By

Paul Cottrell, BSc, MBA, ABD

Chaos Theory and Modern Trading

Author Complexity Science, Behavioral

Finance, Dynamic Hedging, Financial Statistics, Chaos Theory

Proprietary Trader Energy and Currency

Dissertation Dynamically Hedging Oil and

Currency Futures Using Receding Horizontal Control and Stochastic Programming

Introduction

The behavior of dynamic systemsMany systems are non-linear

Unpredictable results can occurDeterministic chaos

Simple chaos where no stochastic functions are in the system

Non-deterministic chaosComplex Chaos where stochastic function are in

the system

What is Chaos Theory?

Simple Chaos

Lorenz System

Double fulcrum Pendulum

Complex Chaos

• Human misbehavior

• Random news events

• Feedback loops

Black Swan vs. Dragon King

Unknowable Knowable

Theory of EmergenceStarted in cosmology

Big Bang leads to further particle evolution and the emergence of materials.Which leads to further complex arrangement

Life Social Organization

Economic or financial emergenceEconomic development Systemic riskContagion

Key takeawayA complex system can evolve into unpredicted pathways

What evolves from Chaos?

Complexity ScienceThe study of complex systems

Using simple rules for agentsSelf organizing behavior Interactions that have a magnifying effect

The Theory of Emergence

How does this relate to trading?

• The “Market”

• Complex organism

• Self organizing• Adam’s invisible

hand

• Price action• Asymmetric

• Information • Asymmetric

How does this relate to trading? (Cont)

• Traders use models

• Models have certain assumptions on price action

• Models can be used incorrectly and cause a system failure• Lehman Crash• Flash Crash

(Maybe?)• Account drawdown• Mass

unemployment• Big Macs too

expensive

The Efficient Market HypothesisAssumptions

Rational investors Information cannot be used to make above normal profitsThe stochastic variations in returns mean to zeroThe market should always be in steady state

ProblemsTraders are greedy and not rational

Due to the Dopamine response mechanismNew information is not completely in the priceProfits can be statistically above average for some groupsStochastic variations in returns can lead to bubbles and

bursts.

Economic Models

Fundamental EquilibriumWhen price is close to “economic value”Could be assumed at a 200 moving average on a

long duration chartFundamental analysis rule the game

Speculative EquilibriumWhen price is above or below “economic value”Chartists or Quants rule the gameMost assets are in Speculative Equilibrium

Evidence in the 50 period moving averageHas mean reverting characteristics

Behavioral Finance

Chaotic Returns• Returns graphed• Daily Returns, Weekly, Monthly• S&P 500• Lower Right Graph

• Dow 30• Monthly

• State Space• X-axis return (t-1)• Y-axis return (t)

• Empirical evidence • That returns are stationary

• In daily returns• Non-stationary

• At larger time scales.• Shows emergence of tend

Fractal Efficiency Ratio

• Ratio to determine level of chaos• “C” is the return at time (t)• Ratio = 1

• Pure trending• Ratio = 0

• Pure Chaos

Mandelbrot Markets

H < 0.5mean reversion

H = 0.5Brownian Motion

H > 0.5Trending

A possible method to describe the market in terms of smoothness.Lower “H” value the smoother the surface of the market.

There is trading time and clock time Clock time is standard time and is constant in velocity Trading time is changing

Velocity (first derivative) depends on the speed of price For example:

During high volatile market days price action is higher Leading to faster time in trade time Lower volatile days have slow trade time

Many traders use terms like Rapid price movement or it was a slow trading day

Time is relative to the level of the price change Can be used to help model discontinuous markets.

Bridge gap with a Brownian motion bridge.

Mandelbrot Time can help frame volatility in terms of delta time. Similar to space-time bending with gravity. Trade-time bends with level of price action.

Mandelbrot Time

The market is a complex system

Usually in speculative equilibrium Volatility and correlations are not

constant

Market participants can profit on average above zero mean

Systems that can monitor the telemetry of the “market” might be able to monitor the endogenous risk in the market (Dragon Kings)

Exogenous risks do exist (Black Swans)

Hedging strategies can, to some degree, mitigate risk factors.

Conclusion