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## Introduction to Algorithmic Trading Strategies Lecture 6

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**Introduction to Algorithmic Trading StrategiesLecture 6**Technical Analysis: Linear Trading Rules Haksun Li haksun.li@numericalmethod.com www.numericalmethod.com**Outline**• Moving average crossover • The generalized linear trading rule • P&Ls for different returns generating processes • Time series modeling**References**• EmmanualAcar, Stephen Satchell. Chapters 4, 5 & 6, Advanced Trading Rules, Second Edition. Butterworth-Heinemann; 2nd edition. June 19, 2002.**Assumptions of Technical Analysis**• History repeats itself. • Patterns exist.**Does MA Make Money?**• Brock, Lakonishok and LeBaron (1992) find that a subclass of the moving-average rule does produce statistically significant average returns in US equities. • Levichand Thomas (1993) find that a subclass of the moving-average rule does produce statistically significant average returns in FX.**Moving Average Crossover**• Two moving averages: slow () and fast (). • Monitor the crossovers. • , • Long when . • Short when .**How to Choose and ?**• It is an art, not a science (so far). • They should be related to the length of market cycles. • Different assets have different and . • Popular choices: • (150, 1) • (200, 1)**AMA(n , 1)**• iff • iff**GMA(n , 1)**• iff • (by taking log) • iff • (by taking log)**Acar Framework**• Acar (1993): to investigate the probability distribution of realized returns from a trading rule, we need • the explicit specification of the trading rule • the underlying stochastic process for asset returns • the particular return concept involved**Empirical Properties of Financial Time Series**• Asymmetry • Fat tails**Knight-Satchell-Tran Intuition**• Stock returns staying going up (down) depends on • the realizations of positive (negative) shocks • the persistence of these shocks • Shocks are modeled by gamma processes. • Persistence is modeled by a Markov switching process.**Knight-Satchell-Tran Process**• : long term mean of returns, e.g., 0 • , : positive and negative shocks, non-negative, i.i.d**Knight-Satchell-Tran**1-q Zt = 0 Zt = 1 p q 1-p**Stationary State**• , with probability • , with probability**GMA(2, 1)**• Assume the long term mean is 0, .**Naïve MA Trading Rule**• Buy when the asset return in the present period is positive. • Sell when the asset return in the present period is negative.**Naïve MA Conditions**• The expected value of the positive shocks to asset return >> the expected value of negative shocks. • The positive shocks persistency >> that of negative shocks.**Period Returns**hold Sell at this time point**Long-Only Returns Distribution**• Proof: make**I.I.D Returns Distribution**• Proof: • make**Expected Returns**• When is the expected return positive? • , shock impact • , shock impact • , if , persistence**MA Using the Whole History**• An investor will always expect to lose money using GMA(∞,1)! • An investor loses the least amount of money when the return process is a random walk.**Optimal MA Parameters**• So, what are the optimal and ?**Linear Technical Indicators**• As we shall see, a number of linear technical indicators, including the Moving Average Crossover, are really the “same” generalized indicator using different parameters.**The Generalized Linear Trading Rule**• A linear predictor of weighted lagged returns • The trading rule • Long: , iff, • Short: , iff, • (Unrealized) rule returns • if • if**Predictor Properties**• Linear • Autoregressive • Gaussian, assuming is Gaussian • If the underlying returns process is linear, yields the best forecasts in the mean squared error sense.**Maximization Objective**• Variance of returns is inversely proportional to expected returns. • The more profitable the trading rule is, the less risky this will be if risk is measured by volatility of the portfolio. • Maximizing returns will also maximize returns per unit of risk.**Truncated Bivariate Moments**• Johnston and Kotz, 1972, p.116 • Correlation:**Expected Returns As a Weighted Sum**a term for volatility a term for drift**Praetz model, 1976**• Returns as a random walk with drift. • , the frequency of short positions**Comparison with Praetz model**• Random walk implies . the probability of being short increased variance**Biased Forecast**• A biased (Gaussian) forecast may be suboptimal. • Assume underlying mean . • Assume forecast mean .**Maximizing Returns**• Maximizing the correlation between forecast and one-ahead return. • First order condition:**First Order Condition**• Let**Fitting vs. Prediction**• If process is Gaussian, no linear trading rule obtained from a finite history of can generate expected returns over and above . • Minimizing mean squared error maximizing P&L. • In general, the relationship between MSE and P&L is highly non-linear (Acar 1993).**Technical Analysis**• Use a finite set of historical prices. • Aim to maximize profit rather than to minimize mean squared error. • Claim to be able to capture complex non-linearity. • Certain rules are ill-defined.**Technical Linear Indicators**• For any technical indicator that generates signals from a finite linear combination of past prices • Sell: iff • There exists an (almost) equivalent AR rule. • Sell: iff • ,**Conversion Assumption**• Monte Carlo simulation: • 97% accurate • 3% error.