• Search in book...
• Toggle Font Controls
64 PASSIVE–AGGRESSIVE MEAN REVERSION
Optimization Problem 3: PAMR-2
b
t+1
= arg min
b∈
m
"
1
2
b b
t
2
+Cξ
2
#
s.t.
(b;x
t
) ξ. (9.4)
We refer to this variant as “PAMR-2.”
Remarks on Loss Function: In our loss function of Equation 9.1, we use the
portfolio return b·x
t
, while it is possible to use log return log(b ·x
t
) (Latané 1959).
With the log utility, optimization problems Equations 9.2 through 9.4 are all noncon-
vex and nonlinear, and thus difﬁcult to solve. One way to solve them is to use logs
ﬁrst-order Taylor expansion at last portfolio and ignore higher order terms, that is,
log(b ·x
t
) log(b
t
·x
t
) +
x
t
b
t
·x
t
(b b
t
).
After the approximation, the nonlinear term becomes linear, and the optimization
problems are thus convex and can be efﬁciently solved. However, such linear approx-
imation may have some drawbacks. First of all, there is no way to justify the goodness
of linear approximation. With log utility, the loss function is ﬂat, then sharply rises
and ﬁnally ﬂattens out. While linear approximation is good in the two ﬂat regimes, it
is terrible at the point of nondifferentiability and subpar in the sharply rising region.
Moreover, linear approximation yields a upper bound on regret of log utility loss
function. For the loss function in the form of Equation 9.1 without log utility or
with log’s linear approximation, the best possible regret, in a minimax sense, is at
most O(
n) (Abernethy et al. 2009), while true log loss minimization algorithm
can routinely achieve O(log n). However, our loss function is not a traditional loss
function maximizing return (or minimizing the loss of log b ·x
t
), but only a tool
to realize mean reversion. Thus, the regret achieved using our loss function does not
represent a regret about return, which may not be meaningful as traditional regret
bound is. Anyway, though on empirical evaluations PAMR works well, anyone who
cares about its theoretical aspects should be notiﬁed about the possible worse bound,
which may not be elicited by the empirical evaluations.
Remarks on Formulations:Although our formulations mainly focus on portfolio
return without explicitly dealing with risk (e.g., volatility of daily returns), the ﬁnal
algorithms can be nicely interpreted as certain trade-offs between risk and return, as
discussed in Section 9.4. Such an interesting observation is further veriﬁed by our
empirical evaluation, which shows that the proposed PAMR algorithms achieve good
risk-adjusted returns in terms of two risk-related metrics (volatility risk and drawdown
risk).
Similar to existing studies, our formulations avoid incorporating transaction cost,
which simpliﬁes and highlights PAMR’s key ingredients. As shown in Sections 2.2
and 13.4, itisstraightforward to evaluate the impact oftransactioncosts. In Chapter 13,
we present results on both cases: with and without transaction costs. The results
Empirically, log utility does not help much, since log(b ·x
t
) and b ·x
t
are both small. However,
theoretically, using log utility may help. We remark on it so as to attract theoretical interest from other
researchers.
T&F Cat #K23731 — K23731_C009 — page 64 — 9/29/2015 — 18:26
ALGORITHMS 65
show that in most markets, the proposed algorithms work well without or even with
moderate transaction costs.
9.3 Algorithms
We now derive the solutions for the three PAMR formulations using standard tech-
niques from convex analysis (Boyd and Vandenberghe 2004) and present the proposed
PAMR algorithms. Speciﬁcally, the following three propositions summarize their
closed-form solutions.
Proposition 9.1 The solution to optimization problem 1 (PAMR) without consider-
ing the nonnegativity constraint (b 0) is expressed as
b = b
t
τ
t
(x
t
−¯x
t
1), (9.5)
where ¯x
t
=
x
t
·1
m
denotes market return, and τ
t
is computed as
τ
t
= max
0,
b
t
·x
t
x
t
−¯x
t
1
2
!
. (9.6)
Proof The proof can be found in Appendix B.2.1.
Proposition 9.2 The solution to optimization problem 2 (PAMR-1) without consid-
ering the nonnegativity constraint (b 0) is expressed as
b = b
t
τ
t
x
t
−¯x
t
1
,
where ¯x
t
=
x
t
·1
m
denotes market return, and τ
t
is computed as
τ
t
= max
0, min
C,
b
t
·x
t
x
t
−¯x
t
1
2
!!
. (9.7)
Proof The proof can be found in Appendix B.2.2.
Proposition 9.3 The solution to optimization problem 3 (PAMR-2) without consid-
ering nonnegativity constraint (b 0) is expressed as
b = b
t
τ
t
x
t
−¯x
t
1
,
where ¯x
t
=
x
t
·1
m
denotes the market return, and τ
t
is computed as
τ
t
= max
0,
b
t
·x
t
x
t
−¯x
t
1
2
+
1
2C
!
. (9.8)
Proof The proof can be found in Appendix B.2.3.
T&F Cat #K23731 — K23731_C009 — page 65 — 9/29/2015 — 18:26
66 PASSIVE–AGGRESSIVE MEAN REVERSION
Algorithm 9.1 details the proposed PAMR algorithms, and Algorithm 9.2 sum-
marizes the OLPS procedure utilizing PAMR. Firstly, with no historical information,
the initial portfolio is set to uniform b
1
=
1
m
,...,
1
m
. At the beginning of period t,
we rebalance the portfolio following the decision made at the end of period t 1.
At the end of t-th period, the market reveals a stock price relative vector, which rep-
resents the market movements. Since both portfolio and price relatives are already
known, the portfolio manager computes the portfolio daily return b
t
·x
t
and the loss
(b
t
;x
t
) as deﬁned in Equation 9.1. Then, we calculate an optimal step size τ
t
based
on last portfolio and stock price relatives. Given an optimal step size τ
t
, we can update
the portfolio for the next period. Finally, by projecting the updated portfolio into the
simplex domain, we normalize the ﬁnal portfolio.
Moreover, our algorithms have two key parameters, viz., the sensitivity param-
eter and the aggressiveness parameter C. In practice, their values could affect the
performance of the proposed algorithms. To achieve a good performance in a spe-
ciﬁc market, the parameters have to be ﬁnely tuned. We will thoroughly examine
the two parameters on real-life datasets and suggest their empirical selections in
Section 13.3.2.
Algorithm 9.1: Passive–Aggressive Mean Reversion: PAMR(,C,b
t
, x
t
1
,t).
Input: ∈[0, 1]: sensitivity parameter; C: aggressiveness parameter; b
t
:
current portfolio; x
t
1
: historical market sequence; t: index of current
Output: b
t+1
: Next portfolio.
begin
Suffer loss:
t
= max{0, b
t
·x
t
};
Set parameters:
τ
t
=
t
x
t
−¯x
t
1
2
(PAMR)
min
C,
t
x
t
−¯x
t
1
2
!
(PAMR-1)
t
x
t
−¯x
t
1
2
+
1
2C
(PAMR-2)
Update portfolio:
b
t+1
= b
t
τ
t
x
t
−¯x
t
1
Normalize portfolio:
b
t+1
= arg min
b
m
b b
t+1
2
end
T&F Cat #K23731 — K23731_C009 — page 66 — 9/29/2015 — 18:26
ANALYSIS 67
Algorithm 9.2: Online portfolio selection with PAMR.
Input: ∈[0, 1]: sensitivity parameter; C: aggressiveness parameter;
x
n
1
: historical market sequence.
Output: S
n
: ﬁnal cumulative wealth.
begin
Initialize b
1
=
1
m
,...,
1
m
, S
0
= 1;
for t = 1,...,ndo
Rebalance the portfolio to b
t
;
t
= (x
t,1
,...,x
t,m
);
Calculate the daily return and cumulative return: S
t
= S
t1
×
b
t
x
t
;
Update the portfolio: b
t+1
= PAMR
,C,b
t
, x
t
1
,t
;
end
end
9.4 Analysis
To reﬂect the mean reversion trading idea, we are interested in analyzing PAMR’s
update rules, which mainly involve portfolio b
t+1
and step size τ
t
. In particular, we
want to examine how the update rules are related to return and risk—the two most
important concerns in a portfolio selection task.
First of all, we analyze the portfolio update rule for the three algorithms, that is,
b
t+1
= b
t
τ
t
x
t
−¯x
t
1
.
The step size τ
t
is nonnegative, and ¯x
t
is mean return or market return. The x
t
−¯x
t
1
represents stock abnormal returns with respect to the market on period t. We can
further interpret it as a directional vector for the weight transfer. The negative sign
before the term indicates that the update scheme is consistent with our motivation, that
is, to transfer weights from outperforming stocks (with positive abnormal returns) to
underperforming stocks (with negative abnormal returns).
It is interesting that the second part of the update,
a
t
=−τ
t
x
t
−¯x
t
1
,
coincides with the general form (Lo and MacKinlay 1990, Eq. (1)) of return-based
contrarian strategies (Conrad and Kaul 1998; Lo 2008), except a changing multi-
plier τ
t
. This part represents an arbitrage (zero-cost) portfolio, since its elements
always sum to 0, that is, a
t
·1 = 0. Adding the arbitrage portfolio to the last portfolio,
b
t
, results in the next portfolio. The long elements of the arbitrage portfolio (a
t,i
> 0)
increase the corresponding elements of the whole portfolio, and the short elements
(a
t,i
< 0) decrease the corresponding elements. Such an explanation is similar to the
analysis in the last paragraph and connects PAMR’s update with the general form of
return-based contrarian strategies.
Besides, another important update is the step size τ
t
calculated as Equations 9.6
through 9.8 for three PAMR methods, respectively. The step size τ
t
T&F Cat #K23731 — K23731_C009 — page 67 — 9/29/2015 — 18:26
68 PASSIVE–AGGRESSIVE MEAN REVERSION
the weights to be transferred by scaling the directional vector. One common term
in τ
t
is
t
x
t
−¯x
t
1
2
. Its numerator denotes the -insensitive loss for period t, which
equals the t-th portfolio return minus a mean reversion threshold, or zero. Assuming
other variables are constant, if the return is high (low), it leads to a large (small)
value of τ
t
, which would aggressively transfer more (less) wealth from outperforming
assets to underperforming assets. The denominator is essentially the market quadratic
variability, that is, the number of assets times market variance of period t. In modern
portfolio theory (Markowitz 1952), the variance of assets returns typically measures
volatility risk for a portfolio. As indicated by the denominator, if the risk is high (low),
the step size τ
t
would be small (large). Consequently, the weight transfer made by the
update scheme will be weakened (strengthened). This is consistent with our intuition
that prediction would not be accurate in drastically dropping markets, and we opt to
make less transfer to reduce risk. Moreover, PAMR-1 caps the step size by a constant
C, while PAMR-2 decreases the step size by adding a constant
1
2C
to its denominator.
Both mechanisms can prevent drastic weight transfers in case of noisy price relatives,
which is consistent with their motivations.
From the above analysis on the updates of portfolio and step size, we can conclude
that PAMR nicely balances between return and risk and clearly reﬂects the mean rever-
sion trading idea. To the best of our knowledge, such an important trade-off has only
been considered by nonparametric kernel-based Markowitz-type strategy (Ottucsák
and Vajda 2007). While the strategy trades off return and risk with respect to a set of
similar historical price relatives, the proposed PAMR explicitly trades off return and
risk with respect to last price relatives. This nice property distinguishes the proposed
approach from most existing approaches that often cater to return, but ignore risk,
and are therefore undesirable.
One objective for PAMR-1 and PAMR-2 is to prevent a portfolio from being
affected too much from noisy price relatives, which might drastically change the
portfolio. In this part, let us exemplify the beneﬁts of PAMR’s variants. Let x
t
=
(1.00, 0.01), whose second value is a noise, and b
t
= (1, 0). Setting = 0.30 and
C = 1.00, we can calculate the next portfolio b
t+1
. This market sequence describes
that certain stocks drop signiﬁcantly, which is common during the ﬁnancial crisis.
Without tuning, PAMR would transfer a large proportion to the second asset. This can
be veriﬁed by calculating PAMR’s portfolio; in other words, PAMR calculates the
update step size τ
t
= 1.43 and obtains the subsequent portfolio b
t+1
= (0.29, 0.71).
However, to avoiding such noises, a natural choice is to transfer less proportion
to the second asset. On the other hand, PAMR-1 and PAMR-2 obtain the step
sizes of τ
t
= 1.00 and τ
t
= 0.71, respectively, which are smaller than the origi-
nal PAMR’s. Accordingly, we obtain the next portfolios b
t+1
= (0.50, 0.50) and
b
t+1
= (0.65, 0.35) for PAMR-1 and PAMR-2, respectively. Clearly, the variants
transfer less wealth to the second asset than the original PAMR does. Thus, PAMR-1
and PAMR-2 suffer less from noisy price relatives, though they cannot completely
avoid such suffering situations.
Finally, let us analyze PAMR’s time complexity. Besides a normalization/
projection step (Step 7 in Algorithm 9.1), PAMR takes O(m) per period. In our
T&F Cat #K23731 — K23731_C009 — page 68 — 9/29/2015 — 18:26
• No Comment
..................Content has been hidden....................