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We consider a two-way trading problem, where investors buy and sell a stockwhose price moves within a certain range. Naturally they want to maximize theirprofit. Investors can perform up to $k$ trades, where each trade must involvethe full amount. We give optimal algorithms for three different models whichdiffer in the knowledge of how the price fluctuates. In the first model, thereare global minimum and maximum bounds $m$ and $M$. We first show an optimallower bound of $\varphi$ (where $\varphi=M/m$) on the competitive ratio for onetrade, which is the bound achieved by trivial algorithms. Perhaps surprisingly,when we consider more than one trade, we can give a better algorithm that losesonly a factor of $\varphi^{2/3}$ (rather than $\varphi$) per additional trade.Specifically, for $k$ trades the algorithm has competitive ratio$\varphi^{(2k+1)/3}$. Furthermore we show that this ratio is the best possibleby giving a matching lower bound. In the second model, $m$ and $M$ are notknown in advance, and just $\varphi$ is known. We show that this only costs usan extra factor of $\varphi^{1/3}$, i.e., both upper and lower bounds become$\varphi^{(2k+2)/3}$. Finally, we consider the bounded daily return model whereinstead of a global limit, the fluctuation from one day to the next is bounded,and again we give optimal algorithms, and interestingly one of them resemblescommon trading strategies that involve stop loss limits.

Optimal Online Two-way Trading with Bounded Number of Transactions