Modelling financial transaction price movements: a dynamic integer count data model

Abstract

In this paper we develop a dynamic model for integer counts to capture fundamental properties of financial prices at the transaction level. Our model relies on an autoregressive multinomial component for the direction of the price change and a dynamic count data component for the size of the price changes. Since the model is capable of capturing a wide range of discrete price movements it is particularly suited for financial markets where the trading intensity is moderate or low. We present the model at work by applying it to transaction data of two shares traded at the NYSE traded over a period of one trading month. We show that the model is well suited to test some theoretical implications of the market microstructure theory on the relationship between price movements and other marks of the trading process. Based on density forecast methods modified for the case of discrete random variables we show that our model is capable to explain large parts of the observed distribution of price changes at the transaction level.

Appendix

This appendix shows that under a correctly specified model for Y _i , the u _i 's drawn from the uniform distributions (4.3) follow a uniform distribution on the interval [0, 1].¹⁵ Consider a discrete random variable Y with support $\Delta \subseteq \mathbb{Z}$, and let u be a continuous random variable with the following conditional uniform distribution

$$u \sim {\user1{\mathcal{U}}}{\left( {u^{l}_{y} ,u^{u}_{y} } \right)},$$

(A.1)

where the boundaries are u _y ^l=Pr(Y≤y−1), u _y ^u=Pr(Y≤y) (for ease of notation we ignore the index i for the variables u and Y). Then, the c.d.f. of the unconditional distribution of u is

$$\Pr {\left( {u \leqslant c} \right)} = {\sum\limits_{y \in \Delta } {\Pr {\left( {\left. {u \leqslant c} \right|Y = y} \right)}\Pr {\left( {Y = y} \right)},\quad c \in {\left[ {0,1} \right]},} }$$

(A.2)

with

$$\Pr {\left( {\left. {u \leqslant c} \right|Y = y} \right)} = \frac{{c - u^{l}_{y} }} {{u^{u}_{y} - u^{l}_{y} }}{\user1{\mathcal{I}}}_{{\left[ {{\user1{}}u^{l}_{y} ,u^{u}_{y} } \right)}} {\left( c \right)} + {\user1{\mathcal{I}}}_{{{\left[ {u^{u}_{y} ,1} \right]}}} {\left( c \right)}$$

(A.3)

$$\Pr {\left( {Y = y} \right)} = u^{u}_{y} - u^{l}_{y} ,$$

(A.4)

where ${\user1{\mathcal{I}}}_{A} {\left( z \right)}$ is an indicator function which is 1 if z∈A and zero for z∉A. Inserting Eqs. (A.3) and (A.4) into Eq. (A.2), we obtain

$${\Pr {\left( {u \le c} \right)} = {\sum\limits_{y \in \Delta } {{\left\{ {{\left( {c - u^{l}_{y} } \right)}{\user1{{\cal I}}}_{{\left[ {{\user1{}}u^{l}_{y} ,u^{u}_{y} } \right)}} {\left( c \right)} + {\left( {u^{u}_{y} - u^{l}_{y} } \right)}{\user1{{\cal I}}}_{{{\left[ {u^{u}_{y} ,1} \right]}}} {\left( c \right)}} \right\}}} }.}$$

(A.5)

Assuming that c∈[u _j ^l, u _j ^u], j∈Δ, we find

$$\begin{array}{*{20}c} {\Pr {\left( {u \leqslant c} \right)} = c{\user1{\mathcal{I}}}_{{{\left[ {0,1} \right]}}} {\left( c \right)} - \Pr {\left( {Y \leqslant j - 1} \right)} + \ldots } \\{ + \Pr {\left( {Y \leqslant j - 3} \right)} - \Pr {\left( {Y \leqslant j - 4} \right)}} \\{ + \Pr {\left( {Y \leqslant j - 2} \right)} - \Pr {\left( {Y \leqslant j - 3} \right)}} \\{ + \Pr {\left( {Y \leqslant j - 1} \right)} - \Pr {\left( {Y \leqslant j - 2} \right)}} \\{ = c{\user1{\mathcal{I}}}_{{{\left[ {0,1} \right]}}} {\left( c \right)},} \\\end{array} $$

(A.6)

which represents the c.d.f. of a uniform distribution on the interval [0, 1].