1 # This function returns a list of returns for each asset 2 def loadChanges (Omega): 3 changes = [] 4 for asset in Omega: 5 # Import time 6 time. sleep (1) 7 # We use the yFinance library to get the data 8 df = yf. download (asset, period = ’2y’, interval = ’1d’, progress = False) 9 if df. empty: 10 continue 11 oneYear = df. loc [’2021-01-01’: ’2022-01-01’] 12 if oneYear. empty or len (df) < 360: 13 continue 14 # Import pandas 15 s = pd. Series (list (oneYear [’Close’])) 16 if not s [s. isin ([0])]. empty: 17 continue 18 else: 19 start = oneYear. iloc [0][’Close’] 20 final = oneYear. iloc [-1][’Close’] 21 change = ((final - start)/ start)*100 22 changes. append (change) 23 print ("len Valid assets : ", len (changes), " (only consider the 1st print!)") 24 return changes ⬇ 1 # This function returns the average of returns 2 def getMeanChanges (Omega): 3 changes = loadChanges (Omega) 4 return sum (changes)/ len (changes) ⬇ 1 # This function randomly selects a portfolio of crypto-assets among those available in Omega 2 def generateRandomPortfolio (Omega, k): 3 randomPortfolio = [] 4 for _ in range (k): 5 # Import random 6 randomPortfolio. append (random. choice (Omega)) 7 return randomPortfolio ⬇ 1 # This function returns the percentage of portfolios with an average return higher than the average return 2 # of the assets in Omega 3 def getPercentageHigherThanAverage (Omega, NbIter, k): 4 nbHigher = 0 5 averageReturns = getMeanChanges (Omega) 6 for _ in range (NbIter): 7 randomPortfolio = generateRandomPortfolio (Omega, k) 8 randomAverage = getMeanChanges (randomPortfolio) 9 if randomAverage > averageReturns: 10 nbHigher += 1 11 perc = round (nbHigher / NbIter *100) 12 print (f "{k} asset(s) in {NbIter} random portfolio(s)") 13 print ("Average returns :", round (averageReturns)) 14 print (f "Percentage of random portfolios above the average : {perc}%") 15 return perc ⬇ 1 # This function returns whether a random portfolio outperforms an average portfolio, 2 # provided that 51% or more random portfolios outperform the average 3 def isRandomBetter (list, NbIter, k): 4 perc = getPercentageHigherThanAverage (list, NbIter, k) 5 if perc < 51: 6 return False 7 else: 8 return True ⬇ 1 # Tests 2 print ("Test 1") 3 print (isRandomBetter (Omega, 10, 10)) 4 print ("Test 2") 5 print (isRandomBetter (Omega, 10, 20)) 6 print ("Test 3") 7 print (isRandomBetter (Omega, 20, 10)) 8 print ("Test 4") 9 print (isRandomBetter (Omega, 20, 20)) 10 print ("Test 5") 11 print (isRandomBetter (Omega, 20, 30)) 12 print ("Test 6") 13 print (isRandomBetter (Omega, 30, 20)) 14 print ("Test 7") 15 print (isRandomBetter (Omega, 30, 30)) 16 print ("Test 8") 17 print (isRandomBetter (Omega, 30, 10)) 18 print ("Test 9") 19 print (isRandomBetter (Omega, 10, 30)) 20 print ("Test 10") 21 print (isRandomBetter (Omega, 40, 5)) Appendix B isSMABetter( $\Omega,n,r$ ) ⬇ 1 # This function returns True if the average return of the SMA strategy 2 # is higher than the average of both hold and random strategies 3 def isSMABetter (Omega, n, r): 4 validAssets = 0 5 SMARets = [] 6 HoldRets = [] 7 RandomRets = [] 8 nbBetter = 0 9 for asset in Omega: 10 sma_return = getSMAReturn (asset, n, r) 11 if not sma_return: 12 continue 13 else: 14 SMARets. append (sma_return) 15 hold_return = getHoldReturn (asset) 16 if not hold_return: 17 continue 18 else: 19 HoldRets. append (hold_return) 20 random_return = getRandomReturn (asset) 21 if not random_return: 22 continue 23 else: 24 RandomRets. append (random_return) 25 if sma_return > hold_return and sma_return > random_return: 26 nbBetter += 1 27 validAssets += 1 28 29 sma_average = round (sum (SMARets) / len (SMARets)) 30 hold_average = round (sum (HoldRets) / len (HoldRets)) 31 random_average = round (sum (RandomRets) / len (RandomRets)) 32 print ("Number of valid assets : ", validAssets) 33 print ("SMA average : ", sma_average) 34 print ("Hold average : ", hold_average) 35 print ("Random average : ", random_average) 36 perc = round (nbBetter / validAssets *100) 37 print (f "{perc}% of assets do better with SMA.") 38 if perc < 50: 39 return False 40 else: 41 return True Appendix C getHoldReturn(asset) ⬇ 1 # This function returns the return of the asset "asset" with the hold strategy 2 def getHoldReturn (asset): 3 df = yf. download (asset, period = ’2y’, interval = ’1d’, progress = False) 4 if df. empty: 5 return False 6 oneYear = df. loc [’2021-01-01’: ’2022-01-01’] 7 s = pd. Series (list (oneYear [’Close’])) 8 if oneYear. empty or len (oneYear) < 360 or not s [s. isin ([0])]. empty: 9 return False 10 else: 11 start = oneYear. iloc [0][’Close’] 12 if start == 0: 13 return False 14 else: 15 final = oneYear. iloc [-1][’Close’] 16 return round (((final - start)/ start)*100) Appendix D getSMAReturn(asset, n, r) ⬇ 1 # This function returns the sum of daily returns 2 # of the asset "asset" with the SMA trading strategy 3 def getSMAReturn (asset, n, r): 4 range = 1+(r /100) 5 df = yf. download (asset, period = ’2y’, interval = ’1d’, progress = False) 6 if df. empty: 7 return False 8 oneYear = df. loc [’2021-01-01’: ’2022-01-01’] 9 s = pd. Series (list (oneYear [’Close’])) 10 if oneYear. empty or len (oneYear) < 360 or not s [s. isin ([0])]. empty: 11 return False 12 else: 13 oneYear [’SMA’] = oneYear [’Close’]. shift (1). rolling (window = n). mean () 14 oneYear [’SMAhigh’] = oneYear [’SMA’]* range 15 oneYear [’SMAlow’] = oneYear [’SMA’]/ range 16 oneYear [’Signal’] = 0 17 oneYear. loc [oneYear [’Close’] > oneYear [’SMAhigh’], ’Signal’] = -1 18 oneYear. loc [oneYear [’Close’] < oneYear [’SMAlow’], ’Signal’] = 1 19 oneYear [’Change’] = ((oneYear [’Close’]- oneYear [’Close’]. shift (1))/ oneYear [’Close’]. shift (1))*100 20 oneYear [’DayReturn’] = oneYear [’Change’]* oneYear [’Signal’] 21 ret = round (oneYear [’DayReturn’]. sum ()) 22 return ret Appendix E getRandomReturn(asset) ⬇ 1 # This function returns the sum of daily returns 2 # of the asset "asset" with a random trading strategy 3 def getRandomReturn (asset): 4 df = yf. download (asset, period = ’2y’, interval = ’1d’, progress = False) 5 if df. empty: 6 return False 7 oneYear = df. loc [’2021-01-01’: ’2022-01-01’] 8 s = pd. Series (list (oneYear [’Close’])) 9 if oneYear. empty or len (oneYear) < 360 or not s [s. isin ([0])]. empty: 10 return False 11 else: 12 oneYear [’Signal’] = 0 13 oneYear [’Random’] = [random. randint (1,9) for _ in oneYear. index] 14 oneYear. loc [oneYear [’Random’] > 6, ’Signal’] = 1 15 oneYear. loc [oneYear [’Random’] < 4, ’Signal’] = -1 16 oneYear [’Change’] = ((oneYear [’Close’]- oneYear [’Close’]. shift (1))/ oneYear [’Close’]. shift (1))*100 17 oneYear [’DayReturn’] = oneYear [’Change’]* oneYear [’Signal’] 18 return round (oneYear [’DayReturn’]. sum ()) Appendix F getRandomPerc( $\Omega$ ) ⬇ 1 # This function returns the percentage of assets that follow a random walk 2 def getPercRandom (Omega): 3 nbRandom = 0 4 nbTotal = 0 5 for asset in Omega: 6 time. sleep (1) 7 df = yf. download (asset, period = ’max’, interval = ’1d’, progress = False) 8 if df. empty: 9 continue 10 s = pd. Series (list (df [’Close’])) 11 if not s [s. isin ([0])]. empty or len (df) < 100: 12 continue 13 else: 14 nbTotal += 1 15 pval = adfuller (df [’Close’])[1] 16 if pval > 0.05: 17 nbRandom +=1 18 perc = nbRandom / nbTotal *100 19 return perc Appendix G getAverageAccuracy( $\Omega,n$ ) ⬇ 1 # This function returns the average accuracy percentage of our machine learning model 2 def getAverageAccuracy (Omega, n): 3 accuracies = [] 4 for asset in Omega: 5 df = yf. download (asset, period = ’1y’, interval = ’1d’, progress = False) 6 df = df. drop (df [df [’Volume’] == 0]. index) 7 df [’RSI’] = ta. RSI (np. array (df [’Close’]. shift (1)), timeperiod = n) 8 df [’SMA’] = df [’Close’]. shift (1). rolling (window = n). mean () 9 df [’Corr’] = df [’Close’]. shift (1). rolling (window = n). corr (df [’SMA’]. shift (1)) 10 df [’SAR’] = ta. SAR (np. array (df [’High’]. shift (1)), np. array (df [’Low’]. shift (1)), 0.2, 0.2) 11 df [’ADX’] = ta. ADX (np. array (df [’High’]. shift (1)), np. array (df [’Low’]. shift (1)), np. array (df [’Close’]. shift (1)), timeperiod = n) 12 df [’ATR’] = ta. ATR (np. array (df [’High’]. shift (1)), np. array (df [’Low’]. shift (1)), np. array (df [’Close’]. shift (1)), timeperiod = n) 13 df [’PH’] = df [’High’]. shift (1) 14 df [’PL’] = df [’Low’]. shift (1) 15 df [’PC’] = df [’Close’]. shift (1) 16 df [’O-O’] = df [’Open’] - df [’Open’]. shift (1) 17 df [’O-C’] = df [’Open’] - df [’PC’]. shift (1) 18 df [’Ret’] = (df [’Open’]. shift (-1) - df [’Open’]) / df [’Open’] 19 for i in range (1, n): 20 df [’r%i’ % i] = df [’Ret’]. shift (i) 21 df. loc [df [’Corr’] < -1, ’Corr’] = -1 22 df. loc [df [’Corr’] > 1, ’Corr’] = 1 23 df = df. dropna () 24 t = 0.8 25 split = int (t * len (df)) 26 df [’Signal’] = 0 27 df. loc [df [’Ret’] > df [’Ret’][: split]. quantile (q =0.66), ’Signal’] = 1 28 df. loc [df [’Ret’] < df [’Ret’][: split]. quantile (q =0.34), ’Signal’] = -1 29 X = df. drop ([’Close’, ’Adj Close’, ’Signal’, ’High’, ’Low’, ’Volume’, ’Ret’], axis =1) 30 y = df [’Signal’] 31 c = [10,100,1000,10000,100000,100000] 32 g = [1 e -4,1 e -3,1 e -2,1 e -1,1 e0] 33 p = {’svc__C’: c, ’svc__gamma’: g, ’svc__kernel’: [’rbf’]} 34 s = [(’s’, StandardScaler ()), (’svc’, SVC ())] 35 pp = Pipeline (s) 36 rcv = RandomizedSearchCV (pp, p, cv = TimeSeriesSplit (n_splits =2)) 37 rcv. fit (X. iloc [: split], y. iloc [: split]) 38 c = rcv. best_params_ [’svc__C’] 39 k = rcv. best_params_ [’svc__kernel’] 40 g = rcv. best_params_ [’svc__gamma’] 41 cls = SVC (C = c, kernel = k, gamma = g) 42 S = StandardScaler () 43 cls. fit (S. fit_transform (X. iloc [: split]), y. iloc [: split]) 44 y_predict = cls. predict (S. transform (X. iloc [split:])) 45 df [’Pred_Signal’] = 0 46 df. iloc [: split, df. columns. get_loc (’Pred_Signal’)] = pd. Series ( 47 cls. predict (S. transform (X. iloc [: split])). tolist ()) 48 df. iloc [split:, df. columns. get_loc (’Pred_Signal’)] = y_predict 49 df [’Ret1’] = df [’Ret’] * df [’Pred_Signal’] 50 cr = classification_report (y [split:], y_predict, output_dict = True) 51 accuracies. append (cr [’accuracy’]) 52 return round (sum (accuracies) / len (accuracies) * 100) Appendix H NLP Trading Bot ⬇ 1 import tweepy 2 import time 3 from textblob import TextBlob 4 import yfinance as yf 5 6 # Authentication 7 key = "" 8 csecret = "" 9 atoken = "" 10 atsecret = "" 11 nb = 500 12 keywords = ["BTC", "#BTC", "Bitcoin"] 13 14 auth = tweepy. OAuthHandler (ckey, csecret) 15 auth. set_access_token (atoken, atsecret) 16 api2 = tweepy. API (auth, wait_on_rate_limit = True, wait_on_rate_limit_notify = True) 17 18 def perc (a, b): 19 temp = 100 * float (a) / float (b) 20 return format (temp, ’.2f’) 21 22 def get_current_price (symbol): 23 ticker = yf. Ticker (symbol) 24 todays_data = ticker. history (period = ’1d’) 25 return todays_data [’Close’][0] 26 27 def get_twitter_BTC (): 28 ratios = 0 29 for keyword in keywords: 30 tweets = tweepy. Cursor (api2. search, q = keyword, lang = "en"). items (nb) 31 pos = 0 32 neg = 0 33 for tweet in tweets: 34 analysis = TextBlob (tweet. text) 35 if 0 <= analysis. sentiment. polarity <= 1: 36 pos += 1 37 elif -1 <= analysis. sentiment. polarity < 0: 38 neg += 1 39 pos = perc (pos, nb) 40 neg = perc (neg, nb) 41 if float (neg) > 0: 42 ratio = float (pos) / float (neg) 43 else: 44 ratio = float (pos) 45 ratios += ratio 46 return ratios 47 48 if __name__ == "__main__": 49 for k in range (1000): 50 score = get_twitter_BTC () 51 min1 = score + (score * 30 / 100) 52 time. sleep (60*5) 53 new_score = get_twitter_BTC () 54 if new_score > min1: 55 btc_price = get_current_price ("BTC-USD") 56 buy = "\nBUY : " + str (btc_price) 57 with open ("output.txt", "a") as f: 58 f. write (buy) 59 time. sleep (60*5) 60 new_new_score = get_twitter_BTC () 61 min2 = new_score - (new_score * 30 / 100) 62 if new_new_score < min2: 63 new_btc_price = get_current_price ("BTC-USD") 64 sell_at = " SELL : " + str (new_btc_price) 65 trade_profit = new_btc_price - btc_price 66 perc_profit = trade_profit / btc_price * 100 67 perc_profit_round = round (perc_profit, 3) 68 sell_message = sell_at + " | " + " Profit = " + str (perc_profit_round) + " %" 69 with open ("output.txt", "a") as f: 70 f. write (sell_message) 71 time. sleep (60*5) 72 else: 73 time. sleep (60*5) References - [Aggarwal, 2019] Aggarwal, D. 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