The Self-Financing Equation in High Frequency Markets

High Frequency Trading (HFT) represents an ever growing pro-portion of all financial transactions as most markets have now switched toelectronic order book systems. The main goal of the paper is to proposecontinuous time equations which generalize the self-financing relationships offrictionless markets to electronic markets with limit order books. We use NAS-DAQ ITCH data to identify significant empirical features such as price impactand recovery, rough paths of inventories and vanishing bid-ask spreads. Start-ing from these features, we identify microscopic identities holding on the tradeclock, and through a diffusion limit argument, derive continuous time equa-tions which provide a macroscopic description of properties of the order book.These equations naturally differentiate between trading via limit and marketorders. We give several applications (including hedging European options withlimit orders, market maker optimal spread choice, and toxicity indexes) to il-lustrate their impact and how they can be used to the benefit of Low FrequencyTraders (LFTs)