Chapter 11
Online Moving Average Reversion
Empirical evidence shows that a stock’s high and low prices are temporary, and stock
price relatives are likely to follow the mean reversion phenomenon. While exist-
ing mean reversion strategies can achieve good empirical performance on many real
datasets, they often make a single-period mean reversion assumption, which is not
always satisfied, leading to poor performance on some real datasets. To overcome the
limitation, this chapter (Li et al. 2015) proposes a multiple-period mean reversion,
or the so-called moving average reversion (MAR), and a new online portfolio selec-
tion (OLPS) strategy named the online moving average reversion (OLMAR), which
exploits MAR by applying powerful online learning techniques. Our empirical eval-
uations in Part IV show that OLMAR can overcome the drawbacks of existing mean
reversion algorithms and achieve significantly better results, especially on the datasets
where existing mean reversion algorithms failed. In addition to superior performance,
OLMAR also runs extremely fast, further supporting its practical applicability to a
wide range of applications.
This chapter is organized as follows. Section 11.1 analyzes existing works and
motivates the proposed strategy. Section 11.2 formulates the strategy, and Section 11.3
solves the formulations and derives the algorithms. Section 11.4 further analyzes the
proposed algorithm. Finally, Section 11.5 summarizes this chapter and indicates future
directions.
11.1 Preliminaries
11.1.1 Related Work
Most existing formulations follow the basic routine of Kelly-based portfolio selec-
tion (Kelly 1956; Thorp 1971). In particular, a portfolio manager predicts
˜
x
t+1
in terms
of k possible values
˜
x
1
t+1
,...,
˜
x
k
t+1
and their corresponding probabilities p
1
,...,p
k
.
Note that each
˜
x
i
t+1
denotes one possible combination vector of individual price rela-
tive predictions. Then, he or she can figure out a portfolio by maximizing the expected
log return,
b
t+1
= arg max
b
m
k
i=1
p
i
log(b ·
˜
x
i
t+1
).
83
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84 ONLINE MOVING AVERAGE REVERSION
Based on the methods to predict
˜
x
i
t+1
and p
i
, most existing algorithms can be clas-
sified into three categories. Table 11.1 summarizes their optimization formulations
and underlying prediction schemes, whose details can be found on their respective
studies. Note that we have transformed certain formulations without changing their
key ideas.
Now let us introduce the three categories according to their empirical perfor-
mance. The second category, which consists of successive constant rebalanced
portfolio (SCRP) and online Newton step (ONS), assumes that the predictions consist
of all historical price relatives with uniform distribution. That is, at period t +1, the
price relative vector may be x
i
,i = 1,...,t with a probability of
1
t
. In other words,
this category aims to model the next price relatives as their historical average. The
algorithms in this category all present good theoretical regret bound and are universal.
However, their empirical results show that such an assumption may be inappropriate
to model the market behaviors. The third category, which mainly consists of the pat-
tern matching–based algorithms, models the next price relatives as a sampled set of
similar price relatives. In particular, denoting the similar index set as C
t
, it models the
next price relative vector as x
i
,i C
t
, with a uniform probability of
1
|C
t
|
. The algo-
rithms in this category (except correlation-driven nonparametric learning [CORN])
enjoy the universal consistency property, and their empirical results also show that
such an assumption can explain the markets well.
Algorithms in the first category, which consists of exponential gradient (EG),
passive–aggressive mean reversion (PAMR), and confidence-weighted mean rever-
sion (CWMR), assume a single prediction value with a probability of 100% and
maintain previous portfolio information via regularization techniques. In particu-
lar, EG assumes
˜
x
1
t+1
= x
t
with p
1
= 100%, while PAMR and CWMR assume
˜
x
1
t+1
=
1
x
t
with p
1
= 100%, which is in essence mean reversion. Note that the formu-
lations of PAMR and CWMR ignore the log utility due to the single-value prediction
and the consideration of convexity and computation. Though all three algorithms
assume that all information is fully reflected by x
t
, their performance diverges and
supports that mean reversion may better explain the markets. On the one hand, even
with a decent theoretical result, EG always performs poorly. On the other hand, though
without theoretical guarantees, PAMR and CWMR have produced the best results in
certain real markets. However, when such a single-period mean reversion assumption
is not satisfied, PAMR and CWMR would suffer from dramatic failures (Li et al.
2012, Table 4, the DJIA dataset), which motivates the following approach.
11.1.2 Motivation
Empirical results (Li et al. 2011b, 2012) show that mean reversion, which assumes
the poor stock may perform well in the subsequent periods, may better explain the
markets. PAMR and CWMR can exploit the mean reversion property well and achieve
good results on most datasets at the time, especially on the New York Stock Exchange
benchmark dataset (Cover 1991). However, they rely on a naïve assumption that next
This assumption requires some transformations.That is, given x
t
R
+
m
, minimizing b·x
t
is equivalent
to maximizing b ·
1
x
t
. The latter follows the analysis framework here.
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PRELIMINARIES 85
Table 11.1 Summary of existing optimization formulations and their underlying predictions
Categories Methods Formulations Prediction (
˜
x
i
t+1
) Probability (p
i
)
In hindsight BCRP b
t+1
= arg max
b
m
n
i=1
1
n
log b·x
i
x
i
,i = 1,...,n 1/n
1EG b
t+1
= arg max
b
m
log b·x
t
λR(b, b
t
) x
t
1.00
PAMR b
t+1
= arg min
b
m
b ·x
t
+λR(b, b
t
) 1/x
t
1.00
CWMR b
t+1
= arg min
b
m
Prob(b ·x
t
) +λR(b, b
t
) 1/x
t
1.00
2 SCRP b
t+1
= arg max
b
m
t
i=1
1
t
log b·x
i
x
i
,i = 1,...,t 1/t
ONS b
t+1
= arg max
b
m
t
i=1
1
t
log b·x
i
λR(b) x
i
,i = 1,...,t 1/t
3B
K
/B
NN
/CORN b
t+1
= arg max
b
m
iC
t
1
|C
t
|
log b·x
i
x
i
,i C
t
1/|C
t
|
Note: R(·) and R(·, ·) denote regularization terms, such as L
2
norm. PAMR/CWMR’s prediction is not of strictly
equivalence, which we do not prove.
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86 ONLINE MOVING AVERAGE REVERSION
price relative
˜
x
t+1
will be inversely proportional to last price relative x
t
. In particular,
they implicitly assume that next price
˜
p
t+1
will revert to last price p
t1
,
˜
x
t+1
=
1
x
t
=⇒
˜
p
t+1
p
t
=
p
t1
p
t
=⇒
˜
p
t+1
= p
t1
.
Note that both x and p are vectors and the above operations are element-wise.
Though empirically effective on most datasets, PAMR and CWMR’s single-
period assumption causes two potential problems. Firstly, both algorithms suffer
from frequently fluctuating raw prices, as they often contain a lot of noise. Secondly,
their assumption of single-period mean reversion may not always be satisfied in the
real world. Even two consecutive declining price relatives, which are common, can
deactivate or fail both algorithms. One real example (Li et al. 2012) is the DJIA
dataset (Borodin et al. 2004), on which PAMR performs the worst among the state of
the art. Thus, traders are more likely to predict prices using some long-term values.
Also on the DJIA dataset, Anticor, which exploits the multiperiod statistical corre-
lation, performs much better than others. However, due to its heuristic nature (Li
et al. 2011b, 2012), Anticor cannot fully exploit the mean reversion property. The
two problems caused by the single-period assumption and Anticors inability to fully
exploit mean reversion call for a more powerful approach to effectively exploit mean
reversion, especially in terms of multiple periods.
Now let us see a classic example (Cover and Gluss 1986) to illustrate the draw-
backs of single-period mean reversion, as shown in Table 11.2.Thetoymarketconsists
of cash and one volatile stock, whose market sequence follows A. It is easy to prove
that best constant rebalanced portfolio (BCRP) (b =
1
2
,
1
2
) can grow by a factor
of
9
8
n/2
, while PAMR can grow by a better factor of
3
2
×2
(n1)/2
. Note that this
virtual sequence is essentially single-period mean reversion, which perfectly fits with
PAMR and CWMR’s assumption. However, if market sequence does not satisfy such
an assumption, both PAMR and CWMR would fail badly. Let us extend the market
sequence to a two-period reversion, that is, market sequence B. In such a market,
BCRP can achieve the same growth as before. Contrarily, PAMR can achieve a con-
stant wealth
3
2
, which has no growth! More generally, if we further extend to k-period
mean reversion, BCRP can still achieve the same growth, while PAMR will grow to
3
2
×
1
2
(n1)×
1
2
1
k
, which definitely approaches bankruptcy if k 3.
To better exploit the (multiperiod) mean reversion property, we proposed a new
type of algorithms, OLMAR, for OLPS. The essential idea is to exploit multiperiod
moving average (mean) reversion via power online machine learning. Rather than
˜
p
t+1
= p
t1
, OLMAR assumes that the next price will revert to a moving average
(MA), that is,
˜
p
t+1
= MA
t
, where MA
t
denotes the MA till the end of period t.In
time-series analysis, MA focuses on long-term trends and is typically used to smooth
short-term price fluctuations, and thus can solve the two drawbacks of existing mean
reversion algorithms.
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PRELIMINARIES 87
Table 11.2 Illustration of the mean reversion strategies on toy markets
Market Sequences BCRP PAMR OLMAR
A:(1, 2),
1,
1
2
,(1, 2),
1,
1
2
,...
9
8
n/2
3
2
×2
n1
2
9
8
B:(1, 2), (1, 2),
1,
1
2
,
1,
1
2
,(1, 2),...
9
8
n/2
3
2
9
16
×2
n4
2
C:(1, 2), (1, 2), (1, 2),
1,
1
2
,
1,
1
2
,
1,
1
2
,(1, 2),...
9
8
n/2
3
2
×
1
2
n1
6
9
8
×2
n5
6
D:(1, 2),...,(1, 2)
-
./ 0
k=4
, (1, 1/2),...,(1, 1/2)
-
./ 0
k=4
, (1, 2),...
9
8
n/2
3
2
×
1
2
n1
4
9
4
×2
n6
8
E:(1, 2),...,(1, 2)
-
./ 0
k=5
, (1, 1/2),...,(1, 1/2)
-
./ 0
k=5
, (1, 2),...
9
8
n/2
3
2
×
1
2
(n1)×
3
10
9
2
×2
n7
10
Note: Since OLMAR is sensitive to the windows size, we set its windows to k. We calculate all
OLMAR values with a mean reversion threshold of 2.
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