• The authors would like to thank Rob Almgren, Tomasz Bielecki, Adrien De Larrard, Jason Ricci, and participants at the SIAM Conference on Financial Mathematics and Engineering 2012, Young Researchers Workshop on Finance Tokyo 2012, and Universidad Carlos III, Madrid. As well, the authors thank two anonymous referees for their comments which ultimately improved this paper. Finally, SJ thanks NSERC and Mprime for partially funding this work.


We propose risk metrics to assess the performance of high-frequency (HF) trading strategies that seek to maximize profits from making the realized spread where the holding period is extremely short (fractions of a second, seconds, or at most minutes). The HF trader maximizes expected terminal wealth and is constrained by both capital and the amount of inventory that she can hold at any time. The risk metrics enable the HF trader to fine tune her strategies by trading off different metrics of inventory risk, which also proxy for capital risk, against expected profits. The dynamics of the midprice of the asset are driven by information flows which are impounded in the midprice by market participants who update their quotes in the limit order book. Furthermore, the midprice also exhibits stochastic jumps as a consequence of the arrival of market orders that have an impact on prices which can give rise to market momentum (expected prices to trend up or down). The HF trader's optimal strategy incorporates a buffer to cover adverse selection costs and manages inventories to maximize the expected gains from market momentum.