Download PDF by Hoi, Steven C. H.; Li, Bin: Online portfolio selection : principles and algorithms

By Hoi, Steven C. H.; Li, Bin

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Example text

If the stock price follows the GBM model, the regret round becomes O(m log Q), where Q is a quadratic variability calculated as n − 1 times the sample variance of price relative vectors. Since Q is typically much smaller than n, the regret bound is significantly improved from previous O(m log n). Besides the improved regret bound, the authors also discussed the relationship between their algorithm and trading frequency. The authors asserted that increasing the trading frequency would decrease the variance of minimum-variance CRP, while the regret stays the same.

Formally, a strategy’s drawdown (DD) at period t is defined as DD(t) = sup[0, supi∈(0,t) Si − St ]. Its maximum drawdown (MDD) is the maximum of drawdowns over all periods and can effectively measure a strategy’s downside risk. Formally, maximum drawdown for a horizon of n, MDD(n), is defined as MDD(n) = sup [DD(t)]. t∈(0,n) Moreover, practitioners also adopt the Calmar ratio (CR) (Young 1991) to measure a strategy’s drawdown risk-adjusted return: CR = APY . MDD The smaller the maximum drawdown, the more drawdown risk the strategy can tolerate.

Suppose we are locating the price relative vectors that are similar to the next vector xt+1 . The basic routine is to iterate all historic price relatives xi , i = w + 1, . . , t and count xi as one similar vector, if its i−1 t preceding market window xi−w is similar to the latest market window xt−w+1 . A set Ct contains the indexes of similar price relatives. Note that the market window is a w × m-matrix and the similarity is typically calculated on the concatenated w × m-vectors. 1 further illustrates the procedure.

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