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Continuous time analysis of fleeting discrete price moves

Neil Shephard, Justin J. Yang

Abstract

This paper proposes a novel model of financial prices where: (i) prices are discrete; (ii) prices change in continuous time; (iii) a high proportion of price changes are reversed in a fraction of a second. Our model is analytically tractable and directly formulated in terms of the calendar time and price impact curve. The resulting càdlàg price process is a piecewise constant semimartingale with finite activity, finite variation and no Brownian motion component. We use moment-based estimations to fit four high frequency futures data sets and demonstrate the descriptive power of our proposed model. This model is able to describe the observed dynamics of price changes over three different orders of magnitude of time intervals.

Continuous time analysis of fleeting discrete price moves

Abstract

This paper proposes a novel model of financial prices where: (i) prices are discrete; (ii) prices change in continuous time; (iii) a high proportion of price changes are reversed in a fraction of a second. Our model is analytically tractable and directly formulated in terms of the calendar time and price impact curve. The resulting càdlàg price process is a piecewise constant semimartingale with finite activity, finite variation and no Brownian motion component. We use moment-based estimations to fit four high frequency futures data sets and demonstrate the descriptive power of our proposed model. This model is able to describe the observed dynamics of price changes over three different orders of magnitude of time intervals.

Paper Structure

This paper contains 29 sections, 8 theorems, 90 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Integer-valued stochastic process in continuous time
  3. Poisson random measure
  4. Lévy basis and Lévy process
  5. Stationary trawl process
  6. Integer-valued price process with fleeting moves
  7. Definition
  8. Distribution of price changes
  9. Autocorrelation structure of price changes
  10. Power variation
  11. Generalized compound representation
  12. Parameterized trawl function
  13. Moment-based inference
  14. Inference of Lévy measure
  15. Inference of permanence and trawl function
  16. ...and 14 more sections

Key Result

Proposition 1

If $leb\left( A\right) <\infty$, then $L\left( A_{t}\right)$ is well-defined and strictly stationary. If $\kappa _{2}\left( L_{1}\right) <\infty$ as well, then it is covariance stationary and for $t>s$ Furthermore, for any $t\geq 0$, is monotonically decreasing as $t$ increases.

Figures (12)

  • Figure 1: Top: Lévy basis $L(\mathrm{d}x,\mathrm{d}s)$, where the horizontal axis $s$ is time and the vertical axis $x$ is height, which plays no rule in this construction of the Lévy process in the lower panel. Black dots denote $1$, white ones $-1$. Bottom: The corresponding Lévy process, which sums up all the effects in the Lévy basis (in the upper panel) from time $0$ to time $t$, while the vertical axis here is the value of the Lévy process, which jumps up by 1 by the effect of black dots and down by 1 by white ones. Code: LpTprocess_Illustration.R.
  • Figure 2: A moving squashed trawl $A_{t}$ is joined by the Lévy basis $L(\mathrm{d}x,\mathrm{d}s)$, where the horizontal axis $s$ is time and the vertical axis $x$ is height. The shaded area is an example of the trawl $A$ generated by the trawl function $d$, while we also show the outlines of $A_{t}$ when $t=1/2$ and $t=1$. Also shown below is the implied stationary process $L(A_{t})$ and the Lévy process $L(B_{t})$ for $t\geq 0$, where $B_{t}=[0,b)\times (0,t]$. Code: LpTprocess_Illurstration.R.
  • Figure 3: $10,000$ Monte Carlo simulation of moment estimations for the price process with exponential trawl $d\left( s\right) =b+\left( 1-b\right) \exp \left( \lambda s\right)$ and the Skellam basis $\nu \left( \mathrm{d}y\right) =\nu ^{+}\delta _{\left\{ 1\right\} }\left( \mathrm{d}y\right) +\nu ^{-}\delta _{\left\{ -1\right\} }\left( \mathrm{d}y\right)$. The vertical lines in each of the histograms mean the true value. The Monte Carlo standard deviations are reported on the scale of the true values. Code: Moment_Inference_ModelBasedBootstrap.R.
  • Figure 4: The complete trace plots for the four data sets during 00:00 to 21:00. The $x$-axis is the calendar time (HH:MM), while the $y$-axis is the price ($). Code: Price_Plots.R.
  • Figure 5: The trace plots for two TNC data sets during 09:00 to 10:00 and for two EUC data sets during 12:46 to 12:48. The $x$-axis is the calendar time (HH:MM:SS), while the $y$-axis is the price ($). Code: Price_Plots.R.
  • ...and 7 more figures

Theorems & Definitions (33)

  • Remark 1
  • Example 1
  • Definition 1
  • Example 2
  • Proposition 1
  • Example 3: Continued from Example \ref{['Ex.: Stationary trawl\nprocess']}
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem \oldthetheorem
  • ...and 23 more