This article provides a comprehensive overview of online portfolio selection algorithms that fall into the aforementioned categories. First, it is important to note that the portfolio selection task in our research differs from a large number of financial engineering studies [Kimoto et al. Finally, this paper focuses on exploring the algorithmic aspects and providing a structural understanding of existing online portfolio selection strategies.
In this section, we model the transaction costs in our formulation, which allows us to evaluate online portfolio selection algorithms. Algorithms in this area formulate the online portfolio selection task as in Section 2 and derive explicit portfolio updating schemes for each period. It is straightforward to see that given the same upper/lower bound assumption of price-related as Cover's UP [Cover 1991, Theorem 6.1], the regret bound is on the same scale as Cover's UP, although the constant term is slightly worse.
Follow-the-Loser Approaches
At the beginning of the 2nd period, the CRP manager rebalances the portfolio to initial uniform portfolio by transferring the wealth from well-performing stock (B) to poor-performing stock (A), which actually follows the mean reversion principle. At the beginning of the 3rd period, the wealth transfer continues with the mean reversion idea. Rather than no distribution assumption like Cover's UP, Anticor strategy assumes that the market follows the mean reversion principle.
To exploit the mean reversion property, it statistically makes a bet on the consistency of positive lagged cross-correlation and negative autocorrelation. On the other hand, although Anticor algorithm achieves good performance outperforming all algorithms at that time, its heuristic nature cannot fully exploit the mean reversion property. 2012] proposed Passive Aggressive Mean Reversion (PAMR) strategy, which exploits the mean reversion property with the Passive Aggressive (PA) online learning [ Shalev-Shwartz et al.
The closed form update scheme clearly reflects the mean reversion trading side by transferring the wealth from the high performing stocks to the poor performing stocks. Similar to the Anticor algorithm, due to the mean reversion nature of PAMR, it is difficult to obtain a meaningful theoretical regret bound. 2011b] proposed the Confidence Weighted Mean Reversion (CWMR) algorithm to further exploit second-order portfolio information, which refers to the variance of portfolio weights (not price or price relative), following the mean reversion trading side via Confidence Weighted (CW) online learning [Dredze et eel.
It is clear that the updating scheme reflects the mean reversion trading idea and can exploit both the first- and second-order information of a portfolio vector. 2012, DJIA dataset], Li and Hoi [2012] defined multiple-period average reversion called Moving Average Reversion and proposed OnLine Moving Average Reversion (OLMAR) to exploit the multiple-period average reversion.
Pattern-Matching Based Approaches
2006], which consists of two steps: the sample selection step and the portfolio optimization step.1 The first step - Sample Selection - selects an index set of similar historical price relatives, the relevant price relatives of which will be used to predict the future relative price. In the following sections, we flesh out the sample selection step in Section 3.4.1 and the portfolio optimization step in Section 3.4.2. Sampling techniques. The general idea in this step is to select similar samples from historical price relatives by comparing the previous market windows of the two price relatives.
Portfolio optimization techniques. The second step in the pattern-matching approaches is to construct an optimal portfolio based on the similar setC. 2007] introduced semi-log-optimal strategy, which approximates log in the log-optimal utility function with the aim of releasing the computation problem, and Vajda [2006] presented theoretical analysis and proved its universal consistency. Ottucs 'ak and Vajda [2007] proposed the nonparametric Markowitz-type strategy, which is a further generalization of the semi-log-optimal strategy.
2006] presented a kernel-based non-parametric log-optimal investment strategy (BK) that combines kernel sample selection and a log-optimal utility function and proved its universal consistency. 2008] then proposed a non-parametric log-optimal nearest-neighbor (BNN) investment strategy that combines nearest-neighbor sample selection and a log-optimal utility function and proved its universal consistency. 2011a] created a correlation nonparametric learning (CORN) approach by combining correlation sample selection and log-optimal utility function and demonstrated its superior empirical performance compared to the previous three combinations.
In addition to the log-optimal utility function, several algorithms using different utility functions have been proposed. Gy¨orfi and Vajda [2008] proposed the non-parametric kernel-based GV-type investment strategy (BGV) by combining the kernel-based sample selection and GV-type utility function to construct portfolios in case of transaction costs.
Meta-Learning Algorithms
Portfolio Optimization Histogram Kernel Nearest Neighbor Correlation Log-optimal BH: CH+UL BK: CK+UL BNN: CN+UL CORN: CC+UL. 2007] proposed the nonparametric kernel-based semi-log-optimal investment strategy (BS) by combining the kernel-based sample selection and the semi-log-optimal utility function to facilitate the calculation of (BK). Ormos and Urb'an [2011] empirically analyzed the performance of log-optimal portfolio strategies with transaction costs.
Note that this section only presents the key steps (or individual expert) in pattern matching-based approaches, while all previously presented algorithms also contain an additional clustering step. Clustering Algorithms. In addition to the algorithms discussed in Section 3.2.5, AA [Vovk 1990; Vovk and Watkins 1998] can also be generalized to more sophisticated base experts. The basic idea of FU is to distribute the wealth evenly among a set of core experts, let these experts operate alone, and ultimately pool their wealth.
FU's update is similar to that of Cover's UP, and it also asymptotically reaches the richness equivalent to an optimal fixed convex combination of basis experts. In addition to the universalization in the continuous parameter space, various discrete BAH combinations have been adopted by various existing algorithms. Moreover, all Pattern Matching-based approaches in Section 3.4 used BAH to combine their underlying experts, also with a finite number of window sizes.
Theoretically, OGU and ONU can achieve the growth rate as the optimal convex combination of the underlying experts. The basic idea is to maintain a working set of finite experts, which are dynamically turned on and off based on their performance, and assign the weights among the actively working experts using an MLA, for example the Herbster-Warmuth algorithm . Herbster and Warmuth 1998].
CONNECTION WITH CAPITAL GROWTH THEORY
Capital Growth Theory for Portfolio Selection
Online Portfolio Selection and Capital Growth Theory
In particular, the first four algorithms in the Follow-the-Winner category (i.e., Universal Portfolios, Exponential Gradient, Follow the Leader, and Follow the Regulated Leader) all release regret bounds whose daily average asymptotically approaches zero as the trading period up to into infinity. Note that some algorithms in the first connection (EG, ONS, etc.) can also be rewritten to this form, although their goals are different from KWB. We present their implied market distributions, denoted by their values (ˆxt+1) and probabilities (Probabilities), in the second and third columns, respectively.
We then rewrite all the algorithms that follow CGT—that is, to maximize the expected log return for the+1st period—in the fourth column. The first category, including EG/PAMR/CWMR/OLMAR/RMR, implicitly or explicitly predicts a single scenario with certainty and tries to select an optimal portfolio. Note that the PAMR and CWMR capital growth forms are copied from their original forms while retaining their essential ideas.
The second category, including model-based approaches, predicts multiple scenarios that are considered similar to the relative future price vector. In particular, it expects the next relative price to be xi,i ∈ C with a uniform probability of |C|1 , where CDenotes the similarity set. Note that some algorithms in model-based approaches, including BS, BM, and BGV, adopt different portfolio optimization approaches, which we do not enumerate here.
Note that unlike the regularization terms in the first category, the regularization terms in this category, such as R(b)= b2, only control the variance of the following portfolio. On the other hand, their motivations follow the first connection, which is validated by their theoretical results.
Underlying Trading Principles
The third category, including FTL and FTRL, implicitly predicts the next scenario like all historical price relationships. Based on such a prediction, strategies in this category aim to maximize the expected log return and derive a regularization term for the FTRL. That is, if the underlying experts are single-stock strategy, which is momentum, then we see AA's trading idea as momentum.
On the other hand, if the underlying experts are CRP strategy which follows the principle of mean reversion, we consider AA's trade idea as mean reversion.
CHALLENGES AND FUTURE DIRECTIONS
Accurate Prediction by Advanced Techniques
Although most existing prediction schemes focus solely on the price ratio (or price), there are other useful side information such as volume, fundamental and expert opinions. Thus, it is an open challenge to incorporate other sources of information to facilitate the prediction of next price relatives.
Open Issues of Portfolio Optimization
CONCLUSIONS
It is our hope that this survey article facilitates researchers to understand the state of the art in this area and may inspire more fruitful future studies. Continuous-Time Mean-Variance Portfolio Selection with Proportional Transaction Costs.SIAM Journal on Financial Mathematics. Security Prices and Stock Exchange Holidays in Relation to Short Selling.Journal of Business of the University of Chicago.
Semi-optimal empirical kernel-based portfolio selection strategies. International Journal of Theoretical and Applied Finance. Capital growth and the mean-variance approach to portfolio selection. Journal of Financial and Quantitative Analysis. CORN: A Correlation-Based Nonparametric Learning Approach for Portfolio Selection. ACM Transactions on Intelligent Systems and Technology.
Confidence Weighted Average Reversion Strategy for Online Portfolio Selection. ACM Transactions on Discovering Knowledge from Data. Long-Term Capital Growth: Pros and Cons of the Kelly and Fractional Kelly Capital Growth Criteria. Quantitative finance. 2005. Fortune's Formula: The Untold Story of the Scientific Betting System That Beat Casinos and Wall Street.
