# The self-financing equation in limit order book markets

## Abstract

The goal of this paper is to present a mathematical framework for trading on a limit order book, including its associated transaction costs, and to propose continuous-time equations which generalise the self-financing relationships of frictionless markets. These equations naturally differentiate between trading via limit and via market orders, as they include a price impact or adverse selection constraint. We briefly mention several possible applications, including hedging European options with limit orders, to illustrate their impact and how they can be used to the benefit of low-frequency traders. Two appendices include empirical evidence for facts which are not universally recognised in the current literature on the subject.

## Keywords

Self-financing equation Limit order book markets## Mathematics Subject Classification (2010)

91G80 91G20## JEL Classification

C6## Notes

### Acknowledgements

We should like to thank the Associate Editor and two anonymous referees for thoroughly reviewing several versions of the paper. Their insightful comments helped us improve dramatically its readability. Moreover, we are grateful to one of the referees for pointing out an error in the original version of Proposition 4.5.

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