Introduction to Algorithmic Trading Strategies Lecture 5 Pairs
- Slides: 43
Introduction to Algorithmic Trading Strategies Lecture 5 Pairs Trading by Stochastic Spread Methods Haksun Li haksun. li@numericalmethod. com www. numericalmethod. com
Outline � First passage time � Kalman filter � Maximum likelihood estimate � EM algorithm 2
References As the emphasis of the basic co-integration methods of most papers are on the construction of a synthetic mean-reverting asset, the stochastic spread methods focuses on the dynamic of the price of the synthetic asset. � Most referenced academic paper: Elliot, van der Hoek, and Malcolm, 2005, Pairs Trading � � Model the spread process as a state-space version of Ornstein-Uhlenbeck process Jonathan Chiu, Daniel Wijaya Lukman, Kourosh Modarresi, Avinayan Senthi Velayutham. High-frequency Trading. Stanford University. 2011 � The idea has been conceived by a lot of popular pairs trading books � � � 3 Technical analysis and charting for the spread, Ehrman, 2005, The Handbook of Pairs Trading ARMA model, HMM ARMA model, some non-parametric approach, and a Kalman filter model, Vidyamurthy, 2004, Pairs Trading: Quantitative Methods and Analysis
Spread as a Mean-Reverting Process � 4
Sum of Power Series � 5
Unconditional Mean � 6
Long Term Mean � 7
Unconditional Variance � 8
Long Term Variance � 9
Observations and Hidden State Process � 10
First Passage Time � 11
A Sample Trading Strategy � 12
Kalman Filter � The Kalman filter is an efficient recursive filter that estimates the state of a dynamic system from a series of incomplete and noisy measurements. 13
Conceptual Diagram as new measurements come in prediction at time t Update at time t+1 correct for better estimation 14
A Linear Discrete System � 15
Observations and Noises � 16
Discrete System Diagram 17
Prediction � 18
Update � 19
Computing the ‘Best’ State Estimate � 20
Predicted (a Priori) State Estimation � 21
Predicted (a Priori) Variance � 22
Minimize Posteriori Variance � 23
Solve for K � 24
First Order Condition for k � 25
Optimal Kalman Filter � 26
Updated (a Posteriori) State Estimation � 27
Updated (a Posteriori) Variance � 28
Parameter Estimation � 29
Likelihood Function � 30
Maximum Likelihood Estimate � 31
Example Using the Normal Distribution � 32
Log-Likelihood � 33
Nelder-Mead � 34
Marginal Likelihood � 35
The Q-Function � 36
EM Intuition � 37
Expectation-Maximization Algorithm � 38
EM Algorithms for Kalman Filter � Offline: Shumway and Stoffer smoother approach, 1982 � Online: Elliott and Krishnamurthy filter approach, 1999 39
A Trading Algorithm � 40
Results (1) 41
Results (2) 42
Results (3) 43
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