FORMULATIONS 89
Moving Average Reversion: MAR-2
˜
x
t+1
(α) =
EMA
t
(α)
p
t
=
αp
t
+(1 −α)EMA
t−1
(α)
p
t
= α1 +(1 −α)
EMA
t−1
(α)
p
t−1
p
t−1
p
t
= α1 +(1 −α)
˜
x
t
x
t
,
(11.2)
where α ∈ (0, 1) denotes the decaying factor and the operations are all element-wise.
Based on the expected price relative vector in Equations 11.1 and 11.2, OLMAR
further adopts the idea of an effective online learning algorithm, that is, passive–
aggressive (PA) (Crammer et al. 2006) learning, to exploit the MAR. Generally
proposed for classification, PA passively keeps the previous solution if the classi-
fication is correct, while aggressively approaches a new solution if the classification
is incorrect. After formulating the proposed OLMAR, we solve its closed-form update
and design specific algorithms.
The proposed formulation, OLMAR, is to exploit MAR via PA online learning.
The basic idea is to maximize the expected return b·
˜
x
t+1
and keep last portfolio
information via a regularization term. Thus, we follow the similar idea of PAMR (Li
et al. 2012) and formulate an optimization as follows.
Optimization Problem: OLMAR
b
t+1
= arg min
b∈
m
1
2
b −b
t
2
s. t. b ·
˜
x
t+1
≥
.
Note that we adopt expected return rather than expected log return. According to
Helmbold et al. (1998), to solve the optimization with expected log return, one can
adopt the first-order Taylor expansion, which is essentially linear. Such discussions
are illustrated in Sections 9.2 and 10.2.
The above formulation explicitly reflects the basic idea of the proposed OLMAR.
On the one hand, if its constraint is satisfied, that is, the expected return is higher than
a threshold, then the resulting portfolio becomes equal to the previous portfolio. On
the other hand, if the constraint is not satisfied, then the formulation will figure out
a new portfolio such that the expected return is higher than the threshold, while the
new portfolio is not far from the last one.
Since OLMAR follows the same learning principle as PAMR, their formulations
are similar. However, the two formulations are essentially different. In particular,
PAMR’s core constraint (i.e., b·x
t
≤ ) adopts the raw price relative and has a dif-
ferent inequality sign. After a certain transformation, PAMR may be written in a
similar form, as shown in Table 11.1. However, the prediction functions are different
(i.e., OLMAR adopts multiperiod mean reversion, while PAMR exploits single-period
mean reversion).
T&F Cat #K23731 — K23731_C011 — page 89 — 9/30/2015 — 16:44