Statistical Arbitrage
Lecture 1: Overview
John Lehoczky
Mark Schervish
Giuseppe Nuti (UBS)
John Lehoczky & Mark Schervish
Statistical Arbitrage Lecture 1a: Overview
August 25, 2014
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Course Administrative Details
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Homework Assignments (50%)
Team Project (50%)
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Teams will be assigned in two weeks
Syllabus
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Course Schedule
Course Description and Learning Objectives
Course Materials
Academic Integrity
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Statistical Arbitrage Lecture 1a: Overview
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Learning Objectives
From the Statistical Arbitrage course, students will gain:
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experience in cleaning and manipulating financial data and
will become aware of possible pitfalls in finding a good
trading strategy and evaluating its performance.
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a mastery of and experience in implementing the major
approaches to statistical arbitrage including pairs trading,
value, momentum and reversal strategies.
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an understanding of the fundamentals of modern
electronic trading including limit-order books,
high-frequency trading and have experience about the
microstructure of stock price paths.
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experience in devising, implementing, and evaluating
statistical arbitrage strategies on market data.
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Course Schedule
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Lectures 1-6 will occur at 5:30pm on the scheduled class
day. Lecture 6 (October 7) will be given by Giuseppe Nuti
of UBS.
Instead of a seventh lecture, there will be a period starting
at 5:30 during which the two instructors will be available to
discuss team projects with team members.
The final exam on Wednesday, October 15 will consist of
team project presentations, given in parallel but separate
sessions in Pittsburgh and New York City. Teams must turn
in both a written report and a set of presentation slides
before the start of the sessions.
There are two milestones leading to the final project
presentations:
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Monday, September 22: project proposals are due
Friday, October 10: draft executive summaries of the
projects are due at noon.
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Statistical Arbitrage Lecture 1a: Overview
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Course Topics: 1
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Lecture 1: Course overview, historical background,
introduction to pairs trading, discussion of data issues, and
homework assignments 1 and 2.
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Lecture 2: Statistical arbitrage by creating long/short
portfolios of winner (long) and loser (short) stocks based
on some principle (e.g Value/contrarian and momentum
strategies). Discussion of classic papers by LSV and JT as
well as new paper by AMP.
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Lecture 3: Continuation of long/short portfolio strategies
(Combining value and momentum, reversals), defining and
identifying statistical arbitrage.
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Statistical Arbitrage Lecture 1a: Overview
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Course Topics: 2
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Lecture 4: Time series background leading up to
cointegration, pairs trading with cointegration, Gatev et al,
Avellaneda and Lee papers.
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Lecture 5: Volatility arbitrage, background on electronic
trading, market microstructure, limit order books and high
frequency trading. Discussion of papers by AHS, “Equity
trading in the 21st century”
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Lecture 6: Lecture by Giuseppe Nuti of UBS on high
frequency finance.
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Lecture 7: Project consultations
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Eras in Stock Market Trading
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Pre 1959: Value investing and Technical Analysis
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1959-1973: The “Golden Age of Quantitative Finance”
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1973-1993: The era of the “efficient market hypothesis”
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1993-2002: The rise of statistical arbitrage
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2002-2014: The rise of electronic trading and
high-frequency finance
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Pre 1959: 1
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Prior to 1962, there were principally two approaches to
investing: value investing and technical analysis.
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Value investing (Benjamin Graham and coauthors), finding
stocks that are undervalued or overvalued relative to their
own fundamentals or their industrial sector.
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Security Analysis, 1934 (B. Graham and D. Dodd)
The Interpretation of Financial Statements, 1937, (B.
Graham and S. Meredith).
The Intelligent Investor, 1949 (Originally B. Graham, latest
editions have W. Buffett and J. Zweig as coauthors)
Note that company financial variables change very slowly.
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Pre 1959: 2
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Technical Analysis (e.g. McGee and Edwards), perhaps
the forerunner of today’s statistical machine learning
approaches.
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No statistical analysis of the likelihood and degree of
success derived from trading on these signals.
Little concern with issues of portfolio construction and
diversification.
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August 25, 2014
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The Golden Age of Mathematical Finance
1959-1973
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Markowitz Portfolio Theory (1952-1959)
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Modigliani-Miller Foundations of Corporate Finance and
Capital Structure (1958, 1961, 1963)
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CAPM (Sharpe, 1964 and others)
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Samuelson: “Properly adjusted stock prices are
martingales” (1965)
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The behavior of stock market prices (non-normal returns)
(Fama, 1965)
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Efficient Market Hypothesis (Fama, 1970)
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Black Scholes Merton option pricing formula (1973)
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CRSP: 1
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The availability of data and computing power is
fundamental to understanding the behavior of equity prices
and other financial instruments.
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CRSP was founded in 1960 at the University of Chicago,
and one of its major projects was the establishment of daily
stock price data (open, high low, close) for all stocks listed
on the NYSE in 1962.
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It began with the NYSE and was expanded to the AMEX
and NASDAQ (and more broadly later).
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This led to the start of a “science” of equity price behavior.
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August 25, 2014
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CRSP: 2
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Before CRSP, the prevailing model of stock price behavior,
if any, dated back to Bachelier in 1900, Theory of
Speculation, namely stock prices behave like geometric
Brownian motion.
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This theory posited that the log-returns of a stock should
have a normal distribution as well as an independent
increments structure.
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CRSP data allowed for a careful study (on a daily time
scale) of the behavior of stock prices. Just like the
microscope revolutionized biology and the telescope
revolutioned astronomy, or fMRI is revolutionizing brain
science, CRSP data revolutionized quantitative stock price
modeling.
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August 25, 2014
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CRSP: 3
Normal Probability Plot for IBM data
0.999
0.997
0.99
0.98
0.95
Probability
0.90
0.75
0.50
0.25
0.10
0.05
0.02
0.01
0.003
0.001
−0.2
−0.15
−0.1
−0.05
Data
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0.05
0.1
Figure: IBM daily log-returns: 1/2/62 to 3/7/12, mean = .0034, std =
.0162
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Efficient Market Hypothesis: 1
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In 1965 Fama published a major empirical study of the
behavior of stock prices. He demonstrated that price
changes (log-returns) have much heavier tails than would
be predicted from a normal distribution.
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In 1970, Fama also formulated the efficient market
hypothesis stating: “... investors can choose among the
securities that represent ownership of firms’ activities
under the assumption that security prices at any time ‘fully
reflect’ all available information. A market in which prices
always ’fully reflects’ available information is called
’efficient’.“
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Efficient Market Hypothesis: 2
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The efficient market hypothesis became standard in
academia. It was popularized in the mainstream by A
Random Walk Down Wall Street, first published in 1973 by
Burton Malkiel. Malkiel said:
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“... A blindfolded chimpanzee throwing darts at the Wall
Street Journal could select a portfolio that would do as well
as the experts.”
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In 2003, he would modify this to encourage buying index
funds with very low expense charges.
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Efficient Market Hypothesis: 3
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The academic community was solidly convinced by the
efficient market hypothesis until the early 1990s; however,
this didn’t prevent the financial services industry from
expanding its stock advisory business during those years.
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The efficient market hypothesis also had some important
skeptics:
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“I’d be a bum in the street with a tin cup if the markets were
efficient.” Warren Buffett, Forbes Magazine, April 3, 1995.
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1990s, Rise of Statistical Arbitrage
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The CAPM of Sharpe coupled with the efficient market
hypothesis argued that there was one risk factor in
investing, market risk. Stocks were characterized by their
“beta coefficient” which measured their risk relative to the
overall market.
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Between 1985 and 1995, three major results were
published that exhibited trading/investment strategies that
resulted in “excess profits” beyond market risk (commonly
called ‘alpha.’ These clearly showed “anomalies” and
suggested trading strategies that were zero cost and
yielded “excess profits.”
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These include pairs trading, value, momentum, and short
term reversals.
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Statistical Arbitrage Lecture 1a: Overview
August 25, 2014
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What is Statistical Arbitrage: 1
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From investwords.com: “An attempt to profit from pricing
inefficiencies that are identified through the use of
mathematical models. Statistical arbitrage attempts to
profit from the likelihood that prices will trend toward a
historical norm. Unlike pure arbitrage, statistical arbitrage
is not riskless.”
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From investopedia.com: “A profit situation arising from
pricing inefficiencies between securities. Investors identify
the arbitrage situation through mathematical modeling
techniques. Statistical arbitrage is not without risk; it
depends heavily on the ability of market prices to return to
a historical or predicted normal.”
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What is Statistical Arbitrage: 2
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From HJTW 2004: “We define statistical arbitrage as a
long horizon trading opportunity that genertes a riskless
profit. As such, statistical arbitrage is a natural extension of
the trading strategies utilized in the existing empirical
literature on persistent anomalies.”
The HJTW definition contains three key ideas:
1. Zero initial cost (initial value of the trading portfolio is 0).
This leads to long-short strategies, not buy-only strategies
2. Long horizon trading. With riskless arbitrage there is a finite
time T at which the value of the trading portfolio will be
non-negative with probability 1 and positive with positive
probability. For statistical arbitrage we must let T → ∞.
3. Riskless profit (in a long horizon context)
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August 25, 2014
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What is Statistical Arbitrage: 3
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HJTW formalize these mathematically, and they develop
statistical methodology to test whether a trading strategy
can be considered to be a statistical arbitrage.
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HJTW definition of statistical arbitrage: v (t) is the value of
a trading portfolio at time t.
A trading strategy with a sequence of portfolio values
{v (t), t ≥ 0} is a statistical arbitrage if it satisfies:
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1.
2.
3.
4.
v (0) = 0,
limt→∞ E(v (t)) > 0,
limt→∞ P(v (t) < 0) = 0,
limt→∞ Var(t)/t = 0.
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What is Statistical Arbitrage: 4
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Statistical arbitrage is not
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Having a “view” of the market and invest in some particular
set of stocks.
Considering a universe of stocks and a benchmark return
(e.g. the S &P 500 index). Create a portfolio with weights
on stocks and compare the return on that portfolio with the
benchmark return.
Neither is zero-cost or market neutral.
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Statistical Arbitrage Lecture 1a: Overview
August 25, 2014
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Structure of Statistical Arbitrage Trading
Strategies
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Determine a universe of stocks
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Select a formation period, a time period over which stocks’
performances are measured
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Find a pair of stocks that exhibit a long-term stationary
pattern (pairs trading), or rank stocks using some
performance measure (value, momentum, reversals)
defining “winners” and “losers.”
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Develop a zero-cost, long-short portfolio (long the winners,
short the losers) and hold it for some given time period.
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Continue to construct and hold portfolios based on the
given criterion (e.g. value, momentum or reversals) and
evaluate the long-run results.
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Statistical Arbitrage Lecture 1a: Overview
August 25, 2014
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Some Basic Strategies: Pairs Trading
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Using statistical methodology, identify two or more stocks
whose co-movement exhibits a long-term relationship
(“mean-reversion”).
Follow these stocks until they diverge by a sufficient
amount (opening signal), i.e. based on the statistical
model, one is priced relatively too high and the other is
priced too low. Go short in the overpriced stock and go
long in the underpriced stock. Hold until convergence.
Need to identify pairs or baskets to trade and need to
specify trading strategy parameters (e.g. opening and
closing signals, bailout criteria)
Strategy is zero-cost and market neutral
We will consider the papers by Gatev et. al. and by
Avellaneda and Lee (traded ETFs versus component
stocks).
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Statistical Arbitrage Lecture 1a: Overview
August 25, 2014
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Gatev et. al.
Figure: From Gatev et. al. illustrating pairs trading
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Value, Momentum and Reversals: 1
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Value investing ranks all stocks in a universe according to
a financial variable (e.g. book value) divided by market
value (BM).
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Lakonishok, Shleifer and Vishny (1993) created long/short
portfolios every 6 months going long in top 10% and short
in bottom 10% and held for some period.
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This zero-cost strategy yielded excess profits of alpha =
10.5% per year over 25 years.
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August 25, 2014
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Value, Momentum and Reversals: 2
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Momentum changes the investing maxim “buy low and sell
high” to
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“Buy high, sell higher” or “Short low, cover lower”
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In 1993 Jegadeesh and Titman introduced a momentum
strategy. Rank all stocks in a universe according to their
returns over a formation period. Every 6 months they
bought the top 10% and shorted the bottom 10%. Over a
23 year period, this zero-cost strategy yielded 1% to 1.5%
per month in excess returns.
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Statistical Arbitrage Lecture 1a: Overview
August 25, 2014
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Value, Momentum and Reversals: 3
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A third strategy, short term reversals, was also
documented by Lehmann (1990) and Cooper (1999) to
show very significant excess returns. It is exactly the
opposite of momentum strategies with holding periods in
the order of 4 or 5 days.
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These three strategies produced major “anomalies” and
led to the rejection of the efficient market hypothesis.
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However, as these strategies became known, and by 2000
it was thought that they were no longer exceptions to the
EMH.
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New paper: “Value and Momentum Everywhere” by
Asness, Moskowitz and Pedersen, July 2013 takes issue
with this. This paper not only argues that these strategies
are still viable, they apply to many different asset classes.
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Statistical Arbitrage Lecture 1a: Overview
August 25, 2014
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2000, The Rise of Electronic Trading: Creative
Destruction of the Traditional Industry
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Just as computers have led to the creative destruction of
industries like the music industry and the publishing
industry (and maybe eventually education), computers
have led to enormous changes in the finance industry.
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New stock exchanges (e.g. BATS, Direct Edge, dark pools)
and mergers of exchanges
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Trading is dominated by electronic trading
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Co-location of traders’ computers at the exchanges to
minimize latency
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Arms race among market players to get the fastest
response to market conditions and news
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Consolidation of Financial Markets
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Market Microstructure: 1
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How are prices determined? If you place an order to buy or
sell some amount, what price will you ultimately pay or
receive?
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On many exchanges the price of a stock is determined
through the limit-order-book.
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1871 was the beginning of the specialist system.
Specialists maintained a list of buy and sell orders for a
particular stock, handled order fulfillment and was charged
with maintaining an orderly market for a stock when there
was a significant order imbalance.
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Statistical Arbitrage Lecture 1a: Overview
August 25, 2014
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Market Microstructure: 2
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Decimalization was introduced on August 28, 2000 and
was fully implemented on January 29, 2001.
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This dramatically reduced bid-ask spreads, thus lowering
the cost of trading. Volumes soared into the billions.
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Specialists began to be replaced by electronic
limit-order-books and market makers.
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From 2001 to 2007 it is estimated that electronic trading in
equities grew from 25% to 80%. Much of that 80% is
coming from algorithmic and high-frequency traders
(computers) making trades at the microsecond level.
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Statistical Arbitrage Lecture 1a: Overview
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Limit-Order-Books: 1
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Limit-order-books are visible, consequently placing a limit
order, especially a large order, will have an impact on the
entire book.
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Computers are tracking order imbalance, so the placement
of a large order will move the market.
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A large sell (buy) order will cause prices to fall (rise)
against the interests of the party placing the order.
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Need a sophisticated strategy to move a large block of
stock without having severe market impact.
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Statistical Arbitrage Lecture 1a: Overview
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A Limit Order Book
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Limit-Order-Books: 2
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We need a sophisticated strategy to move a large block of
stock without having severe market impact.
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This problem of market impact has led to the estblishment
of new exchanges called “dark pools.” These are
exchanges in which the “lights are out”, i.e. the
limit-order-book is not visible to market participants.
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This has led to a whole new set of issues concerning how
to gather information about a market in which the
limit-order-book is not visible.
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Statistical Arbitrage Lecture 1a: Overview
August 25, 2014
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New Exchanges
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The new era of electronic trading has been fostered by the
SEC through two important regulatory changes:
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Reg ATS in 1998 that permitted the establishment of ECNs
(Electronic Communication Networks)
Reg NMS (National Market System) that unified those
ECNs into a single market system and assured that
investors would obtain the NBBO (National Best Bid Offer)
One of the first and most important ECNs was Island
founded by Joshua Levine, a Carnegie Mellon student.
Island was sold to Instinet (later merged into Nasdaq) for
$500M.
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Statistical Arbitrage Lecture 1a: Overview
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High Frequency Trading:
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There has been an “arms race” among investment banks
in acquiring the fastest possible computing and networking
to 1) react to changes in the limit-order-books and 2) react
to news.
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In 2007, the CTO of the NYSE reported that one firm
estimated a reduction of 1 millisecond in latency was worth
$100 million/year to the bottom line.
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Communications systems to minimize latency between
exchanges (e.g. New York and Chicago or New York and
London) are being built.
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