lecture 5: mean-reversion
TRANSCRIPT
Stationarity/ Non Stationarity
Definition: a stochastic process is stationary if
( )( ){ } ( ){ }AXXXAXXX
Attm
hththtttt
nm
mm∈=∈
∈∀∀∀
+++ ,...,,.Pr,...,,.Pr
,,..., ,
2121
1 R
A stationary process is a process that is ``statistically invariant under translations’’
Examples: the Ornstein-Uhlembeck process is stationary, Brownian motionis not.
The Ornstein-Uhlenbeck process
( )
( ) ( )( ) ( )
( ) ( ) ( ) noise hiteGaussian w ,
1
=+=
+−+=
+−=
∫
∫
∞−
−−
−−−−−−
sdssemX
dWemeXeX
dWdtXmdX
tst
t
u
t
s
utsts
stt
ttt
ηησ
σ
σκ
κ
κκκ
Exponentially-weighted moving average of uncorrelated Gaussianrandom variables.
Statistics of the OU process
( ) ( ) ( ) ( )( ) ( ) ( )
( ) ( )
( )
( )khtht
kh
t stkkh
t shtkstk
t ht shtkstk
t ht shtkstkhtt
ek
XX
k
e
dsee
dsee
dsdsssee
dssedsseXX
−+
−
∞−
−−−
∞−
−+−−−
∞−
+
∞−
−+−−−
∞−
+
∞−
−+−−−+
−=−
=
=
=
−=
⋅=
∫
∫
∫ ∫
∫ ∫
1
2
''
''
22
2
22
2
'2
'2
σ
σ
σ
σ
δσ
ηησ
Structure Function
Random Walk, Fractional BM
( )( )
>
=
<<
≈
+=
>−+
=
==−
==
−
+
∞
−+
∞−
++
∫
∫
1
1 )ln(
12/1
)1(
2/1 1
motionBrownian ,
2
2
12
2
112
2
22
ph
ph
h
ph
XX
uu
du
hXX
pst
dssX
tXXhXX
WWX
p
p
htt
h
ppphtt
t
pt
thttht
ttt
σ
σ
σ
σ
ησ
σ
σ Brownian motion (non-stationary)Structure fn grows linearly
Geometric Brownian motion
Correlations decay like power-laws(large h)
Autoregressive Models
( )
10
0
0
1
1
11101
21
1
...
0 ,
...
0 ,
01...
...01
...
,...
1,0~ ,...
,...,...,,
+
+−
++−+
++=
=
=
=
=
++++=
+ n
m
mn
n
inmnmnn
n
nn
n
aaa
X
X
NXaXaaX
XXX
ν
σ
νσν
ΣAYAY
ΣAAY
AR-n model corresponds to a vector AR-1 model
Structure function: SPY Jan 1996-Jan 2009
Use log prices as time series. Structure function with lags 1 day to 2 yrs
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
1 27 53 79 105 131 157 183 209 235 261 287 313 339 365 391 417 443 469 495
SPY is highly non stationary, as shown in the chart.Look for mean-reversion in relative value, i.e. in terms of two or more assets.
Structure function log (SLB/OIH)Data: Apr 2006 to Feb 2009
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
1 28 55 82 109 136 163 190 217 244 271 298 325 352 379 406 433 460 487
days
S(x
)
Saturation
OIH: Oil Services ETF, SLB: Schlumberger-Doll Research
Structure Function: long-shortequal dollar weighted SLB-OIH
0
0.005
0.01
0.015
0.02
0.025
1 24 47 70 93 116 139 162 185 208 231 254 277 300 323 346 369 392 415 438 461 484
days
S(x
)
( ) nnnn PXRRPP ln ,1 oihslb1 =−+×=+
Structure Function for Beta-Neutrallong-short portfolio SLB-Beta*OIH
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
1 36 71 106 141 176 211 246 281 316 351 386 421 456 491
days
S(x
)
( ) nnnn PXRRPP ln ,1 oihd60slb1 =⋅−+×=+ β
Structure Function log (GENZ/IBB)
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
1 35 69 103 137 171 205 239 273 307 341 375 409 443 477
days
S(x
)
Structure function ln (DNA/GENZ)
00.020.040.060.080.1
0.120.140.160.18
1 29 57 85 113 141 169 197 225 253 281 309 337 365 393 421 449 477
days
S(x
)
DNA: Genentech Inc.GENZ; Genzyme Corp.
Mean-reversion: large negativecurvature here.
Structure Fn for Beta-Neutral GENZ-DNA Spread
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
1 24 47 70 93 116 139 162 185 208 231 254 277 300 323 346 369 392 415 438 461 484
days
S(x
)
Poor reversion for the beta adjusted pair. Beta is low ~ 0.30
Systematic Approach for looking for MR in Equities
Look for stock returns devoid of explanatory factors, and analyze thecorresponding residuals as stochastic processes.
( )( )
( )( ) ( )tdXtP
tdP
tS
tdS
XX
FR
k
km
kk
t
sst
tkt
m
kkt
+=
+=
+=
∑
∑
∑
=
=
=
1
10
1
β
ε
εβ Econometric factor model
View residuals as increments of aprocess that will be estimated
Continuous-time model for evolutionof stock price
Interpretation of the model
The factors are either
A. eigenportfolios corresponding to significant eigenvaluesof the market
B. industry ETF, or portfolios of ETFs
Questions of interest:
Can residuals be fitted to (increments of) OU processes orother MR processes?
If so, what is the typical correlation time-scale?
Estimation of Ornstein-Uhlenbeck models
( ) ( )
{ }
−++=
+−+=
∆−
++
∆+−−∆−∆−
∆+ ∫
k
eNbaXX
dWeemXeX
tk
nnnn
tt
t
sstktk
ttk
tt
2
1,0 i.i.d.
1
22
11 σνν
σ
( ) ( )( )( ) ( )( )
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∆−−−=
−=
∆=
==
+
−−−−
−−−−
ata
baXX
a
bm
atk
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XXXXa
nn
nlnnln
nlnnln
1ln
12
1
STDEV ,
1 ,
1ln
1
,...,;,...,INTERCEPT
,,...,;,...,SLOPE
2
1
11
11
σ
Portfolio Strategy
portfolio of exposuredollar net :
factor along exposure beta-dollarnet :
costs) ransaction(neglect t
prices adjusted-dividend ,.....,,
short)or (long stocksdifferent in invested $ ,....,,
1
1
11 11
111
11
21
21
−
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−
+=
−=Π
∑
∑
∑∑ ∑∑
∑∑∑
∑∑
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=
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===
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N
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m
k k
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Q
kQ
rdtQP
dPQdXQ
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rdtQS
dSQd
SSS
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β
β
β
Market-Neutral Portfolio
( ) { }
( )( )
( )( )
( ) ( )( )
( ) dtQdVar
dtrXmkQdE
dWQdtrXmkQ
rdtQdWdtXmkQ
rdtQdXQd
dWdWdtXmkdX
i
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iiii
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iiiiii
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iii
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iiiiiii
2
1
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1
11
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|
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eduncorrelat Assume
σ
σ
σ
σ
∑
∑
∑∑
∑∑
∑∑
=
=
==
==
==
=
=Π
−−=Π
∴
+−−=
−+−=
−=Π
+−=
X
X
Mean-Variance Optimal Portfolio
( ) ( )
( )
22Ratio Sharpe Annualized
2 ;
2
2 22
)0or ,0 if(
2
1max
222
2
2
22
222
kN
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kN
dtN
kN
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kN
d
kXm
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dWXmk
dtXmk
d
Qr
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ii
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i
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ii i
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i i
ii
i
i
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i iiiiiQ
=
⋅=
=Π−Π
=Π
−=+=Π
−+−=Π
==
=∴
−
∑
∑∑
∑∑
∑∑
∑
∑ ∑
λλ
σξξλξλ
σλ
σλ
σµλσ
λµ
ETF Abs(Alpha) Beta Kappa Reversion days EquiVol Abs(m)
HHH 0.20% 0.69 38 7 4% 3.3%
IYR 0.11% 0.90 39 6 2% 1.8%
IYT 0.18% 0.97 41 6 4% 3.0%
RKH 0.10% 0.98 39 6 2% 1.7%
RTH 0.17% 1.02 39 6 3% 2.7%
SMH 0.19% 1.01 40 6 4% 3.2%
UTH 0.09% 0.81 42 6 2% 1.4%
XLF 0.11% 0.83 42 6 2% 1.8%
XLI 0.15% 1.15 42 6 3% 2.4%
XLK 0.17% 1.03 42 6 3% 2.7%
XLP 0.12% 1.01 42 6 2% 2.0%
XLV 0.14% 1.05 38 7 3% 2.5%
XLY 0.16% 1.03 39 6 3% 2.5%
Total 0.15% 0.96 40 6 3% 2.4%
Statistics on the Estimated OU Parameters
Average over 2006-2007