Random Walks, Efficient Markets & Stock Prices
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Transcript of Random Walks, Efficient Markets & Stock Prices
Random Walks,
Efficient Markets &
Stock Prices
Luigi Cenatti Gianni NEO Empresarial
Why is it so hard to BEAT THE MARKET?
What should be the STRATEGY
of a SMALL INVESTOR?
How to forecast
the RISK and RETURN of an asset?
Table of Contents
Random Walks
Efficient Market Hypothesis
Playing with Wolfram Mathematica
ยป
ยป
ยป
Table of Contents
Random Walks
Efficient Market Hypothesis
Playing with Wolfram Mathematica
ยป
What makes a process random?
1. Sequence of random variables
2. independent from each other
3. and determined by a distribution
outcome
time
f(t)
Heads or tails?
Flip a coin 10 times
If heads, +1
If tails, -1
Heads or tails?
- 2
t
F(t)
Is this a random process?
Heads or tails?
Whatโs the expected outcome?
- 2
t
F(t)
Heads or tails?
Whatโs the expected outcome?
We have a feeling that, if we play it
many times, in most of them we will
end up with 0
Heads or tails?
Whatโs 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 weโre right
Heads or tails?
But what if the distribution looks like this?
Heads or tails?
What is the expected outcome?
But what if the distribution looks like this?
Heads or tails?
If we know the distribution, we can
simulate the process
Heads or tails?
If we know the distribution, we can
simulate the process
Heads or tails?
If we know the distribution, we can
simulate the process
Heads or tails?
This is commonly referred to as a
Monte Carlo Simulation
Table of Contents
Random Walks
Efficient Market Hypothesis
Playing with Wolfram Mathematica
ยป
Efficient Markets
Prices reflect all relevant information
Efficient Markets
Prices reflect all relevant information
If information is immediately reflected on
stock prices, tomorrowโs price change will
reflect only tomorrowโs news
Efficient Markets
Prices reflect all relevant information
Tomorrowโs price change is independent
of the price changes today
If information is immediately reflected on
stock prices, tomorrowโs price change will
reflect only tomorrowโs news
Efficient Markets
The Efficient Market hypothesis is
associated with the idea of a โrandom
walkโ
Efficient Markets
The Efficient Market hypothesis is
associated with the idea of a โrandom
walkโ
Therefore, itโs impossible to consistently
beat the market
Efficient Markets
Private investment funds canโt beat the
market
Source: Varga, G., รndice de Sharpe e outros indicadores de performance aplicados a fundos de aรงรตes brasileiros
Efficient Markets
Private investment funds canโt beat the
market
Source: Varga, G., รndice de Sharpe e outros indicadores de performance aplicados a fundos de aรงรตes brasileiros
Efficient Markets
BOVA11 beat 60% of active funds and
100% of passive funds, prior to 2009
According to Bloomberg:
Efficient Markets
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
According to Bloomberg:
Non-Efficient Markets?
Behavioral Finances: imperfections in financial
markets due to overconfidence, overreaction, and
other biases
Non-Efficient Markets?
Behavioral Finances: imperfections in financial
markets due to overconfidence, overreaction, and
other biases
Economic Bubbles
Non-Efficient Markets?
Behavioral Finances: imperfections in financial
markets due to overconfidence, overreaction, and
other biases
Economic Bubbles
Markets are efficient for small investors
Table of Contents
Random Walks
Efficient Market Hypothesis
Playing with Wolfram Mathematica
ยป
Problem
Today is January 1st, 2011. We want to
figure out the price of GOOG in one year
$ 593.97
Assumptions
1. Markets are efficient, so daily returns
are random variables, independent from
each other
2. Daily returns follow a determined
probability distribution
Framework
1. Fit a distribution to past returns
Framework
1. Fit a distribution to past returns
2. Simulate n random walks
Framework
1. Fit a distribution to past returns
2. Simulate n random walks
3. Price of stock will be mean of
outcomes
Fitting data to a distribution
๐๐๐๐๐๐๐ญ๐๐๐๐
= ๐ ๐ข๐ง๐๐ง๐๐ข๐๐ฅ๐๐๐ญ๐["๐๐๐๐", "๐๐๐ญ๐ฎ๐ซ๐ง", ๐๐๐๐, ๐, ๐ , ๐๐๐๐, ๐, ๐ , "๐๐๐ฅ๐ฎ๐" ;
{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, ...
Fitting data to a distribution
๐๐๐๐๐๐๐ญ๐๐๐๐
= ๐ ๐ข๐ง๐๐ง๐๐ข๐๐ฅ๐๐๐ญ๐["๐๐๐๐", "๐๐๐ญ๐ฎ๐ซ๐ง", ๐๐๐๐, ๐, ๐ , ๐๐๐๐, ๐, ๐ , "๐๐๐ฅ๐ฎ๐" ;
{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, ...
NormalDistribution[0.0005029, 0.0227045
๐๐๐๐๐๐ข๐ฌ๐ญ
= ๐๐ฌ๐ญ๐ข๐ฆ๐๐ญ๐๐๐๐ข๐ฌ๐ญ๐ซ๐ข๐๐ฎ๐ญ๐ข๐จ๐ง[๐๐๐๐๐๐๐ญ๐๐๐๐, ๐๐จ๐ซ๐ฆ๐๐ฅ๐๐ข๐ฌ๐ญ๐ซ๐ข๐๐ฎ๐ญ๐ข๐จ๐ง[๐, ๐
Fitting data to a distribution
Is the normal distribution a good fit?
Fitting data to a distribution
Is the normal distribution a good fit?
๐ = DistributionFitTest[GOOGRet2006, GOOGDist, "HypothesisTestData"]
Fitting data to a distribution
Problem of โfat tailsโ
Fitting data to a distribution
The stable distribution allows us to solve
this problem, because of two additional
parameters (alpha & beta)
Fitting data to a distribution
๐๐๐๐๐๐ญ๐๐๐ข๐ฌ๐ญ
= ๐๐ฌ๐ญ๐ข๐ฆ๐๐ญ๐๐๐๐ข๐ฌ๐ญ๐ซ๐ข๐๐ฎ๐ญ๐ข๐จ๐ง[๐๐๐๐๐๐๐ญ๐๐๐๐, ๐๐ญ๐๐๐ฅ๐๐๐ข๐ฌ๐ญ๐ซ๐ข๐๐ฎ๐ญ๐ข๐จ๐ง[๐, ๐, ๐, ๐, ๐
StableDistribution[1, 1.5313, โ0.0097, 0.0004, 0.0110
Fitting data to a distribution
๐๐๐๐๐๐ญ๐๐๐ข๐ฌ๐ญ
= ๐๐ฌ๐ญ๐ข๐ฆ๐๐ญ๐๐๐๐ข๐ฌ๐ญ๐ซ๐ข๐๐ฎ๐ญ๐ข๐จ๐ง[๐๐๐๐๐๐๐ญ๐๐๐๐, ๐๐ญ๐๐๐ฅ๐๐๐ข๐ฌ๐ญ๐ซ๐ข๐๐ฎ๐ญ๐ข๐จ๐ง[๐, ๐, ๐, ๐, ๐
StableDistribution[1, 1.5313, โ0.0097, 0.0004, 0.0110
๐ = DistributionFitTest[GOOGRet2006, GOOGStbDist, "HypothesisTestData"]
Fitting data to a distribution
The stable distribution is a better fit.
Simulating future prices
๐ฌ๐ข๐ฆ๐๐๐ญ๐ฌ = ๐๐๐ง๐๐จ๐ฆ๐๐๐ซ๐ข๐๐ญ๐[๐๐๐๐๐๐ญ๐๐๐ข๐ฌ๐ญ, ๐๐๐ ;
๐ฅ๐๐ฌ๐ญ๐๐ซ๐ข๐๐ = ๐๐๐๐๐๐ซ๐ข๐๐๐๐๐๐โฆโ๐ ;
Simulating future prices
๐๐๐๐๐ ๐๐ก ๐๐๐ฆ 1 = ๐๐๐ ๐ก๐๐๐๐๐ โ ๐๐๐ก1
๐๐๐๐๐ ๐๐ก ๐๐๐ฆ 2 = ๐๐๐ ๐ก๐๐๐๐๐ โ ๐(๐๐ก1+๐๐ก2)
Simulating future prices
๐๐ข๐ฌ๐ญ๐๐ข๐ง๐๐๐ฅ๐จ๐ญ[๐ฅ๐๐ฌ๐ญ๐๐ซ๐ข๐๐ โ ๐๐ฑ๐ฉ[๐๐๐๐ฎ๐ฆ๐ฎ๐ฅ๐๐ญ๐[๐ฌ๐ข๐ฆ๐๐๐ญ๐ฌ
๐๐๐๐๐ ๐๐ก ๐๐๐ฆ 1 = ๐๐๐ ๐ก๐๐๐๐๐ โ ๐๐๐ก1
๐๐๐๐๐ ๐๐ก ๐๐๐ฆ 2 = ๐๐๐ ๐ก๐๐๐๐๐ โ ๐(๐๐ก1+๐๐ก2)
Simulating future prices
๐ฆ๐๐๐ง๐๐๐๐๐๐ซ๐ข๐๐ = ๐๐๐๐ง[ ๐๐๐๐ง[ ๐๐ซ๐๐ฉ๐๐ง๐[
๐ฅ๐๐ฌ๐ญ๐๐ซ๐ข๐๐ โ ๐๐ฑ๐ฉ[๐๐๐๐ฎ๐ฆ๐ฎ๐ฅ๐๐ญ๐[๐๐๐ง๐๐จ๐ฆ๐๐๐ซ๐ข๐๐ญ๐[๐๐๐๐๐๐ญ๐๐๐ข๐ฌ๐ญ, ๐๐๐, ๐๐
, ๐๐จ๐ง๐ฌ๐ญ๐๐ง๐ญ๐๐ซ๐ซ๐๐ฒ[๐ฅ๐๐ฌ๐ญ๐๐ซ๐ข๐๐, ๐๐ ] ] ]
Simulating future prices
The price of GOOG will be the mean of
the means of each random walk
How close were we?
GOOG traded at $ 645.90 on
December 30, 2011
An idea of risk & return
www.wolframalpha.com
An idea of risk & return
An idea of risk & return
GOOG traded at $ 727.44 on
September 20, 2012
An idea of risk & return
GOOG traded at $ 727.44 on
September 20, 2012
In one year, thereโs a 95% chance its
price is going to be between $ 454.11
and $ 1294.98
An idea of risk & return
Would you buy it today?
Why is it so hard to BEAT THE MARKET?
What should be the STRATEGY
of a SMALL INVESTOR?
How to forecast
the RISK and RETURN of an asset?
Luigi Cenatti Gianni
[email protected] br.linkedin.com/in/luigigianni
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
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
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
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
References
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