Testing for Heavy-Tails
Use Kurtosis test Hill estimator for Power-Law distribution: )-1 ( K 1 X (i) (i+1) (log r - log r ) , HK ,T = K i=1
1 K T.
Plot HK ,T against `K '; Power-Law coef is inferred from the stable region.
Some Basics (Contd)
Rt = t Pt t11 log returns: rt = log(1 + Rt ) = log Pt log Pt 1 rt (K ) = rt + rt 1 + + rt K +1 Adjusting for dividends: 1 + Rt =
Pt +Dt Pt 1 P P
Returns:
(easy to use)
Random Walk: Let Z1 , Z2 , . . . be iid with mean , std dev ; Let S
Some Basics
Gaussian Distribution; Skewness=0; Kurtosis=3 Jarque-Bera Test: JB = n (K 3)2 SK 2 + 6 24 2 2
Lognormal: Mean > Median, because of skewness Heavy-Tailed Distribution:
Double exponential t-distribution - polynomial tail Mixture distribution Par
Course Outline: Topics
Stylized Facts of Asset Returns High Frequency Data Analysis Asset Pricing Models Mechanics of Trading Trading Strategies: Technical Analysis, Momentum and Pairs trading, etc Execution Strategies
Trends
Computing is becoming cheaper by the day Pressure to reduce trading costs Multiple instruments are traded together; Joint optimization is now possible with the computing power!
Algorithm Trading: Some Components
Easy access to historical and real time market data Perform statistical analysis; Identify trading opportunities; Determine the execution strategies; Measure against benchmarks. Order management/processing
Filter Rule: (Alexander, 1961, 1964, IMR)
x % Filter
If a daily closing price moves up at least x %, buy and hold until its price moves down at least x % from a subsequent high, sell and go short. Maintain the short position until the price rises at least
Random walk with zero or positive drift, trading rule cannot outperform buy and hold strategy But there is some persistence in successive price changes RW vs Martingale Martingale: E (pt ) is independent of pt-1 , pt-2 , . . . But the distribution of pt d
Modeling: Historical Perspective
Louis Bachelier (1900): Brownian Motion Markowitz (1952): Risk/Return Tradeo Sharpe, Lintner, Mossin and Black (1962-1964): CAPM Key Conclusions:
log(pt ) is a random walk Risk and return are related; risk is measured by v
Statistical Properties of Returns (rt )
No autocorrelation except at small time scales Heavy tails Asymmetry Volatility clustering Volume/volatility correlation (Cont, 2001)
Testing for Autocorrelations
Ljung-Box test: Q (m) = T (T + 2)
m X n=1
eh /(T h ) 2 2 m
H0 : e1 = e2 = = em = 0 For heavy-tails, rate of convergnece of ACF is slower than T ; hence asymptotic condence intervals are wider than above; models for volatility
Temporal dependence: High frequency data display some dependence. Dependence is due to
price discreteness bid-ask bounce execution of orders
volatility clustering similar to low frequency data. ACF must be calculated after adjusting for diurnal pattern. U
Diurnal patterns: Periodic pattern; U-shape. Volatility, volume, spread are higher near the open and close. Time between trades, shorter near open and close.
Discreteness: Price changes fall on multiples of ticks. (1/8-th of a $ to decimilization.) Most changes are within a dime. Induces a high degree of Kurtosis.
Modeling Volatility
Estimates:
1 r 1. t = K 1 i =1 (rt i K )2 , 2 K - horizon for investment 2. EWA: K
K = r
K i =1 rt i /K
t 2 ut 1
Models:
2 = t 1 + (1 )ut 1 , 2
= rt 1 t r
2 Garch: rt = + t t ; t ARMA(p , q ) Garch(1,1) - Most popular IGARCH, EGARCH, e
Volume-Volatility Correlation
Debate on what is information! Why do stock prices vary? Tauchen and Pitts Model (1983) r v = = 1 I Z1 2 I + 2 I Z2
2 where Z s are unit normal; Note cov (r 2 , v ) = 1 2 var (I ) > 0. I is the mixing variable.
Asymmetry and Volatility Clustering
Asymmetry
Response to positive returns is generally smaller than that of negative returns. How do we test this?
Volatility Clustering
Strong autocorrelations in |rt | and in rt2 . Asymptotic condence intervals are wider
Portfolio Rebalancing
Why? New information How? Balance between tracking error and transcation costs Let x = w w0 , where w0 is current portfolio weights and w is target weights. Constraints: w 0, X |xi | u , li wi ui ,
N X i =1 N X i =1
Long-only Turn-ov
Interpreting / Testing for the number of Factors
Usually the rst few factors are market / industry related. As N , K ; let Q = matrix: 1; the pdf of eigenvalues of the correlation p (max )( min ) Q p () = , 2 2 1 +2 Q r 1 , Q
T N
where max min = = 1+ 1+
r
Blind-Factor Models
Linear factor models: Rt = Bft + at ; both B and f are unknown. P Let Cov (Rt ) = ; Use PCA: = N i vi vi , where i s are eigenvalues i =1 and vi s are corresponding eigenvectors; yi = vi R , i = 1, 2, . . . , N are the transformed vari
Portfolio Theory with a Risk free asset Minimizing w w s.t. w + (1 w 1)Rf = p . Lagrangian: L = w w + (p w (1 w 1)Rf ) Solution: wp = p R f 1 ( Rf 1) = Cp w ( Rf 1) 1 ( Rf 1)
For any asset or portfolio, a, Sharpe ratio: aaRf : Expected Excess Return per
Portfolio Theory R is N 1 returns on risky assets; E (R) = ; Var (R) = . Minimize w w =risk, s.t., w = p , w 1 = 1. Objective Function: L(w , 1 , 2 ) = w w + 1 (p - w ) + 2 (1 - w 1). Let A = 1 -1 , B = -1 and C = 1 -1 1. Then wp = g + p h A C A B where g
Estimating volatility using high frequency data: Intraday data: Realized volatility :
nt
RVt =
i =1
rt2,i .
impacted by market microstructure noise overlooks overnight volatilities as 0, RVt .
Q Optimal sampling interval or n = 6.5hrs 2 3 , where Q = M 1
Modeling tick time and marks: How does private information becomes impounded in asset prices? Specialist can learn from the characteristics of transactions. Key: f (yi +1 |past tick times , past marks ). Simpler Models: f (yi +1 |past marks ); pi = change