Price signatures

Abstract

Price signatures are statistical measurements that aim to detect systematic patterns in price dynamics localised around the point of trade execution. They are particularly useful in electronic trading because they uncover market dynamics, strategy characteristics, implicit execution costs, or counter-party trading behaviours that are often hard to identify, in part due to the vast amounts of data involved and the typically low signal to noise ratio. Because the signature summarises price dynamics over a specified time interval, it constitutes a curve (rather than a point estimate) and because of potential overlap in the price paths it has a non-trivial dependence structure which complicates statistical inference. In this paper, I show how recent advances in functional data analysis can be applied to study the properties of these signatures. To account for data dependence, I analyse and develop resampling-based bootstrap methodologies that enable reliable statistical inference and hypothesis testing. I illustrate the power of this approach using a number of case studies taken from a live trading environment in the over-the-counter currency market. I demonstrate that functional data analysis of price signatures can be used to distinguish between internalising and externalising liquidity providers in a highly effective data driven manner. This in turn can help traders to selectively engage with liquidity providers whose risk management style best aligns with their execution objectives.

Keywords:

• FX liquidity
• Execution costs
• Functional data analysis
• Bootstrap methods

Acknowedgment

The Author would like to thank three anonymous referees, Anthony Neuberger, Andrew Patton, Jin-Ting Zhang, colleagues at Deutsche Bank, and the participants at TradeTech FX 2018 in Miami and Barcelona, the DB Quant Seminar Series, the Social and Economic Data Science seminar at the London School of Economics, the Financial Mathematics Practitioners Seminar at University College London, the Profit & Loss Forex Network 2018 in London and Chicago, Duke University, and Trinity College Cambridge for helpful comments.

Disclaimer

The views and opinions rendered in this paper reflect the authors' personal views about the subject and do not necessarily represent the views of Deutsche Bank AG or any part thereof. This article is necessarily general and is not intended to be comprehensive, nor does it constitute legal or financial advice in relation to any particular situation.

Disclosure

The author is employed as a Managing Director in the FX trading division at Deutsche Bank A.G. Deutsche Bank (DB) is an industry recognised world leader in the foreign exchange business, and offers a full spectrum of foreign exchange products and services, including the trading of foreign exchange products through its Autobahn electronic trading platform. This paper was prepared within the Sales and Trading function of DB, and was not produced, reviewed or edited by the DB Research Department. The views and opinions rendered in this paper reflect the author's personal views about the subject. No part of the author's compensation was, is, or will be directly related to the views expressed in this paper.

Notes

† Alongside the opportunities this vast amount of data offers, it also poses a variety of challenges, including legal questions regarding ownership, privacy and security considerations, and the search for sustainable and efficient storage solutions ranging from eco-friendly data centres in the arctic circle (e.g. http:/kolos.com) to data encoding into bacterial DNA sequences (e.g. Extance 2016).

‡ I assume that a page of text stores 10 KB, and that a sheet of paper is 0.1 mm thick. The distance to the sun is 149.6e6 km or 1 astronomical unit (AU). One ZB is thus equivalent to $n=\left(1024{)}^7/\right(10240)$ of pages, which when stacked would measure a height of $n/1e7/149.6e6=77$ AU.

† Appendix 2 provides a few more examples in figures A1 and A2 on global warming, mortality rates, and fertility rates.

† Two other important recommendations are for the trader to execute each ticket for the full amount with a single selected dealer (rather than splitting it across multiple dealers simultaneously), and to leave a sufficient amount of time in between consecutive executions to allow the dealer to internalise the risk. Both are controllable by the trader and practical guidance on the latter is found in Butz and Oomen (2018).

† In practice, one may incorporate the paid or earned bid-offer spread into the signature so that $S\left(0\right)$ measures the volume weighted average spread. I abstract away from this for simplicity of exposition, but all the methods presented in this paper directly apply to this case.

† This of course assumes that the trader holds on to their trades for longer than δ. If the trades are closed out prior to that, the incremental variation in the signature value represents the opportunity costs of not having held on to the positions longer.

† The post-trade adverse selection within this model is given as $-(1-{\beta }^{\delta })\omega \sqrt{1-\rho }{c}_M$ where δ is the signature horizon, β measures the dependence in the LP's measurement error of the unobserved efficient price, ρ denotes the correlation of measurement error across LPs, ω is the efficient price volatility, and $c_M$ is a constant that depends on the number of LPs M that compete in the aggregator. In the simulations I set $\sigma =0.5,\beta =0.9,\omega =0.5,\rho =0.5$ and M=3 or 10.

‡ The signatures themselves, on the other hand, are independent from one another provided that they are calculated over non-intersecting time periods displaced by more than δ and the price process is a martingale. Functional data analysis when there is dependence in the underlying price process (rather than in the set of sampled price paths) is studied by, for instance, Horváth et al. (2014) and Zhang (2016).

† With non-IID data, the decision whether to apply the L2-norm test or say the integrated F test is similar to whether one would bootstrap the sample mean or the t-statistic. Typically the former is the more natural one, but depending on the data properties both may be valid. Also note that the weighting by $N_k$ of SSH in equation (10(10) $\begin{array}{rl}& \mathrm{SSH}\left(\delta \right)=\sum _{k=1}^K{N}_k\left(S_k\right(\delta )-S(\delta ){)}^2,\mathrm{and}\mathrm{SSE}(\delta )\\ & =\sum _{k=1}^K\sum _{n=1}^{N_k}\left(s_n^{\left(k\right)}\right(\delta )-S_k(\delta ){)}^2,\end{array}$(10) ) also relies on the IID assumption, and may require adjustments particularly when there is significant between group dependence, or strong heterogeneity in trade arrival frequency across groups. This is beyond the scope of this paper, and of no concern for the empirical analysis below.

† Assuming a 5.5 day / 24 hour trading session, this translates into an average duration between trades of about 8 minutes, but regularly going as low as 10 seconds and as high as 2 hours.