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

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»

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Table of Contents

Random Walks

Efficient Market Hypothesis

Playing with Wolfram Mathematica

»

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?

What’s the expected outcome?

- 2

t

F(t)

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

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

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

»

Efficient Markets

Prices reflect all relevant information

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

Efficient Markets

Prices reflect all relevant information

Tomorrow’s price change is independent

of the price changes today

If information is immediately reflected on

stock prices, tomorrow’s price change will

reflect only tomorrow’s news

Efficient Markets

The Efficient Market hypothesis is

associated with the idea of a “random

walk”

Efficient Markets

The Efficient Market hypothesis is

associated with the idea of a “random

walk”

Therefore, it’s impossible to consistently

beat the market

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

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

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

»

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

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, ...

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, ...

NormalDistribution[0.0005029, 0.0227045

𝐆𝐎𝐎𝐆𝐃𝐢𝐬𝐭

= 𝐄𝐬𝐭𝐢𝐦𝐚𝐭𝐞𝐝𝐃𝐢𝐬𝐭𝐫𝐢𝐛𝐮𝐭𝐢𝐨𝐧[𝐆𝐎𝐎𝐆𝐑𝐞𝐭𝟐𝟎𝟎𝟔, 𝐍𝐨𝐫𝐦𝐚𝐥𝐃𝐢𝐬𝐭𝐫𝐢𝐛𝐮𝐭𝐢𝐨𝐧[𝝁, 𝝈

Fitting data to a distribution

Is the normal distribution a good fit?

Fitting data to a distribution

Is the normal distribution a good fit?

𝓗 = DistributionFitTest[GOOGRet2006, GOOGDist, "HypothesisTestData"]

Fitting data to a distribution

Problem of “fat tails”

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

𝐆𝐎𝐎𝐆𝐒𝐭𝐛𝐃𝐢𝐬𝐭

= 𝐄𝐬𝐭𝐢𝐦𝐚𝐭𝐞𝐝𝐃𝐢𝐬𝐭𝐫𝐢𝐛𝐮𝐭𝐢𝐨𝐧[𝐆𝐎𝐎𝐆𝐑𝐞𝐭𝟐𝟎𝟎𝟔, 𝐒𝐭𝐚𝐛𝐥𝐞𝐃𝐢𝐬𝐭𝐫𝐢𝐛𝐮𝐭𝐢𝐨𝐧[𝟏, 𝛂, 𝛃, 𝛍, 𝛔

StableDistribution[1, 1.5313, −0.0097, 0.0004, 0.0110

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"]

Fitting data to a distribution

The stable distribution is a better fit.

Simulating future prices

𝐬𝐢𝐦𝐑𝐞𝐭𝐬 = 𝐑𝐚𝐧𝐝𝐨𝐦𝐕𝐚𝐫𝐢𝐚𝐭𝐞[𝐆𝐎𝐎𝐆𝐒𝐭𝐛𝐃𝐢𝐬𝐭, 𝟐𝟓𝟎 ;

𝐥𝐚𝐬𝐭𝐏𝐫𝐢𝐜𝐞 = 𝐆𝐎𝐎𝐆𝐏𝐫𝐢𝐜𝐞𝟐𝟎𝟎𝟔⟦−𝟏 ;

Simulating future prices

𝑃𝑟𝑖𝑐𝑒 𝑎𝑡 𝑑𝑎𝑦 1 = 𝑙𝑎𝑠𝑡𝑃𝑟𝑖𝑐𝑒 ∗ 𝑒𝑟𝑡1

𝑃𝑟𝑖𝑐𝑒 𝑎𝑡 𝑑𝑎𝑦 2 = 𝑙𝑎𝑠𝑡𝑃𝑟𝑖𝑐𝑒 ∗ 𝑒(𝑟𝑡1+𝑟𝑡2)

Simulating future prices

𝐋𝐢𝐬𝐭𝐋𝐢𝐧𝐞𝐏𝐥𝐨𝐭[𝐥𝐚𝐬𝐭𝐏𝐫𝐢𝐜𝐞 ∗ 𝐄𝐱𝐩[𝐀𝐜𝐜𝐮𝐦𝐮𝐥𝐚𝐭𝐞[𝐬𝐢𝐦𝐑𝐞𝐭𝐬

𝑃𝑟𝑖𝑐𝑒 𝑎𝑡 𝑑𝑎𝑦 1 = 𝑙𝑎𝑠𝑡𝑃𝑟𝑖𝑐𝑒 ∗ 𝑒𝑟𝑡1

𝑃𝑟𝑖𝑐𝑒 𝑎𝑡 𝑑𝑎𝑦 2 = 𝑙𝑎𝑠𝑡𝑃𝑟𝑖𝑐𝑒 ∗ 𝑒(𝑟𝑡1+𝑟𝑡2)

Simulating future prices

𝐦𝐞𝐚𝐧𝐆𝐎𝐎𝐆𝐏𝐫𝐢𝐜𝐞 = 𝐌𝐞𝐚𝐧[ 𝐌𝐞𝐚𝐧[ 𝐏𝐫𝐞𝐩𝐞𝐧𝐝[

𝐥𝐚𝐬𝐭𝐏𝐫𝐢𝐜𝐞 ∗ 𝐄𝐱𝐩[𝐀𝐜𝐜𝐮𝐦𝐮𝐥𝐚𝐭𝐞[𝐑𝐚𝐧𝐝𝐨𝐦𝐕𝐚𝐫𝐢𝐚𝐭𝐞[𝐆𝐎𝐎𝐆𝐒𝐭𝐛𝐃𝐢𝐬𝐭, 𝟐𝟓𝟎, 𝟓𝟎

, 𝐂𝐨𝐧𝐬𝐭𝐚𝐧𝐭𝐀𝐫𝐫𝐚𝐲[𝐥𝐚𝐬𝐭𝐏𝐫𝐢𝐜𝐞, 𝟓𝟎 ] ] ]

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, there’s 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

lcgianni@gmail.com 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