Table 3.2. Ilelatiiv frequencies of juice changes for lick ilata of five slocks.

Relative frequency count, in percent, lor all 1991 transaction price changes in ticks tor live NYSK slocks: AACAnacouip; Atl)=Air Products and Chemicals; CHS=Cotiuuhia Broadcasting System; CCU=Capital t.iiics ABC; KAb=K.incl> Services.

lire 3.2 using all ol ilie stocks transactions (luring the 1091 calendar year. Tltc lower-pi u ed slocks-КЛН and AAC-have very lew transaction price changes beyond the -1 tick to +1 tick range; these three values account lor 99.0% and 99.3% ol all the trades Cor KAR and AAC, respectively. In contrast, lot a higher-priced stock like (ICR, with an average price of $468 (luting 1991, lite range from -I lick to +1 tick accounts for 03.6% of its Hades. While discreteness is relatively less pronounced for CCB, it ;s nevertheless still present. К ven when we turn lo daily data, the histograms of daily price changes in figure 3.2b show that discreteness can still be important, especially for lower-priced stocks such as KAB and AAC.

Moreover, discreteness may be more evident in the conditional and joint distribution ol high frequency returns, even if it is difficult to delect in the unconditional or marginal distributions. For example, consider the graphs in Figure 3.3a in which pairs of adjacent daily simple returns (/? R,+ i) are plotted for each of the live stocks in fable 3.1 over the three-year sample period. These m-histories (here, m = 2) are often used to detect structure in nonlinear dynamical systems (see Chapter 12). The scales of the two axes are identical for all live stocks to make cross-stock comparisons meaningful, and range from -5% to 5% in Figure 3.3a, -10% lo 10% in Figure 3.3b, and -20% to 20% in Figure 3.3c.

Figure 3.3a shows that (here is considerable structure in the returns of the lower-priced stocks, KAB and AA(this is a radially symmetric structure that is solely attributable lo discreteness. In contrast, no slruclurc is evident in die 2-lusiorics of the higher-priced stocks, CBS and CCB. Since APDs initial price is in between those of the other lourslocks, it displays less sirur-

2-History of KAB Returns, P = $4.665 (a) (b) (c)

figure 3.3. 2-Histories of Daily Slock Returns for Five NYSE Slocks from January 2, 1990

to December 31, 1992

tttre (ban the lower-priced stocks but more than the higher-priced stocks. Figures 3.3b and 3.3c show that changing the scale of the plots can ofjten reduce and, in the case of APD, completely obscure the regularities associated with discreteness. For further discussion of these 2-histories, see Crack and I.edoit (1996).

These empirical observations have motivated several explicit models of trice discreteness, and wc shall discuss the strengths and weaknesses of each >f these models in the following sections.

3.3.2 Rounding and Barrier Models

SJcveral models of price discreteness begin with a true but unobserved continuous-state price process Pt, and obtain the observed price process P by discretizing P, in some fashion (see, for example, Ball [1988], Cho and Frees [1988], and Gottlieb and Kalay [1985]). Although this maybe a convenient starting point, the use of the term true price for the continuous-state price process in this literature is an unfortunate choice of terminology-il implies that the discrete observed price is an approximation to the true price when, in fact, the reverse is true: continuous-stale models are approximations to actual market prices which arc discrete. When the approximation errors inherent in continuous-slate models are neglected, this can yield misleading inferences, especially for transactions data.32

Rounding Errors

To formalize this notion of approximation error, denote by X, ihe gross return of the continuous-state process P, between /-1 ami I, i.e., X, = P,/P,-\. Wc shall measure the impact of discreteness by comparing X, to the gross returns process X° = P°/P l corresponding to a discrclized price process P°.

The most common method of discretizing P, is to round il lo a multiple of d, the minimum price variation increment. To formalize this, wc shall require the floor and ceiling functions

[xj = greatest integer < x (floor function) (3.3.1) \x \ = least integer > x (ceiling function), (3.3.2)

for any real number x.33 Using (3.3.1) and (3.3.2), wc can express the three most common methods of discretizing compactly as

Pi =

(3.3.3)

sThc question of which price is the true prire may not he crucial for the statistical aspects of itlotlels of discreteness-alter all, whether one is an approximation lo the other or vie e-versa aliens only the sign of the approximation error, not its absolute magnitude-but it is central to t te motivation and interpretation of the results (see lite discussion at the end of Section 3.3.2 for examples). Therefore, although we shall adopt the terminology of this literature lot the pioment, the reader is asked to keep this ambiguity in mind while reading this section.

IFor further properties and applications of these integer functions, see Graham, Knutli, andPatashhik (1989. Chapter 3).

i < 1

P = - + -1 d 2

(3.3.4)

(3.3.5)

where the first method rounds down, the second rounds up, and the third rounds lo the nearest multiple of d. For simplicity, wc shall consider only (3.3.3), although our analysis easily extends to the other two methods.

At the heart оГthe discreteness issue is the difference between the return X, based on continuous-slate prices and the return X based on discrclized prices. To develop a sense of just how different these two returns can be, we shall construct an upper bound for ihe quantity X - X, = \R° - R,{, where R, and R denote the simple net return of ihe continuous-stale anil discrclized price processes, respectively. Let x and у be any two arbitrary nonnegative real numbers such dial у > 1, and observe that

x- 1

Subtracting х/у from (3.3.6) then yields

(3.3.6)

(3.3.7)

i- I

(3.3.8)

which implies the inequality

I Ы У \ У

Assuming thai P, > d for alU, we may set x S P,/d, у = P,-,/rfand substitute these expressions into (3.3.8) to obtain ihe following upper bound:

\r;-r,\

5,-i

1-5,-,

Max [ X, , 1

= ЦХ !,-,), (3.3.9)

where 5, ; = rf/P, i is defined lo be the grid size at lime I-I.

Although die upper bound (3.3.9) is a strict inequality, it is in fact the least upper bound, i.e., for any fixed d and any e > 0, there always exists some combination of and X, for which \R° - R,\ exceeds !.{&, X f, i) - (. Therefore, (3.3.9) measures the worst-case deviation of R/ from R and it is the tightest of all such measures.

Note dial (3.3.9) does not yield a uniform tippet bound in r since /.

depends on :

ЦХ <5, )

1-5,.

I + Max [ R, . 1

(3.3.10)

Never! lu-lcss, il still provides a useful guideline for the impact of discreteness on returns as prices and returns vary, for example, (3.3.9) formalizes the intuition that discreteness is less problematic for higher-priced slocks, since /. is an increasing function of A, i and, therefore, a decreasing function ol/V,.

It is important to keep in mind that (3.3.9) is only an upper bound, and while il does provide a measure of the worst-case discrepancy between , and , it is not a measure of the discrepancy itself. This distinction is best understood by grappling with die fact that the expected upper bound l-l /.(A ($, , )\&, , I is an increasing function of the mean and variance of .V,- the larger the expected return and volatility, the larger is the average value of the upper bound. This seems paradoxical because it is generally presumed that discreteness is less problematic for longer-horizon returns, hut these have higher means and variances by construction. The paradox is readily resolved by observing that although the expected upper bound increases as the mean and variance increase, the probability mass of , - R,\ near the upper bound may actually decline. Therefore, although the expected worst-case discrepancy increases with the mean and variance, the probability that such discrepancies are realized is smaller. Also, as we shall see below, the expected upper bound sccius to lie relatively insensitive to changes in the mean and variant с of Л so that when measured as a percentage ol die expected return K .Y U. expected upper bound does decline for longer-horizon returns.

By specifying a particular prot ess foi /> we can evaluate the expectation ol /.( ) lo develop some sense for (he magnitudes of expected discreteness bias E[ , - ( that are possible. For example, let /, follow a geometric random walk with drift /t and diffusion coefficient a so (hat log P,/P,-1 are 111) normal random variables with mean ц and variance a-. In this case, we have

/ <</. Л,.i) I /,

log(I-(5) - ц - a-

+ Ф K< !-*)-/<

(3.3.11)

where <t>() is the normal С.ПЕ:,Г

Ideally, we would like to t Ii.u.k tcii/c\Щ - /f, directly. 1)41 it is surprisingly difficult lo do so Willi .my dcgicc ol gcnci alily. I lowevi г, see tile discussion below regarding the rounding and lianic! models-nude, spent paiaincliM assumptions loi ,Y more precise characlcri/alions ol I lit- dist reteness bias are available.

Note the siiuilaiiiv between (:i.:s.l) : i.l tbe Шаск-Sclmles call-option pri< ing formula.

Tables 3.3a-c report numerical values of (3.3.9) for price levels P,-\ = $1, $5, $10, $50, $100, and $200, and for values of Ц and a corresponding i annual means and standard deviations for simple returns ranging from 10% to 50% each, respectively, and then rescaled to represent daily returns in Table 3.3a, monthly returns in Table 3.3b, and annual returns in Table 3.3c.

Table 3.3a shows that for stocks priced at $1, the expected upper bound for the discreteness bias is approximately 14 percentage points, a substantial bias indeed. However, this expected upper bound declines to approximately 0.25 percentage points for a $50 stock and is a negligible 0.06 percentage points for a $200 stock. These upper bounds provide the rationale for the empirical examples of Figures 3.3a-c and the common intuition that discreteness has less of an impact on higher-priced stocks. Table 3.3a also shows that for daily returns, changes in the mean and standard deviation of returns have relatively little impact on the magnitudes of the upper bounds.

Tables 3.3b and 3.3c indicate that the potential magnitudes of discreteness bias are relatively stable, increasing only slightly as the return-horizon increases. Whereas the expected upper bound is about 2.5 percentage points for daily returns when = $5, it ranges from 2.8% to 3.9% for annual returns. This implies that as a fraction of the typical holding pejriod return, discreteness bias is much less important as the return horizon increases. Not surprisingly, changes in the mean and standard deviation of returns have more impact with an annual return-horizon.

Rounding Models

Even if E[ ° - Rt\] is small, the statistical properties of P° can still differ in subtle but important ways from those of P,. If discreteness is an unavoidable aspect of the data at hand, it may be necessary to consider a more explicit statistical model of the discrete price process. As we suggested abovje, a rounding model can allow us to infer the parameters of the continuous-slate process from observations of the rounded process. In particular, in much of the rounding literature it is assumed that P, follows a geometric Browi(iian motion dP = tiPdt + aPdW, and the goal is to estimate /X and a fom the observed price process P°. Clearly, the standard volatility estimator a based on continuously compounded observed returns will be an inconsistent

estimator of a, converging in probability to yEKlog/, - log/?)2] rather

than to v/E[(logP,+i -logP,)2] . Moreover, it can be shown that a will be an overestimate of cr in the presence of price-discreteness (see Ball [1988, Table I] and Gottlieb and Kalay [1985, Table I] for approximate magnitudes of this upward bias). Ball (1988), Cho and Frees (1988), Gottlieb and Kalay

Ibis is no accident, since MaxX may be rewritten as Max[X, - (1 -5).0J + 1-J; hence the upper bound may be recast as the payoff ol a call option on X, with strike price 1.