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.

1. Random Walks,
Efficient Markets &
Stock Prices
Luigi Cenatti Gianni
NEO Empresarial

2. Why is it so hard
to BEAT THE MARKET?

3. What should be the STRATEGY
of a SMALL INVESTOR?

4. How to forecast
the 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 distribution
f(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 distribution
𝐆𝐎𝐎𝐆𝐑𝐞𝐭𝟐𝟎𝟎𝟔
= 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 distribution
𝐆𝐎𝐎𝐆𝐑𝐞𝐭𝟐𝟎𝟎𝟔
= 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, 𝝈
NormalDistribution[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

59. An idea of risk & return
GOOG traded at $ 727.44 on
September 20, 2012

60. 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

61. An idea of risk & return
Would you buy it today?

62. Why is it so hard
to BEAT THE MARKET?

63. What should be the STRATEGY
of a SMALL INVESTOR?

64. How to forecast
the RISK and RETURN of an asset?

65. Luigi Cenatti Gianni
lcgianni@gmail.com
br.linkedin.com/in/luigigianni

66. References
Random Walks and Finance:
http://sas.uwaterloo.ca/~dlmcleis/s906/chapt1-6.pdf
http://www.norstad.org/finance/ranwalk.pdf
Random Walks and Efficient Markets:
http://www.duke.edu/~rnau/411georw.htm
http://www.amazon.com/Random-Walk-Down-Wall-Street/dp/0393325350
Wolfram Mathematica:
http://reference.wolfram.com/mathematica/howto/PerformAMonteCarloSimulation.html

67. References
Online classes on Finance:
https://www.coursera.org/course/compfinance
https://www.coursera.org/course/introfinance
Others:
http://www.scientificamerican.com/article.cfm?id=can-math-beat-financial-markets
http://www.scientificamerican.com/article.cfm?id=after-the-crash
http://www.scientificamerican.com/article.cfm?id=trends-in-economics-a-calculus-of-risk

68. References
Quick readings on Wikipedia:
http://en.wikipedia.org/wiki/Monte_Carlo_methods_for_option_pricing
http://en.wikipedia.org/wiki/Black%E2%80%93Scholes
http://en.wikipedia.org/wiki/Geometric_Brownian_motion
http://en.wikipedia.org/wiki/Random_walk
http://en.wikipedia.org/wiki/Exchange-traded_fund

69. References
In Portuguese:
http://br.ishares.com/content/stream.jsp?url=/content/br/pt/repository/material/5-Min-
Guide_PT.pdf&mimeType=application/pdf
http://www.scielo.br/pdf/rac/v5n3/v5n3a11.pdf
http://www.lume.ufrgs.br/bitstream/handle/10183/29661/000769163.pdf?sequence=1

70. Images
http://www.thedigeratilife.com/images/january_effect_graph.png
http://forexachievers.com/wp-content/uploads/2010/09/beh.jpg
http://stockcharts.com/freecharts/historical/images/SPX1960s.png
http://204.143.68.15/file.php/400/quarter.jpg
http://www.wolframalpha.com/
http://stockcharts.com/school/data/media/chart_school/overview/random_walk_theory/
rw-5-fattails.png
http://blog.wolfram.com/data/uploads/2010/11/m8-logo.jpg
http://zoonek2.free.fr/UNIX/48_R/g606.png