View
217
Download
0
Tags:
Embed Size (px)
Citation preview
MS&E 444 Kay Giesecke, April 7 2010
Project 4. Statistical Arbitrage
MS&E 444 Investment Practice
Spring 2010
Jeff Blokker [[email protected]]Emile Chamoun[[email protected]]
Ibrahim Jreige[[email protected]]Paris Georgoudis[[email protected]]
Sameh Galal[[email protected]]
MS&E 444 – Investment Practice
2 Factor Model
• Statistical Arbitrage– A standard model for the dynamics of stock price is
– This model can be enhanced by expanding the noise term
– Where are risk factors associated with the market
– In discrete time
– Assume that , , , and that
F and are independent.
( )
1
pjt
j t tjt
dSdt F d
S
( )jtF
tt
t
dSdt d
S
t
( )
1
(log )n
jt j t t
j
d S dt F d
1log logi i i i ir P P t βF
t tdt d βF
( ) 0E F cov( ) F I ( ) 0E
MS&E 444 – Investment Practice
3 Covariance of Log Returns
– If we have n observations and p factors:
– Or in matrix form
– Using
(1) (2) ( )1 1 11 12 1 1
(1) (2) ( )2 2 21 22 2 2
(1) (2) ( )1 2
...
...
...
pt t p t
pt t p t
pn n n t n t np t n
r t F F F
r t F F F
r t F F F
t r μ βF ε( )( ) ( )T T T T Tt t r μ r μ βF βF εβF βF
cov( ) ( )( )TE t t r r μ r μ
( ) ( ) ( ) ( )T T T T TE E E E β FF β εF β β F
T ββ Ψ
MS&E 444 – Investment Practice
4 Principal Component Analysis
• Principal Component Analysis – Spectra decomposition of matrix
where are the Eigen value, Eigen vector pair
• Noise Reduction– We can approximate the model with a limited set of m Eigen vectors or Principal
Components
– Using the largest Eigen vectors will add the components that contribute most to the variance in the data
1
cov( )p
T Ti i i
i
r ββ Ψ e e Ψ
( , )i i e
( )
1
ˆm p
ji j t t
j
r dt F d
MS&E 444 – Investment Practice
5 Stability of Principal Components
• Comparison of the Stability/Evolution of the PCA– 30 day initial data sample– Moved forward one day at a time.
– 10 largest Eigen cectors compared to the first sample using dot product
• Two Subtle Problems– 1. The Eigen vectors returned by PCA may be the inverse of the first set.
– 2. Since the Eigen vectors are given in descending order, a change in the relative magnitude of any components may swap their position. Therefore, comparisons must be made carefully.
• Results– Eigen vectors are relatively stable over time.
– After 10 Eigen vectors they become more unstable.
0cos Tn n e e
MS&E 444 – Investment Practice
6 Stability of Principal Components
-0.5 0 0.5 1 1.5 2 2.5 3 3.50
5
10
15
20
25
30Distribution of Eigen Vector #1
Cos(theta), Mean=0.99382
Num
ber
of V
ecto
rs
-0.5 0 0.5 1 1.5 2 2.5 3 3.50
1
2
3
4
5
6
7
8
9Distribution of Eigen Vector #2
Cos(theta), Mean=0.9595
Num
ber
of V
ecto
rs
-0.5 0 0.5 1 1.5 2 2.5 3 3.50
1
2
3
4
5
6
7Distribution of Eigen Vector #3
Cos(theta), Mean=0.91897
Num
ber
of V
ecto
rs
-0.5 0 0.5 1 1.5 2 2.5 3 3.50
1
2
3
4
5
6Distribution of Eigen Vector #4
Cos(theta), Mean=0.89915
Num
ber
of V
ecto
rs
MS&E 444 – Investment Practice
7 Stability of Principal Components
-0.5 0 0.5 1 1.5 2 2.5 3 3.50
1
2
3
4
5
6
7
8
9
10Distribution of Eigen Vector #5
Cos(theta), Mean=0.71333
Num
ber
of V
ecto
rs
-0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.5
1
1.5
2
2.5
3
3.5
4Distribution of Eigen Vector #6
Cos(theta), Mean=0.77994
Num
ber
of V
ecto
rs
-0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.5
1
1.5
2
2.5
3
3.5
4Distribution of Eigen Vector #8
Cos(theta), Mean=0.58858
Num
ber
of V
ecto
rs
-0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.5
1
1.5
2
2.5
3
3.5
4Distribution of Eigen Vector #7
Cos(theta), Mean=0.77283
Num
ber
of V
ecto
rs
MS&E 444 – Investment Practice
8 Statistical Distance vs Time of Day
• Mahanalobis Distance– The distance a data point is from the center of the distribution
• Procedure– The training set of 15 minute log return data was for 100 days.
– The distance of the next 10 data points was calculated.
– The training set was then shifted forward and the next 10 points measured.
– The data was sorted by time of day to analyze the time of day that generated the most outliers.
1( ) ( ) ( )TMD x x Σ x
MS&E 444 – Investment Practice
9 Distance of new Test Data form the Training Data
0 5 10 15 20 25 300
1
2
3
4
5
6
7
8
9x 10
4 Mahalanobis Distance of new Data Throughout the day
Mag
natu
de o
f D
ista
nce
Number of 15 Minute Intervals in Day
Mahalanobis Distance1( ) ( ) ( )T
MD x x Σ x
Conclusion – We can separate the market into two distinct time periods where the returns are generated by two different processes.
MS&E 444 – Investment Practice
10 Generation of Residuals• Partial Least Squares
– If X is the data set and Y is the component desired to regress from the data
– then PCA analyzes
– And PLS analyzes 1. PLS finds the matrix information associated with the first Eigen vector
2. Subtracts this information from the covariance matrix
3. Then finds the information for the second Eigen vector, etc.
• Procedure– Test data : 100 day sample of 15 minute log returns on 500 stocks
– Predict the next 10 points of data using PLS with largest 9 Eigen vectors
– Test data moved forward
• Results– Measure of fit
2 1( ) ( )
T
TR
ε ε
y y
( )TE X X( )TE X Y
MS&E 444 – Investment Practice
11 PLS First 45 Minutes of Market Removed
-0.015 -0.01 -0.005 0 0.005 0.01 0.0150
100
200
300
400
500
600
700
800Out of Sample Distribution of Residuals
Deviation of Residuals =0.0016586 R2 =0.87011
Num
ber
of
Sam
ple
s
-4 -3 -2 -1 0 1 2 3 4-0.015
-0.01
-0.005
0
0.005
0.01
0.015
Standard Normal Quantiles
Quantile
s o
f In
put
Sam
ple
Q-Q Plot Out of Sample Residuals
0 1000 2000 3000 4000 5000 6000 7000 8000 9000-0.015
-0.01
-0.005
0
0.005
0.01
0.015Out of Sample Residuals over time
Residuals
Tim
e
MS&E 444 – Investment Practice
12 PLS First 45 Minutes of the Market
0 200 400 600 800 1000 1200-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03Out of Sample Residuals over time
Residuals
Tim
e
-4 -3 -2 -1 0 1 2 3 4-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
Standard Normal Quantiles
Quantile
s o
f In
put
Sam
ple
Q-Q Plot Out of Sample Residuals
-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.030
50
100
150
200
250
300Out of Sample Distribution of Residuals
Deviation of Residuals =0.004329 R2 =0.7535
Num
ber
of
Sam
ple
s
MS&E 444 – Investment Practice
13 Calibrating OU Process: Problem Setup• Need to estimate κ, μ and σ in the OU-Process Equation:
• The discrete form of the solution of the SDE can be written as:
κ: coefficient of mean reversion
∆: discretization time step
μ: long term mean of the residuals
ttt dWXdX *)(
)1,0(*2
1
)1(
:
*
2
1
Ne
eb
ea
where
bXaX tt
MS&E 444 – Investment Practice
14 Calibrating OU Process: OLS and MLE• Least Squares:
Basic idea: Fit parameters by minimizing sum of square of error terms.
• Maximum Likelihood Estimation: Basic idea: Find parameters by maximizing log-likelihood of the data.
MS&E 444 – Investment Practice
15 Main Issue• OLS and MLE tend to produce similar results.• However, MLE is known for overestimating the mean
reversion speed κ:example: Johnson, Thomas. “Approximating Optimal Trading Strategies Under Parameter Uncertainty: A Monte Carlo Approach”. Kellog Business School. 2009.• Main idea: MLE typically overestimates the mean reversion speed and as
a result, underestimates the noise σ.• Paper compares filtering trading strategy to MLE. • Filtering outperforms MLE every time.
• Reason: Boguslavsky, Boguslavskaya. “Arbitrage Under Power”. February 2009.• MLE model suggests overly aggressive positions that can quickly lead
the trader to bankruptcy.
MS&E 444 – Investment Practice
16 Kalman Filtering• Idea: mathematical method to use noisy measurements to
produced results that tend to be closer to the true value of the variable of interest.
MS&E 444 – Investment Practice
17 Comparison of Estimation Methods
• Parameter estimation by Kalman Filtering Produces produces more accurate estimates of the OU process parameters than either MLE or OLS.
• Major disadvantage of EM Algorithm: Might take a long time to converge, computationally intensive for large window sizes.
• Solution: Use MLE/OLS to produce initial guesses then use EM to refine estimation.
MS&E 444 – Investment Practice
18 Optimal Trading of the Residuals-1• Implement the Boguslavsky/ Boguslavskyaya strategy
described in: “Optimal Arbitrage Trading” (2003).
• O-U process:
• Conditional Distribution:
• Utility Function
• Normalization Process : Let α be the control variable and W the wealth at time t:
• Value Function:
MS&E 444 – Investment Practice
19 Optimal Trading of the Residuals-2• Solve for optimal control parameter using HJB equation:
• Reduces to the PDE:
• Solution: Let τ be the time left for trading,
MS&E 444 – Investment Practice
20 Results on EvA residuals• ∆ ~ 1 min, γ = -0.5, initial wealth = 100,000
Cumulative Wealth, Optimal Trading PositionPeak ~ 4,300,000End ~ 3,700,000
MS&E 444 – Investment Practice
21 Results on Our residuals using EvA’s data-XOM
• ∆ ~ 15 min, initialWealth = 100,000
Cumulative Wealth, γ = 0 Cumulative Wealth,γ = -0.5Peak ~ 530,000 Peak ~ 520,000End ~ 490,000 End ~ 450,000
MS&E 444 – Investment Practice
22 Incorporating TC-Separate Fund Allocation• All wealths curves will • lie between the red and• green curves.• • Blue curve = no fixed cost• peak = 530,000, End = 490,000• • Green curve• peak = 470,000, end = 420,000
Blue = no costGreen = 10*fixed costRed = 1*fixed cost
MS&E 444 – Investment Practice
23 Trading Residuals in Practice• Look at historical 15 minute data for ~500 stocks using a
100 days sliding window
• For every stock i at time t– Generate partial least square representation using 10 components using
the remaining 499 stocks last 100 days return sliding window
– Generate a residual return by removing the PLS approximation from the stock return
– Generate residue replicating portfolio weights• Pi = [-β1 –β2 …. -βi-1 1 -βi+1 …. -βn]
MS&E 444 – Investment Practice
24 Available Data at Time t• Stock returns vector R(t)
• Residuals returns Vector Rresidue(t)
• Residuals means Vector μresidue(t)
• Residuals standard deviations Vector σresidue(t)
• Residuals replication matrix P(t)– Pij(t) is the weight of the jth stock in the portfolio replicating ith residue
– If we have residuals positions vector V(t), the final investment portfolio will be V(t)P(t)
MS&E 444 – Investment Practice
25 The Trading Strategy• Evaluate the market every 15 minutes to look for strong
deviations of residuals from mean– Enter positions that exceed a entering threshold
– Leave positions that cross the leaving threshold
– Allocate money in a certain defined percentage equally between all opportunities invested in given a certain minimum cash position percentage
• The dynamic rebalancing of portfolio is based on log optimal portfolio growth strategy of volatility pumping
MS&E 444 – Investment Practice
26 The Secret Sauce: Trading Parameters• 6 parameters
– Long Enter threshold, Short Enter threshold
– Long Exit Threshold, Short exit threshold
– Minimum Cash percentage
– Maximum single position percentage
• Trading algorithm is robust with trading parameters (at least as far as I tested!)
• Divided data sets into a training period and used matlab optimization toolbox to find parameters that maximizes sharpe ratio and applied the resulting parameters into a testing period
• This strategy can be applied continuously to periodically recalibrate the trading parameters
MS&E 444 – Investment Practice
27
0 1000 2000 3000 4000 5000 60000.5
1
1.5
2
2.5
3
Wea
lth M
ultip
les
Long Enter threshold=-2.698336, Long Exit threshold=-1.500553,Short Enter Threshold=2.698336, Short Exit Threshold=1.500553,
Minimum Cash Position=84.418854%, Maximum Investment=4.071198%, sharpe ratio=0.349356
training period
test period