Chaos Theory and the Markets

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Chaos Theory and the Markets

• Erik Long, President
• Tetrahex, Inc.

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RISK DISCLOSURE STATEMENT

• THE RISK OF LOSS IN TRADING COMMODITIES CAN BE
  SUBSTANTIAL. YOU SHOULD THEREFORE CAREFULLY
  CONSIDER WHETHER SUCH TRADING IS SUITABLE FOR YOU
  IN LIGHT OF YOUR FINANCIAL CONDITION. IN
  CONSIDERING WHETHER TO TRADE OR TO AUTHORIZE
  SOMEONE ELSE TO TRADE FOR YOU, YOU SHOULD BE
  AWARE OF THE FOLLOWING
• IF YOU PURCHASE A COMMODITY OPTION, YOU MAY
  SUSTAIN A TOTAL LOSS OF THE PREMIUM AND OF ALL
  TRANSACTION COSTS.
• IF YOU PURCHASE OR SELL A COMMODITY FUTURE, OR
  SELL A COMMODITY OPTION, YOU MAY SUSTAIN A TOTAL
  LOSS OF THE INITIAL MARGIN FUNDS AND ANY
  ADDITIONAL FUNDS THAT YOU DEPOSIT WITH YOUR
  BROKER TO ESTABLISH OR MAINTAIN YOUR POSITION.
  IF THE MARKET MOVES AGAINST YOUR POSITION, YOU
  MAY BE CALLED UPON BY YOUR BROKER TO DEPOSIT A
  SUBSTANTIAL AMOUNT OF ADDITIONAL MARGIN FUNDS, ON
  SHORT NOTICE, IN ORDER TO MAINTAIN YOUR POSITION.
  IF YOU DO NOT PROVIDE THE REQUESTED FUNDS WITHIN
  THE PRESCRIBED TIME, YOUR POSITION MAY BE
  LIQUIDATED AT A LOSS, AND YOU WILL BE LIABLE FOR
  ANY RESULTING DEFICIT IN YOUR ACCOUNT.
• UNDER CERTAIN MARKET CONDITIONS, YOU MAY FIND IT
  DIFFICULT OR IMPOSSIBLE TO LIQUIDATE A POSITION.
  THIS CAN OCCUR, FOR EXAMPLE, WHEN THE MARKET
  MAKES A LIMIT MOVE.
• THE PLACEMENT OF CONTINGENT ORDERS BY YOU OR YOUR
  TRADING ADVISOR, SUCH AS A STOP-LOSS OR
  STOP-LIMIT ORDER, WILL NOT NECESSARILY LIMIT
  YOUR LOSSES TO THE INTENDED AMOUNTS, SINCE MARKET
  CONDITIONS MAY MAKE IT IMPOSSIBLE TO EXECUTE SUCH
  ORDERS.
• A SPREAD POSITION MAY NOT BE LESS RISKY THAN A
  SIMPLE LONG OR SHORT POSITION.
• THE HIGH DEGREE OF LEVERAGE THAT IS OFTEN
  OBTAINABLE IN
• COMMODITY TRADING CAN WORK AGAINST YOU, AS WELL
  AS FOR YOU. THE USE OF LEVERAGE CAN LEAD TO
  LARGE LOSSES AS WELL AS GAINS.
• HYPOTHETICAL RESULTS SHOWN HERE DO NOT REPRESENT
  ACTUAL TRADING. YOUR RESULTS MAY VARY
  SUBSTANTIALLY FROM THOSE SHOWN HERE DUE TO
  COMMISSIONS, MANAGEMENT FEES, SLIPPAGE AND OTHER
  UNFORSEEN MARKET FACTORS.
• IN SOME CASES, MANAGED COMMODITY ACCOUNTS ARE
  SUBJECT TO SUBSTANTIAL CHARGES FOR MANAGEMENT AND
  ADVISORY FEES. IT MAY BE NECESSARY FOR THOSE
  ACCOUNTS THAT ARE SUBJECT TO THESE CHARGES TO
  MAKE SUBSTANTIAL TRADING PROFITS TO AVOID
  DEPLETION OR EXHAUSTION OF THEIR ASSETS. THIS
  DISCLOSURE DOCUMENT CONTAINS, AT PAGE 3, A
  COMPLETE DESCRIPTION OF EACH FEE TO BE CHARGED TO
  YOUR ACCOUNT BY THE COMMODITY TRADING ADVISOR.
• THIS BRIEF STATEMENT CANNOT DISCLOSE ALL THE
  RISKS AND OTHER SIGNIFICANT ASPECTS OF THE
  COMMODITY MARKETS.

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An Organized Approach to Chaos

• What is Chaos?
• Why Euclidean Geometry Doesnt Work for Traders
• What are Fractals
• Market Applications
• Fractal Finance Tools
• Trading with Fractals

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Examples of Chaos

• Lightning
• Weather Patterns
• Earthquakes
• Financial Markets
• Social and Natural Systems
• Governmental and Financial Institutions

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Constant Bewilderment

• Chaos Theory is a way to describe or quantify
  nonlinear, random events or systems
• Analyze events or systems that are influenced by
  their own outcomes, taking on a life of their own
• Order and randomness can coexist allowing
  predictability

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Why is Chaos so Confusing?

• Euclidean Geometry assumes a symmetrical world
• Mountains are not cones
• Clouds are not spheres
• Coastlines are not circles
• Lightning doesnt travel in a straight line
• Markets chop and correct

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Fractals Bringing Order to Chaos

• Assumes an infinite complexity in everything
• Worldly objects are a collection of many similar
  curves combined
• Each curve is made up of identical smaller curves
  making for infinite length
• Each curve has self-similar smaller curves or
  Fractal Dimensions within it
• Fractals identify order in apparent randomness
• Patterns exist within a markets underlying
  noise

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Fractals Measure the Noise

• Divide the top line by half
• Divide all subsequent lines by half
• What ultimately looks like noise is actually a
  clearly discernable pattern called Cantor Dust

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Order Out of Randomness

• Draw 3 points on a page
• Label the points (1,2), (3,4) (5,6)
• Draw a point P anywhere within the three points
• Roll a fair die for random results
• Draw a point halfway between P and the numbered
  point on the die
• Repeat 10,000 times for a Sierpinski Triangle
• Random die rolls result in a stable, self-similar
  natural system with each points position
  dependent on a previous point and all points
  bounded by the perimeter of the triangle

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Fractals in Finance

• Markets are a function of nonlinear human
  activity
• Technical traders are at a disadvantage because
  traditional technical analysis techniques are
  based upon linear equations and Euclidean
  Geometry
• Most analysis techniques cannot quantify
  nonlinear noise and attempt to merely filter it
  out
• Market reversals are nonlinear
• Technical Analysis is a poor indicator for the
  trend vs range trading decision
• Fractals quantify what Euclid could not

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Measuring the Unmeasurable

• 3 dimensional Whiffle Ball
• Actual dimension is a fraction between the 2 and
  3 dimensional realm
• Check your Whiffle Ball for your TradeStation
  free trial. 5 free trials will now be raffled off
  to Cornerstone members. Let us know if we draw
  your number!

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Measuring Chaos

• Mandlebrot measured Englands irregular, chaotic
  coastline more accurately using fractal measuring
  techniques
• The Koch Snowflake demonstrates how using
  infinitely finer fractals increases measurement
  accuracy
• Mandlebrot applied these same nonlinear measuring
  techniques to the cotton market

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Long-term Market Nonlinearity

• Elliot Wave describes the inherent difficulty in
  long-term forecasting
• Each cycle is influenced by the feedback of
  component cycles reducing long-term forecast
  accuracy
• This feedback is why Elliot Wave analysis is an
  art and its counts often subject to revision
• Fractals measure feedback components and
  forecasts apparently random short, medium and
  long-term outcomes

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Market Applications of Chaos Theory and Fractal
Analysis

• Market prices tend to seek natural levels or
  ranges of balance
• These levels or ranges can be described as
  attractors
• These ranges (attractors) are determinant.
• However, data within these ranges remains random

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Types of Attractors

• Point Attractors
• Limit Cycles
• Strange or Fractal Attractors

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Point Attractors

• Simplest form of attractor
• Point at which a pendulums swing reverses
• Theoretically comparable to an economic
  supply/demand balance or market equilibrium

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Limit Cycle Attractors

• Friction and mechanical limitations are removed
• Pendulum freely swings a full 360 indefinitely
• This represents the market volatility around
  equilibrium or market noise

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Chaotic or Fractal Attractors

• Maps multiple pendulum swings of various
  magnitudes but never completing a full 360
  circle
• These random swings are always within a subset of
  the range of the Limit Cycle attractor called a
  phase space
• Phase Space is comparable to the perimeter of the
  Sierpinski triangle
• These attractors have fractal dimensions

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Fractal Attractors in Finance

• Equity and Futures markets are classic examples
• Buying or selling interest is comparable to the
  swing of the pendulum
• Measurable in any time increment
• Each equity or futures contract would be its own
  phase space
• Long-term forecast are very dependent on accurate
  measurement of initial market conditions

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Trading With Fractal Finance

• Fractal Finance evaluates market variables of
  price, volume, liquidity and time
• Fractal Pivot Points are described as price
  attractors on multiple time frames that establish
  starting points for market trends
• Fractal Dimension Index (FDI) determines the
  persistence or antipersistence of market trends

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Liquidity and Volume over Time

• Liquidity ? Volume
• Rapidly moving markets are indicative of low
  liquidity against high volume
• A markets fair, equilibrium price is where
  supply meets demand
• With efficient, stable markets fair price
  actual price
• Low liquidity and high volume creates market
  instability and trading opportunities
• Liquidity disappears when long, medium and short
  term investors all share the same market
  perspective eliminating a two sided market

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Efficient verses the Fractal Market Hypothesis

• Efficient Market Hypothesis
• Gaussian assumption of normally distributed
  prices
• Weak-form EMH with purely random price
  distributions has been widely discounted
• Semi-strong form EMH where all public information
  is reflected in the prices is favored by the
  professional community
• Long-term prices exhibit no memory
• Crash of 87 was an outlier

• Fractal Market Hypothesis
• Prices exhibit a leptokurtic distribution
• Similar price patterns found at different time
  increments i.e. Daily, weekly, monthly (Fractal
  Structure)
• Decreasing reliability as forecast extends out
  into the future (Sensitive Dependence)
• Prices exhibit short and long-term correlations
  and trends (Feedback Effects)
• Erratic market activity under certain conditions
  (Critical Levels)

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Price over Time

• Money Flow is the energy behind a market move
• The high and low of a price bar determines the
  relative aggressiveness of bulls and bears,
  respectively
• The relative open and close of a price bar
  determines the dominant players
• Volume determines if the market is attracting new
  players, the rate of change and direction in
  Money Flow
• Immediate Trend is the midpoint of the current
  bar relative to previous bars
• The comparison of Money Flow and Immediate Trend
  is a confirming indicator for Fractal Pivot
  Points

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Fractal Pivot Points

• Fractal Pivot Points are attractors on multiple
  time frames
• Lower time frame Pivot Points are shorter term
  indicators
• Fractal Pivot points are adjustable depending on
  the time frame you wish to trade
• Confidence Levels are adjustable according to the
  number of time frames that are attracted to the
  same price

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Fractal Pivot Points
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Fractal Pivot Points with Confidence
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Fractal Dimension Index

• Determines the persistence or antipersistence of
  a market
• A persistent market closely follows a market
  trend
• An antipersistent market results in substantial
  volatility around the trend with a low r2
• An antipersistent market has a more jagged price
  plot and is more vulnerable to price reversals

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FDI and the Nile River

• Early 20th Century H.E. Hurst had to determine
  the capacity for the Nile River Dam reservoir
• A dam too high wasted , too low allowed for
  flooded farmland
• He considered correlations between annual
  rainfall, water level extremes and reservoir
  levels
• All calculations yielded random and statistically
  insignificant results
• However, flood cycles actually did exist with a
  series of unequal flood events
• Standard statistical techniques failed to reveal
  them due to their non-periodic characteristics

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Using FDI

• Results range between 1.0 and 2.0
• FDI 1.2 is a strongly directional rocket
  shot. Our geography analogy would be the border
  between Kansas and Missouri
• FDI 1.8 is a very volatile wide range trade.
  Our geography analogy here would be the English
  coastline
• FDI 1.5 is a purely random unpredictable market
• FDI 1.4 is for a longer-tem, position trading,
  trend following strategy
• FDI 1.6 is for a active, shorter-term, range
  trading strategy

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FDI and Technical Analysis

• FDI may be used as confirmation for many other
  market trend or oscillating indicators
• 1.6FDI2.0 will confirm stochastic, RSI,
  Bollinger Band and reversal pattern signals
• 1.0FDI1.4 will confirm moving average
  cross-over and continuation pattern signals

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Fractals and Technical Analysis
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More Fractals and Technical Analysis
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Derivation of FDI

• Hursts work was rooted in Brownian Motion and
  the fact that the length of the random path of a
  particle is the square root of the amount of time
  used to measure its movement.
• Otherwise, known as the T ½ Rule.
• If you double the amount of time spent observing
  a particle in motion, the distance it travels
  will be half that of the previous shorter
  observation period
• This is the benchmark value for our FDI where a
  purely random market has an FDI 1.5

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FDI Proof as Rocket Science

• Xt , N cumulative deviation over N periods
• Eu return in year u
• MN average annual returns over N periods
• R range of X
• Max (X) maximum value of X over N periods
• Min (X) minimum value of X over N periods
• St logarithmic return at time t
• Pt price at time t
• N number of observations
• H Hurst exponent
• a numerical constant

• Max(Xt ,N) ? ( Eu MN )
• Min(Xt ,N) ? ( MN Eu)
• R Max(Xt ,N) Min(Xt ,N)
• St ln ( Pt / P(t-1))
• These equations then go into our rescaled range
  identity below
• R/S (a N)H
• Our rescaled range increases at the rate of T ½
• Setting H .50 assumes a random walk. Deviations
  from .50 implies profit opportunities exist
  within the markets

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FDI Estimate as Rocket Science

• R/S rescaled range
• H Hurst exponent
• N number of observations
• a numerical constant .50
• FDI Fractal Dimension Indicator
• log (R/S) H log(N) log(a)
• Reconfigure to solve for H log(R/S) / log(N/2)
• FDI 2 - H

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Using the FDI
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Fraclets

• Combination of a Fractal Tool and a Wavelet
  Filter
• A Wavelet Filter allows one to remove extraneous
  data (white noise) from a data or price stream
  allowing for a more focused analysis

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Fraclet Trading Tool
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Thank you for your time.

• Erik T. Long
• President
• Tetrahex, Inc.
• 111 West Jackson Boulevard
• Suite 948
• Chicago, IL 60604
• Toll free 877.808.7233
• email elong_at_fractalfinance.com
• Web URL www.fractalfinace.com

• Joseph A. Skibinski
• Principal
• Ceres Capital, LLC
• 111 West Jackson Boulevard
• Suite 948
• Chicago, IL 60604
• Toll free 877.808.7233
• email jskibinski_at_cerescapital.net
• Web URL www.cerescapital.net

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Erik T. Long

• Bachelors Bio-Anthropology and Economics from
  UCLA, 1992
• Founder of Tetrahex, 1996
• Post Graduate Diploma in Economics from
  University of London, 1998
• Developer of proprietary trading systems for
  Altea Trading, 2000 - 2002
• Consultant in non-linear methodologies for
  Nexbridge Inc., 2000 2002
• Founding member of the System Select Group at
  Peregrine Financial Group, 2002 2004
• Founding Principal of Ceres Capital LLC, 2004