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Random Walks, Efficient Markets & Stock Prices

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The famous financial theory of Efficient Markets is associated with the idea of a Random Walk. If the theory holds true, that makes prices unpredictable, and therefore it'd be impossible to …

The famous financial theory of Efficient Markets is associated with the idea of a Random Walk. If the theory holds true, that makes prices unpredictable, and therefore it'd be impossible to consistently beat the market.

The seminar discusses the mathematical idea of a random walk, then moves on to understand what makes a market efficient.

Finally, we conduct a Monte Carlo Simulation on Wolfram Mathematica, to forecast the behaviour of Google's stock price one year from now.

Published in: Economy & Finance

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  • 1. Random Walks,Efficient Markets &Stock PricesLuigi Cenatti GianniNEO Empresarial
  • 2. Why is it so hard to BEAT THE MARKET?
  • 3. What should be the STRATEGY of a SMALL INVESTOR?
  • 4. How to forecastthe RISK and RETURN of an asset?
  • 5. Table of Contents» Random Walks» Efficient Market Hypothesis» Playing with Wolfram Mathematica
  • 6. Table of Contents» Random Walks Efficient Market Hypothesis Playing with Wolfram Mathematica
  • 7. What makes a process random? 1. Sequence of random variables 2. independent from each other 3. and determined by a distributionf(t) outcome time
  • 8. Heads or tails? Flip a coin 10 times If heads, +1 If tails, -1
  • 9. Heads or tails? Is this a random process?F(t) t -2
  • 10. Heads or tails? What’s the expected outcome?F(t) t -2
  • 11. Heads or tails? What’s the expected outcome? We have a feeling that, if we play it many times, in most of them we will end up with 0
  • 12. Heads or tails? What’s the expected outcome? We have a feeling that, if we play it many times, in most of them we will end up with 0 And we’re right
  • 13. Heads or tails? But what if the distribution looks like this?
  • 14. Heads or tails? But what if the distribution looks like this? What is the expected outcome?
  • 15. Heads or tails? If we know the distribution, we can simulate the process
  • 16. Heads or tails? If we know the distribution, we can simulate the process
  • 17. Heads or tails? If we know the distribution, we can simulate the process
  • 18. Heads or tails? This is commonly referred to as a Monte Carlo Simulation
  • 19. Table of Contents Random Walks» Efficient Market Hypothesis Playing with Wolfram Mathematica
  • 20. Efficient Markets Prices reflect all relevant information
  • 21. Efficient Markets Prices reflect all relevant information If information is immediately reflected on stock prices, tomorrow’s price change will reflect only tomorrow’s news
  • 22. Efficient Markets Prices reflect all relevant information If information is immediately reflected on stock prices, tomorrow’s price change will reflect only tomorrow’s news Tomorrow’s price change is independent of the price changes today
  • 23. Efficient Markets The Efficient Market hypothesis is associated with the idea of a “random walk”
  • 24. Efficient Markets The Efficient Market hypothesis is associated with the idea of a “random walk” Therefore, it’s impossible to consistently beat the market
  • 25. Efficient Markets Private investment funds can’t beat the market Source: Varga, G., Índice de Sharpe e outros indicadores de performance aplicados a fundos de ações brasileiros
  • 26. Efficient Markets Private investment funds can’t beat the market Source: Varga, G., Índice de Sharpe e outros indicadores de performance aplicados a fundos de ações brasileiros
  • 27. Efficient Markets According to Bloomberg: BOVA11 beat 60% of active funds and 100% of passive funds, prior to 2009
  • 28. Efficient Markets According to Bloomberg: BOVA11 beat 60% of active funds and 100% of passive funds, prior to 2009 With lower volatility (risk) than 78% of active funds and 100% of passive
  • 29. Non-Efficient Markets? Behavioral Finances: imperfections in financial markets due to overconfidence, overreaction, and other biases
  • 30. Non-Efficient Markets? Behavioral Finances: imperfections in financial markets due to overconfidence, overreaction, and other biases Economic Bubbles
  • 31. Non-Efficient Markets? Behavioral Finances: imperfections in financial markets due to overconfidence, overreaction, and other biases Economic Bubbles Markets are efficient for small investors
  • 32. Table of Contents Random Walks Efficient Market Hypothesis» Playing with Wolfram Mathematica
  • 33. Problem Today is January 1st, 2011. We want to figure out the price of GOOG in one year $ 593.97
  • 34. Assumptions 1. Markets are efficient, so daily returns are random variables, independent from each other 2. Daily returns follow a determined probability distribution
  • 35. Framework 1. Fit a distribution to past returns
  • 36. Framework 1. Fit a distribution to past returns 2. Simulate n random walks
  • 37. Framework 1. Fit a distribution to past returns 2. Simulate n random walks 3. Price of stock will be mean of outcomes
  • 38. Fitting data to a distribution𝐆𝐎𝐎𝐆𝐑𝐞𝐭𝟐𝟎𝟎𝟔= 𝐅𝐢𝐧𝐚𝐧𝐜𝐢𝐚𝐥𝐃𝐚𝐭𝐚["𝐆𝐎𝐎𝐆", "𝐑𝐞𝐭𝐮𝐫𝐧", 𝟐𝟎𝟎𝟔, 𝟏, 𝟏 , 𝟐𝟎𝟏𝟏, 𝟏, 𝟏 , "𝐕𝐚𝐥𝐮𝐞" ;{0.0229993, 0.0134759, 0.0319564, 0.00266289, 0.00612551,0.00398076, -0.0169624, 0.00565106, 0.0018445, -0.0475263,-0.0190151, -0.084752, 0.0701948, 0.0363275, -0.0226396, ...
  • 39. Fitting data to a distribution𝐆𝐎𝐎𝐆𝐑𝐞𝐭𝟐𝟎𝟎𝟔= 𝐅𝐢𝐧𝐚𝐧𝐜𝐢𝐚𝐥𝐃𝐚𝐭𝐚["𝐆𝐎𝐎𝐆", "𝐑𝐞𝐭𝐮𝐫𝐧", 𝟐𝟎𝟎𝟔, 𝟏, 𝟏 , 𝟐𝟎𝟏𝟏, 𝟏, 𝟏 , "𝐕𝐚𝐥𝐮𝐞" ;{0.0229993, 0.0134759, 0.0319564, 0.00266289, 0.00612551,0.00398076, -0.0169624, 0.00565106, 0.0018445, -0.0475263,-0.0190151, -0.084752, 0.0701948, 0.0363275, -0.0226396, ...𝐆𝐎𝐎𝐆𝐃𝐢𝐬𝐭= 𝐄𝐬𝐭𝐢𝐦𝐚𝐭𝐞𝐝𝐃𝐢𝐬𝐭𝐫𝐢𝐛𝐮𝐭𝐢𝐨𝐧[𝐆𝐎𝐎𝐆𝐑𝐞𝐭𝟐𝟎𝟎𝟔, 𝐍𝐨𝐫𝐦𝐚𝐥𝐃𝐢𝐬𝐭𝐫𝐢𝐛𝐮𝐭𝐢𝐨𝐧[𝝁, 𝝈NormalDistribution[0.0005029, 0.0227045
  • 40. Fitting data to a distribution Is the normal distribution a good fit?
  • 41. Fitting data to a distribution Is the normal distribution a good fit? 𝓗 = DistributionFitTest[GOOGRet2006, GOOGDist, "HypothesisTestData"]
  • 42. Fitting data to a distribution Problem of “fat tails”
  • 43. Fitting data to a distribution The stable distribution allows us to solve this problem, because of two additional parameters (alpha & beta)
  • 44. Fitting data to a distribution 𝐆𝐎𝐎𝐆𝐒𝐭𝐛𝐃𝐢𝐬𝐭 = 𝐄𝐬𝐭𝐢𝐦𝐚𝐭𝐞𝐝𝐃𝐢𝐬𝐭𝐫𝐢𝐛𝐮𝐭𝐢𝐨𝐧[𝐆𝐎𝐎𝐆𝐑𝐞𝐭𝟐𝟎𝟎𝟔, 𝐒𝐭𝐚𝐛𝐥𝐞𝐃𝐢𝐬𝐭𝐫𝐢𝐛𝐮𝐭𝐢𝐨𝐧[𝟏, 𝛂, 𝛃, 𝛍, 𝛔 StableDistribution[1, 1.5313, −0.0097, 0.0004, 0.0110
  • 45. Fitting data to a distribution 𝐆𝐎𝐎𝐆𝐒𝐭𝐛𝐃𝐢𝐬𝐭 = 𝐄𝐬𝐭𝐢𝐦𝐚𝐭𝐞𝐝𝐃𝐢𝐬𝐭𝐫𝐢𝐛𝐮𝐭𝐢𝐨𝐧[𝐆𝐎𝐎𝐆𝐑𝐞𝐭𝟐𝟎𝟎𝟔, 𝐒𝐭𝐚𝐛𝐥𝐞𝐃𝐢𝐬𝐭𝐫𝐢𝐛𝐮𝐭𝐢𝐨𝐧[𝟏, 𝛂, 𝛃, 𝛍, 𝛔 StableDistribution[1, 1.5313, −0.0097, 0.0004, 0.0110 𝓗 = DistributionFitTest[GOOGRet2006, GOOGStbDist, "HypothesisTestData"]
  • 46. Fitting data to a distribution The stable distribution is a better fit.
  • 47. Simulating future prices 𝐬𝐢𝐦𝐑𝐞𝐭𝐬 = 𝐑𝐚𝐧𝐝𝐨𝐦𝐕𝐚𝐫𝐢𝐚𝐭𝐞[𝐆𝐎𝐎𝐆𝐒𝐭𝐛𝐃𝐢𝐬𝐭, 𝟐𝟓𝟎 ; 𝐥𝐚𝐬𝐭𝐏𝐫𝐢𝐜𝐞 = 𝐆𝐎𝐎𝐆𝐏𝐫𝐢𝐜𝐞𝟐𝟎𝟎𝟔⟦−𝟏 ;
  • 48. Simulating future prices 𝑃𝑟𝑖𝑐𝑒 𝑎𝑡 𝑑𝑎𝑦 1 = 𝑙𝑎𝑠𝑡𝑃𝑟𝑖𝑐𝑒 ∗ 𝑒 𝑟 𝑡1 𝑃𝑟𝑖𝑐𝑒 𝑎𝑡 𝑑𝑎𝑦 2 = 𝑙𝑎𝑠𝑡𝑃𝑟𝑖𝑐𝑒 ∗ 𝑒 (𝑟 𝑡1 +𝑟 𝑡2)
  • 49. Simulating future prices 𝑃𝑟𝑖𝑐𝑒 𝑎𝑡 𝑑𝑎𝑦 1 = 𝑙𝑎𝑠𝑡𝑃𝑟𝑖𝑐𝑒 ∗ 𝑒 𝑟 𝑡1 𝑃𝑟𝑖𝑐𝑒 𝑎𝑡 𝑑𝑎𝑦 2 = 𝑙𝑎𝑠𝑡𝑃𝑟𝑖𝑐𝑒 ∗ 𝑒 (𝑟 𝑡1 +𝑟 𝑡2) 𝐋𝐢𝐬𝐭𝐋𝐢𝐧𝐞𝐏𝐥𝐨𝐭[𝐥𝐚𝐬𝐭𝐏𝐫𝐢𝐜𝐞 ∗ 𝐄𝐱𝐩[𝐀𝐜𝐜𝐮𝐦𝐮𝐥𝐚𝐭𝐞[𝐬𝐢𝐦𝐑𝐞𝐭𝐬
  • 50. Simulating future prices 𝐦𝐞𝐚𝐧𝐆𝐎𝐎𝐆𝐏𝐫𝐢𝐜𝐞 = 𝐌𝐞𝐚𝐧[ 𝐌𝐞𝐚𝐧[ 𝐏𝐫𝐞𝐩𝐞𝐧𝐝[ 𝐥𝐚𝐬𝐭𝐏𝐫𝐢𝐜𝐞 ∗ 𝐄𝐱𝐩[𝐀𝐜𝐜𝐮𝐦𝐮𝐥𝐚𝐭𝐞[𝐑𝐚𝐧𝐝𝐨𝐦𝐕𝐚𝐫𝐢𝐚𝐭𝐞[𝐆𝐎𝐎𝐆𝐒𝐭𝐛𝐃𝐢𝐬𝐭, 𝟐𝟓𝟎, 𝟓𝟎 , 𝐂𝐨𝐧𝐬𝐭𝐚𝐧𝐭𝐀𝐫𝐫𝐚𝐲[𝐥𝐚𝐬𝐭𝐏𝐫𝐢𝐜𝐞, 𝟓𝟎 ] ] ]
  • 51. Simulating future prices The price of GOOG will be the mean of the means of each random walk
  • 52. How close were we? GOOG traded at $ 645.90 on December 30, 2011
  • 53. An idea of risk & return www.wolframalpha.com
  • 54. An idea of risk & return
  • 55. An idea of risk & return GOOG traded at $ 727.44 on September 20, 2012
  • 56. An idea of risk & return GOOG traded at $ 727.44 on September 20, 2012 In one year, there’s a 95% chance its price is going to be between $ 454.11 and $ 1294.98
  • 57. An idea of risk & return Would you buy it today?
  • 58. Why is it so hard to BEAT THE MARKET?
  • 59. What should be the STRATEGY of a SMALL INVESTOR?
  • 60. How to forecastthe RISK and RETURN of an asset?
  • 61. Luigi Cenatti Giannilcgianni@gmail.combr.linkedin.com/in/luigigianni
  • 62. References Random Walks and Finance:http://sas.uwaterloo.ca/~dlmcleis/s906/chapt1-6.pdfhttp://www.norstad.org/finance/ranwalk.pdf Random Walks and Efficient Markets:http://www.duke.edu/~rnau/411georw.htmhttp://www.amazon.com/Random-Walk-Down-Wall-Street/dp/0393325350 Wolfram Mathematica:http://reference.wolfram.com/mathematica/howto/PerformAMonteCarloSimulation.html
  • 63. References Online classes on Finance:https://www.coursera.org/course/compfinancehttps://www.coursera.org/course/introfinance Others:http://www.scientificamerican.com/article.cfm?id=can-math-beat-financial-marketshttp://www.scientificamerican.com/article.cfm?id=after-the-crashhttp://www.scientificamerican.com/article.cfm?id=trends-in-economics-a-calculus-of-risk
  • 64. References Quick readings on Wikipedia:http://en.wikipedia.org/wiki/Monte_Carlo_methods_for_option_pricinghttp://en.wikipedia.org/wiki/Black%E2%80%93Scholeshttp://en.wikipedia.org/wiki/Geometric_Brownian_motionhttp://en.wikipedia.org/wiki/Random_walkhttp://en.wikipedia.org/wiki/Exchange-traded_fund
  • 65. References In Portuguese:http://br.ishares.com/content/stream.jsp?url=/content/br/pt/repository/material/5-Min-Guide_PT.pdf&mimeType=application/pdfhttp://www.scielo.br/pdf/rac/v5n3/v5n3a11.pdfhttp://www.lume.ufrgs.br/bitstream/handle/10183/29661/000769163.pdf?sequence=1
  • 66. Imageshttp://www.thedigeratilife.com/images/january_effect_graph.pnghttp://forexachievers.com/wp-content/uploads/2010/09/beh.jpghttp://stockcharts.com/freecharts/historical/images/SPX1960s.pnghttp://204.143.68.15/file.php/400/quarter.jpghttp://www.wolframalpha.com/http://stockcharts.com/school/data/media/chart_school/overview/random_walk_theory/rw-5-fattails.pnghttp://blog.wolfram.com/data/uploads/2010/11/m8-logo.jpghttp://zoonek2.free.fr/UNIX/48_R/g606.png

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