Integer-valued Lévy processes and low latency financial econometrics

Abstract

Motivated by features of low latency data in financial econometrics we study in detail integer-valued Lévy processes as the basis of price processes for high-frequency econometrics. We propose using models built out of the difference of two subordinators. We apply these models in practice to low latency data for a variety of different types of futures contracts.

Keywords:

• Futures markets
• High frequency econometrics
• Low latency data
• Negative binomial
• Skellam
• Tempered stable

JEL Classification :

• C01
• C14
• C32

Acknowledgements

We are grateful to the low latency data providers QuantHouse (http://www.quanthouse.com) for allowing us to use their data in this paper. We would also like to acknowledge useful and informative discussions with Richard Adams, Giuliana Bordigoni, Holger Fink, Tom Kelly, Anthony Ledford, Andrew Patton and Kevin Sheppard. The comments of three referees and the Editor, Frederi Viens, were also helpful.

Notes

†By low latency data we mean high-frequency data that are not conflated and that contain very few data errors such as zero or negative bid–ask spreads. ‘Conflation’ is a process by which a number of consecutive data updates are aggregated with only the aggregate result being reported by the data provider. This process produces an incorrect record of market activity. Apart from conflation, some data feeds that are not fast enough may also simply drop or miss data updates. Such data losses can result in inconsistent bid and ask prices that may present as zero or negative bid–ask spreads. With good low latency data we go as far as current technology allows to getting an accurate record of market activity.

‡The ‘active day’ is restricted to the period 12.30–18.00 GMT, for each of the data sets we study.

†By ‘returns’ here and elsewhere we mean the tick-size normalised raw price changes as opposed to the log-returns or arithmetic-returns more conventionally used in lower-frequency financial analysis.

†Note that both and . Hence a very basic factor model for the bid and ask in continuous time is to model a discrete-valued martingale and two stationary non-negative discrete-valued processes and .

†Discrete infinite divisibility for distributions on ℕ₀ = {i: i = 0, 1, 2,…} is discussed briefly by Bondesson (1992) and more extensively by Steutel and Van Harn (2004).

†The latter point is easy to fix by allowing the intensity process of price changes to vary with the price level.