SUMMARY 69
implementation, we adopt a linear projection method (Duchi et al. 2008),
∗
which
takes O(m) per period. In total, the time complexity is O(mn). Thus, PAMR has
the same time complexity as the EG algorithm and is more superior to other meth-
ods. Linear time complexity enables the proposed algorithm to handle transactions
in scenarios in which low latency is of crucial importance, such as high-frequency
trading (Aldridge 2010).
9.5 Summary
In this chapter, we proposed a novel online portfolio selection (OLPS) strategy,
passive–aggressive mean reversion (PAMR). Motivated by the idea of mean rever-
sion and passive–aggressive online learning, PAMR either aggressively updates the
portfolio following mean reversion, or passively keeps the previous portfolio. PAMR
executes in linear time, making it suitable for online applications. We also find that
its update scheme is based on the trade-off between return and volatility risk, which
is ignored by most existing strategies. This interesting property connects the PAMR
strategy with modern portfolio theory, which may provide further explanation from
the aspect of finance.
The proposed algorithms are still far from perfect and may be improved in the
following aspects. First of all, though the universality property may not be required
in real investment, PAMR’s universality is still an open question. Second, PAMR
sometimes fails if mean reversion does not exist in the market components. Thus, it is
crucial to locate asset sets exhibiting mean reversion. Finally, PAMR’s formulations
ignore transaction costs. Thus, directly incorporating the issue into formulations may
improve PAMR’s practical applicability.
∗
The precise MATLAB
routine ProjectOntoSimplex can be found on http://www.cs.
berkeley.edu/∼jduchi/projects/DuchiShSiCh08/
T&F Cat #K23731 — K23731_C009 — page 69 — 9/29/2015 — 18:26
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