Table 3.3a. Expected upper bounds for discreteness bios: daily returns

Kxlrcctcd upper bounds for discreteness bins in simple returns 1H - H,\x 100 under a geometric rat: dom walk for prices /, with drill and diffusion parameters ft and о calibrated to annual mean am standard deviation of simple returns m and д, respectively, each ranging from 10% to 50%, am then rescaled to match daily data, i.e., /t/360, (T/V.ltiO. Oiscreti/.ed prices Г, = [Il/tljrf. 4 = 0.125, arc used to calculate returns K s (/, () - I.

Table З.Я. 1Ы*сМ Чфгг bounds for disc,*..... bias: m , M, rc,ums.

--, --ii-ГЗОХ 7=10% a = 50%

s = io% * = Я* . LnillfL -----

I /< - ii I х UK) uiulcr ii eeouuiiif fxpcc.eduppc, b( ,n,ls r,.r<l,sc,e.e essbiasi s ,.!,-rctmns I - lliinnllllllI1,.all

and standard devtation of stmple returns and ,., . aml (1,n rescaled to match mon dy .lata ...... <</. ;. -

d = 0.125, are used to calculate returns It, = I,/, ,

Table 3. Jr. I.xjieilrd n/i/iei Imiimls fin discreteness Inns: uiiiiii/il returns.

I.xpcrtcd uppci hounds lot disc icicncsshias ill .simple i elm ns /( - /(, x 100 under а geometric random walk lot pi ii eswiili ill ill and dilliisiihi parameters m and о calibrated mammal mean and standard devi.iiiiin ul simple returns in and >. respectively, each ranging Ггош ID, in 50%. Disrrrti/cd ptii es - (/,/<<<<. - .125, are used ui calculate returns Л, s (/, , . ,1-1.

3.3. Modeling Transactions Data

(1985), and Harris (1990) all provide methods for estimating a consistently from the observed price process P .M

liimier Models

A slightly different but closely related set of models of price discreteness has been proposed by Cho and Frees (1988) and Marsh and Rosenfeld (1980) which we shall call barrier models. In these models, the continuous-state line price process P, is also a continuous-time process, and trades are observed whenever P, reaches certain levels or barriers.

Marsh and Rosenfeld (1980) place these barriers at multiples of an eighth, so that conditional on the most recent trade at, say 40, the wailing time until the next trade is the first-passage time of P, to two barriers, one at 40 j and the other at 40 (assuming that P, has positive drift).

Cho and Frees (1988) focus on gross returns instead of prices and define slopping limes т as

Therefore, according to their model a stock which has just traded al time r i at $10.000 a share will trade next at time r when the unobserved continuous-stale gross returns process P,/$10.000 reaches either 1.125 or 1/1.125, or when P, reaches either $10.125 or $8.888. If P, reaches $8.888, the stock will trade next when P, reaches either $10.000 or $7.901, and so

This process captures price-discreteness of a very different nature smce the price increments defined by the stopping times are not integer Multiples of any fixed quantity (for example, the lower barrier 1/1.125 does not correspond to a one-eighth price decline). However, such an unnatural]definition of discreteness does greatly simplify the characterization of stopping times and the estimation of the parameters of P since the first-difference of r is IID. I

Under the more natural specification of price discreteness, not considered by Cho and Frees (1988), the stopping time becomes

т* = infb > T -,: -- i (l---- , 1 + ---И (3.3l )

I /(r. ,) r \ f(T -l> Р(т ,)Л j

which reduces to the Marsh and Rosenfeld (1986) model in which thej increments of stopping times are not IID.

lowever, see ihe discussion at (he end of Section 3.3.2 for some caveats about the motivation lui these models.

Limitations

Although all of ihe previous rounding and barrier models do capture price discreteness and admit consistent estimators of the parameters of the unol>-served continuous-slate price process, they sulfcr from at least three important limitations.

First, for unobserved price processes other than geometric Brownian motion, these models and their corresponding parameter estimators become intractable.

Second, the rounding and barrier models focus exclusively on prices and allow no role for other economic variables that might influence price behavior, e.g., bid-ask spreads, volatility, trading volume, etc.

Third, and most importantly, the distinction between the true and observed price is artificial at best, and the economic interpretation of the two quantities is unclear. For example, Ball (1988), Clio and Frees (1988), Gottlieb and Kalay (1985), and Harris (1990) all provide methods for estimating the volatility of a continuous-time price process from discrete olv served prices, never questioning the motivation of this arduous task. If die continuous-time price process is an approximation to actual market prices, why is the volatility of the approximating process of interest? One might argue that derivative pricing models such as the Black-Scholes/Mcrton formulas depend on the parameters of such continuous-time processes, but thoscy models are also approximations to market prices, prices which exhibit iiscreteness as well. Therefore, a case must be made for the economic relevance of the parameters of continuous-stale price processes to properly motivate the statistical models of discreteness in Section 3.3.2.

hit the absence of a well-articulated model of true price, it seems unnatural to argue that the true price is continuous, implying that observed discrete market prices are somehow less genuine. After all, the economic clefiimion of price is that quantity of numeraire at which two mutually consenting economic agents arc willing to consummate a trade. Despite the fact that institutional restrictions may require prices to fall on discrete values, as Ion ;as both buyers and sellers arc aware of this discreteness in advance and aie still willing lo engage in trade, then discrete prices corresponding to maikct trades arc true prices in every sense.

3.3.3 The Ordered Probit Model

To adtlrcss the limitations of the rounding and barrier models, Hausman, Lo, an rl MacKinlay (1992) propose an alternative in which price changes are modeled directly using a statistical model known as ordered probit, a technique used most frequently in empirical studies of dependent variables that take on only a finite number of values possessing a natural ordering.37 Heuristically,

For example, the dependent variable might be the level ol education, as measured by three categories: less than high school, high school, and college education. The dependent

ordered probit analysis is a generali/.ation of the linear regression model to cases where the dependent variable is discrete. As such, among the existing models of stock price discreteness-e.g., Ball (1988), Clio and Frees (1988), Gottlieb and Kalay (1985), I larris (1990), and Marsh and Rosenfeld (1986)-ordered probit is the only specification that can easily capture the impact of explanatory variables on price changes while also accounting for price discreteness and irregular transaction intervals.

The liasir Specification

Specifically, consider a sequence of transaction prices l(k\), l\t\).....I\t )

sampled at times / , Ц,..., / , and denote by ), ) >. ..., Y the corresponding price changes, where Yk s P(lk) - P(tk~\) is assumed to be an integer multiple of some divisor, e.g., a lick. Let )/ denote an unobsei vable continuous random variable such that

Y- = Xji + c, К[е*Х4] = 0, с INlDAAO.a/), (3.3.H)

where the (i/xl) veclor X* = [ Au X,(t ] is a vector of explanatory variables that determines the conditional mean of Yk and INID indicates that the eVs are independently but not identically distributed, an important difference from standard econometric models which we shall return to shortly. Note that subscripts are used to denote transaction time, whereas time arguments lk denote calendar or r/or7< time, a convention wc shall follow throughout Section 3.3.3.

The heart of the ordered probit model is the assumption that observed price changes Yk are related to the continuous variables Yk in the following manner:

if г; 6 Лi I if € Л,

Y =

(3.3.15)

where the sets A, form a partition oilhc stale space S* of Yk , i.e., 5* - Ц=1 Ai and At П A, = 0 for i ф j, and the s/s arc the discrete values dial comprise the state space S of Yk.

The motivation for the ordered probit specification is lo uncover the mapping between 5* and 5 and relate il to a set ol economic variables. In Hausman, l.o, and MacKinlay (1992), the .v,s are defined as: 0, -й, +s,

variable is discrete and is naturally ordered since college education always lollows high .school (seeNf.ul<lalallW:qit.rfurlherdetails).lhe...xle,e<lp.<.bit..uKlelwas<level..pe<lbyA.tclus<.u

and Silvey (1957) and Ashlord (1959), and generalized to nonnormal disturbances by (.mailt I.

l.ee,andDahm (191,0). For more recent extensions, see Maddala (I9H: ). McCullagh (19Я0),

andThisicd (1991).