Relative Volume As a Doubly Stochastic Binomial Point Process QUANTITATIVE FINANCE RESEARCH CENTRE

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QUANTITATIVE FINANCE RESEARCH CENTRE

Research Paper 146 October 2005

Relative Volume as a Doubly Stochastic Binomial Point Process

James McCulloch

ISSN 1441-8010 www.qfrc.uts.edu.au Relative Volume as a Doubly Stochastic Binomial Point Process∗

James McCulloch October, 2005

Abstract Relative Intra-day Volume on the NYSE Modelled as a Doubly Sto- chastic Binomial Point Process. Relative intra-day cumulative volume is intra-day cumulative volume divided by final total volume. If intra-day cumulative volume is modelled as a Cox (doubly stochastic Poisson) point process, then using initial enlargement of filtration with the filtration of the Cox process enlarged by knowledge of final volume, it is shown that relative intra-day volume conditionally has a binomial distribution and is a novel generalization of a binomial point process; the doubly stochastic binomial point process. Re-scaling the intra-day traded volume to a relative volume between 0 (no volume traded) and 1 (daily trading completed) allows empirical intra-day volume distribution information for all stocks to be used collectively to estimate and identify the random intensity component of the doubly stochastic binomial point process and closely related Cox point process.

Keywords: Binomial, Point Process, Doubly Stochastic, Relative Volume, Cox Process, Initial Enlargement of Filtration, NYSE, New York Stock Exchange

∗°c Copyright James McCulloch 2005. Contact email [email protected].

1 1 Introduction

The Cox1 (Doubly Stochastic Poisson) point process is used to model trade by trade market behaviour by a number of Financial market researchers including Engle and Russell [18], Engle and Lunde [18], Jasiak et al. [13] and Rydberg and Shephard [19].

If intra-day trade arrival is modelled as a Cox Point Process then relative volume is a Doubly Stochastic Binomial Point Process.

The conditional marginal distribution of the Doubly Stochastic Binomial point process, R(t) = N(t)/N(T ), is derived from the underlying Cox process by a straightforward application of the theory of initial enlargement of filtrations developed by Jeulin [15], Jacod [14] and Yor [22]. This distribution is shown to be equivalent to the conditional distribution of the underlying Cox Process under a change of measure due to the initially enlarged filtration produced by augmenting the natural filtration of the Cox process by the sigma algebra produced σ(N(T )) by knowledge of the the final value of the Cox process, N(T ) = K.

µ ¶ · ¸ · ¸ K Λ(t) K Λ(t) K−aK P ( R(t) = a | Λ(T )) = 1 − aK Λ(T ) Λ(T )

The result that the Doubly Stochastic Binomial point process is conditionally binomially distributed implies that the unconditional distributions of R(t) are a binomial mixture (Daley and Vere-Jones [8]) where the mixing distributions are the ﬁnite dimensional (ﬁ-di) distributions of the self-normalized integrated intensity of the underlying Cox process. µ ¶ Λ(t) Φ (s) = P ≤ s , s ∈ [0, 1] t Λ(T )

Z 1 P ( R(t) = a) = Binomial(aK; K, s) dΦt(s) 0 1Named a Cox process in recognition of David Cox’s 1955 [7] paper in which he introduced the Doubly Stochastic Poisson Point Process.

2 Re-scaling the intra-day traded volume to a relative volume between 0 (no volume traded) and 1 (daily trading completed) allows the use of stocks with different final trade counts to conveniently estimate the empirical unconditional (binomial mixture) fi-di distributions of the Doubly Stochastic Binomial point process in a 2 dimension histogram. This convenient statistical modelling is a major advantage of the Doubly Stochastic Binomial point process analysis proposed in this paper. Using the binomial mixture model, the histogram estimates of the fi-di distributions of R(t) are used to estimate the moments of the fi-di distributions of the self-normalized integrated intensity Λ(t)/Λ(T ) of the underlying Cox process. It is readily shown that the expectation of the relative volume process R(t) is equivalent to the expectation of the self-normalized integrated intensity for stocks of all trade counts. · ¸ Λ(t) E[R(t)] = E Λ(T ) By definition, this is a model of the ‘U’ shaped deterministic intra-day trading seasonality characteristic of equity markets2 (NYSE) and the actual seasonality is readily estimated from the 2-d histogram. This is useful for Volume Weighted Average Price (VWAP) traders who require a stochastic model of relative intra-day cumulative volume to implement risk optimal VWAP trading strategies3. Examination of the higher moments of the self-normalized integrated intensity, Λ(t)/Λ(T ), permits an analysis of models of the integrated intensity Λ(t) of the underlying Cox process. The integrated intensity is the compensator (Brémaud[4]) of the random counting measure N(t) of the Cox process, so Λ(t) = E[N(t)] and in particular, Λ(T ) = E[N(T )]. Since stocks have different final trade counts the integrated intensity for stock i,Λi(t), can be modelled using a baseline integrated intensity Λ0(t) in two ways. (i) The integrated intensity for stock i is modelled as a baseline integrated intensity scaled by a stock specific constant, αi, related to final trade count. Λi(t) = αiΛ0(t). (ii) The integrated intensity for stock i is modelled as a baseline integrated intensity where the time parameter is multiplied by a stock specific constant, αi, related to final trade count. Λi(t) = Λ0(αit) 2For further discussion of intra-day market seasonality see Brock and Kleidon [5], Ad- mati and Pfleiderer [1] and Coppejans, Domowitz and Madhavan [6]. 3Relative volume modelling and VWAP are discussed in a paper by the author (Mc- Culloch [16]).

3 In the first case, the self-normalized integrated intensity is the same for all stocks irrespective of final trade count and therefore all higher moments of the self-normalized integrated intensity will be the same irrespective of trade count. This is empirically tested by dividing stocks into trade count bands and calculating the variances of the self-normalized integrated intensities for different trade counts. The variances of the self-normalized integrated intensities for stocks of different trade count are different and case (i) is rejected.

This is an interesting insight since it suggests that the integrated intensities of different stocks are unique only up to a time scaling proxied by final trade trade count. This idea is developed further in a related paper by the author (McCulloch [17]) where it is shown the scaling of the variances of the self-normalized integrated intensities of stocks with different trade counts is closely related to the Hurst exponent (Embrechts and Maejima [10]) of the integrated intensity of the underlying Cox process of trade arrivals.

2 Trade Arrival as a Cox Process

A point process on the time index is a set of discrete events that can be ordered in time, t1 < t2 < ··· < tm <. If the time ordering is strict then there are no co-occurring points and the point process is simple. A time indexed point process is formally deﬁned by its random counting measure N(.). This is a random measure deﬁned on a probability space (Ω, F,P ) that maps subsets of a Borel algebra of non-negative real numbers onto the set of non-negative integers.

N(B, ω) → I,I ∈ Z+,B ∈ B(R+), ω ∈ Ω The value of the random counting measure on an interval including zero N([0, t], ω) is generally of interest and the notation will be truncated to N(t) where this is clear. Likewise the formal probability space notation will be suppressed wherever it is not explicitly required. Trade arrivals can be modelled as a point process with the cumulative trade count deﬁned by the random counting measure N(t). In particular, trade arrivals can be modelled as a Cox process, a simple point process directed by a stochastic intensity. The following limit4 deﬁnes the intensity

4Formally, a random integrated intensity may exist that does not have an integral representation (the intensity process limit does not exist), see Segall and Kailath [20] (page 136) for an example of such a point process. In all cases considered in this paper, an intensity process will be deﬁned for the point process.

4 process of the Cox process.

P [N(t + ∆) − N(t) > 0 | HΛ] λ(t) = lim t ∆→0+ ∆ The intensity process of a Cox process, λ(t), is a non-negative Lebesque Λ integrable stochastic process adapted to the filtration Ht at time t, this filtration may be larger than the natural filtration (internal history) of the point N Λ process, Ft ⊆ Ht . The integral of the intensity process is the integrated intensity.

Z t Λ(t) = λ(s) ds (1) 0 Λ The integrated intensity is a Ht -compensator of the random counting measure N(t) (Br´emaud[4]).

Λ N(t) − Λ(t) = Mt a Ht -martingale Conditional on a realization of the random integrated intensity, the Cox process is a Poisson point process (Daley and Vere-Jones [9] deﬁnition 6.2.I).

Λ(t)k P (N(t) = k | Λ(t)) = e−Λ(t) k! 3 The Doubly Stochastic Binomial Point Process

The intra-day trade count can be scaled to between 0 and 1 by the simple expedient of dividing the intra-day count (N(t) = aK) by the ﬁnal trade count (N(T ) = K). This deﬁnes the self-normalized trade count process R(t) which is formally named the random relative counting measure.

N(t) aK 1 K − 1 R(t) = = = a, a ∈ {0, ..., , 1} N(T ) K K K This section formalizes the relationship of the random relative counting measure R(t) and the self-normalized integrated intensity, Λ(t)/Λ(T ). It is unsurprising that the relationship between these measures is described by a binomial point process directed by the self-normalized integrated intensity. This point process is related to a binomial point process in way analogous to the relationship between a Cox point process and the Poisson point process.

Theorem 3.1. The distribution of the Doubly Stochastic Binomial point process conditional on the integrated intensity measure of the underlying Cox

5 process at time T , Λ(T ) and ﬁnal trade count N(T ) = K is a binomial distribution. The Doubly Stochastic Binomial point process is also shown to be semi-martingale.

P ( R(t) = a | Λ(T )) = P ( N(t) = aK | N(T ) = K, Λ(T ))

µ ¶ · ¸ · ¸ K Λ(t) aK Λ(t) (1−a)K = 1 − aK Λ(T ) Λ(T ) (2)

1 K − 1 K ∈ Z+, a ∈ {0, ..., , 1}, t ∈ [0,T ] K K

Proof. The proof is a straightforward application of the theory of initial enlargement of ﬁltrations. For further reading see [15], [14], [22], [2], [11], [12] with a summary on ﬁnancial applications by Baudoin [3].

Let (Ω, F, F,P ) be a ﬁltered probability space with ﬁltration F = (Ft)t≥0 satisfying the usual conditions. Assume ξ is a random variable on (Ω, F) taking values in a measurable Polish space (X, X ) where X is a sigma algebra on X. The σ-algebra of ξ is non-trivial with σ(ξ) * F0. The following is subject to some non-restrictive formal conditions to exclude pathological cases that do not arise in the context of this paper, see Jacod [14]. + Let t ∈ R and Φt(x) be the conditional distribution of random variable ξ given information Ft.

Φt(x) = P (ξ ∈ dx|Ft) dx ∈ X

Then by Jacod [14] there exists a process (πt(x))t≥0 deﬁned by the nor- malization of the conditional distribution with the unconditional distribution Φ(x) of the random variable ξ which is a (P, F)-martingale with π0(x) = 1.

Φ(x) = P (ξ ∈ dx)

Φ (x) π (x) = t , E[π (x)] = E[π (x)] = 1, ∀s 6= t t Φ(x) t s

The martingale πt(x) is the (Baudoin [3], page 47) density process for the measure change to the disintegrated probability measure P x(.) = P ( . | ξ = x)

x P = πt(x)P

6 This disintegrated probability measure is precisely what is required to calculate the distribution of P ( N(t) = aK | N(T ) = K, Λ(T )) given the distribution of P ( N(t) = aK | Λ(T )).

Noting that the distribution of the Cox process conditional on Λ(t) is the same as the distribution conditioned on Λ(T ).

P (N(t) = aK | Λ(t)) = P (N(t) = aK | Λ(T ))

Λ(t)aK = e−Λ(t) aK! Then using the disintegrated probability measure and replacing ξ = x with N(T ) = K.

P ( R(t) = a | Λ(T )) = P (N(t) = aK | N(T ) = K, Λ(T ))

= P K (N(t) = aK | Λ(T )),P K (.) = P (. | N(T ) = K)

= πt(K) P (N(t) = aK | Λ(T ))

Λ(t)aK = π (K) e−Λ(t) t aK!

So ﬁnding the conditional distribution of R(t) reduces to ﬁnding the change of measure martingale πt(x). The distribution of the Cox process N(T ) conditional on the σ-algebra Ft is equivalent to the distribution of N(T ) conditional on the outcome of N(t).

Φt(x) = P (N(T ) = x | Λ(T ), Ft)

= P (N(T ) = x | Λ(T ),N(t))

£ ¤ (Λ(T ) − Λ(t))x−N(t) = exp − (Λ(T ) − Λ(t)) (x − N(t))!

7 The distribution of the Cox process N(T ).

Φ(x) = P (N(T ) = x | Λ(T ))

£ ¤ Λ(T )x = exp − Λ(T ) x! So the change of measure martingale for the Cox process is:

Φ (x) £ ¤(Λ(T ) − Λ(t))x−N(t) x! π (x) = t = exp Λ(t) t Φ(x) Λ(T )x (x − N(t))! Therefore the distribution of R(t) conditional on Λ(T ) is:

P ( R(t) = a | Λ(T )) = P ( N(t) = aK | N(T ) = K, Λ(T ))

= P K ( N(t) = aK |Λ(T )),P K (.) = P (. | N(T ) = K)

= πt(K)P ( N(t) = aK |Λ(T ))

£ ¤(Λ(T ) − Λ(t))K−aK K! £ ¤ Λ(t)aK = exp Λ(t) exp − Λ(t) Λ(T )K (K − aK)! aK!

µ ¶ · ¸ · ¸ K Λ(t) K Λ(t) K−aK = 1 − aK Λ(T ) Λ(T )

Finally by Jacod’s [14] well-known theorem, a Ft semi-martingale is a Gt = Ft ∨ σ(ξ) semi-martingale (under non-restrictive formal conditions). Therefore since a Cox process is semi-martingale, the Doubly Stochastic Bi- nomial point process is also semi-martingale.

4 The Unconditional Distribution of R(t)

The unconditional distribution of a point process directed by a stochastic measure can be modelled as a mixture distribution of the point process (bi-

8 nomial) with the marginal distribution of the directing measure as the mixing distribution (see page 235 of Daley and Vere-Jones [8] for a full discussion). This implies that the unconditional distribution of the doubly stochastic binomial point process R(t) is a mixture distribution of the distribution of the self-normalized integrated intensity at time t,Ψt(s) (the mixing distribution), and the Binomial distribution of the probability of aK trades given K ﬁnal trades, B(aK|K, s). Thus if Θt(aK|K) is the probability of aK trades at time t given K trades at time T , then this distribution is the following mixture distribution:

Z 1 P (R(t) = a) = Θt(aK|K) = B(aK|K, s)Ψt(s) ds µ ¶ 0 K ¡ ¢ B(aK|K, s) = saK 1 − s K−aK aK µ ¶ Λ(t) Φ (s) = P ≤ s , ∀s ∈ [0, 1] t Λ(T ) dΦ (s) Ψ (s) = t t ds © 1 2 K − 1 ª K ∈ Z+, a ∈ , ,..., , 1 , t ∈ [0,T ] K K K

4.1 The Moments of the Self-Normalized Integrated Intensity Since the the distribution of relative intra-day trade count is a binomial mixture distribution, it is easy to extract the moments of the mixing distribution by using the following relation to relate moments of the relative trade count (observed data) to the moments of the mixing distribution.

µ ¶ Z · ¸ XK k n 1 1 XK Θ (k|K) = kn B(k|K, s) Ψ (s) ds (3) K t Kn t k=0 0 k=0

The first four non-central moments of the mixing distribution are calculated by substituting the non-central moments of the mixing distribution. Let δi be the ith non-central moment of the (observed binned data) relative trade count distribution, Θt(k|K), and λi be the ith non-central moment for the mixing distribution Ψt(s), with final trade count K, then the first 4 non-central mixing distribution moments are:

9 (i) λ1 = δ1

(ii) Kδ − λ λ = 2 1 2 K − 1 (iii) K2δ − 3Kλ + 3λ − λ λ = 3 2 2 1 3 K2 − 3K + 2 (iv) K3δ − 6K2λ + 18Kλ − 12λ − 7Kλ + 7λ − λ λ = 4 3 3 3 2 2 1 4 K3 − 6K2 + 11K − 6

4.2 Moments with Different Trade Counts This section shows that although self-normalized integrated intensity moments higher than 1 are dependent on final trade count K, a scaling can be introduced so that moments can be extracted from a collection of stocks with different trade counts. The distribution of trade counts of an observed collection of stocks can be represented by a discrete distribution, zn = P r(K = n), n ∈ {1,...,B}, where B is the maximum number of trades (B = 5022 in the data).

XB zi = 1 i=1 Then the non-central moment of the relative trade count of an observed collection of stocks with different final trade counts is the weighted sum of the moments for each different final trade count.

µ ¶ Z · µ ¶ ¸ XB XK k n XB 1 XK k n z Θ (k|K) = z B(k|K, s) Ψ (s) ds K K t K K t K=1 k=0 K=1 0 k=0 As an example, the second non-central moment of an observed collection of stocks with diﬀerent ﬁnal trade counts is used to calculate the second non-central moment of the mixing distribution.

10 µ ¶ Z · µ ¶ ¸ XB XK k 2 XB 1 XK k 2 z Θ (k|K) = z B(k|K, s) Ψ (s) ds K K t K K t K=1 k=0 K=1 0 k=0

Z µ ¶ XB 1 s (K − 1)s2 = z + Ψ (s) ds K K K t K=1 0

Z Z Z XB z 1 1 XB z 1 = K s Ψ (s) ds + s2 Ψ (s) ds − K s2 Ψ (s) ds K t t K t K=1 0 0 K=1 0

(4)

Since the trade count distribution of an observed collection of stocks is known, the following substitution is convenient:

XB z V = K n Kn K=1 Substituting the above into equation 4:

µ ¶ Z Z Z XK k 2 1 1 1 V Θ (k|K) = V s Ψ (s) ds + s2 Ψ (s) ds − V s2 Ψ (s) ds 0 K t 1 t t 1 t k=0 0 0 0

Again, let δi be the ith non-central moment of the (observed binned data) relative trade count distribution, Θt(k|K), and λi be the ith non-central moment for the mixing distribution and noting that V0 = 1:

δ2 = V1λ1 + λ2 − V1λ2 So the ﬁrst 4 non-central moments of the mixing distribution calculated using a collection of stocks with diﬀerent trade counts can be written using Vn constants (which can be readily pre-computed).

(i) λ1 = δ1

11 Frequency Count of Total Trades per Day (9:30-16:00) on the NYSE from 1 Jun 2001 to 31 Aug 2001 10000

1000

100 Number of Stocks

1 1 10 100 1000 10000 Daily Trades

Figure 1: The frequency count of stocks with diﬀerent daily trade counts. The counts were generated for stocks trading on the NYSE between 9:30 and 16:00 from 1 June 2001 to 31 August 2001.

(ii) δ2 − V1λ1 λ2 = 1 − V1 (iii) δ3 − λ1V2 − λ2(3V1 − 3V2) λ3 = 1 − 3V1 + 2V2 (iv) δ4 − λ3(6V1 − 18V2 + 12V3) − λ2(7V2 − 7V3) − λ1V3 λ4 = 1 − 6V1 + 11V2 − 6V3 The machinery is now available to extract the moments of the mixing distribution (the moments of the ﬁ-di distributions of the self-normalized integrated intensity) by observing the relative trade counts of a collection of stocks.

12 4.3 The Mean is Intra-day Seasonality

£ Λ(t) ¤ The self-normalized integrated intensity Mean - E Λ(T )

NYSE Mean Trading Intensity (RPM Mean) Daily Trade Count Linear Trend Removed 1 - 5 0.1 6 - 10

0.08 11 - 20 21 - 50 0.06 51 - 100 101 - 200 0.04 201 - 400 401 - 5022 0.02

0 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00 15:30 16:0 -0.02

-0.04

-0.06

-0.08

-0.1

Figure 2: The mean for the self-normalized integrated intensity for stocks within diﬀerent trade count bands. To aid£ visual¤ comparison, the constant Λ(t) t time component has been subtracted, E Λ(T ) − T (all of these means are actually monotonically increasing from 0 to 1). Mean market trading intensity is boosted by market opening for low trade count stocks and attenuated by market opening for high trade count stocks.

The mean trading intensity is examined for stocks split into 8 diﬀerent trade count bands of approximately 30,000 relative volume trajectories in each band. The trading intensity of low trade count stocks is boosted by market open because these stocks are much more likely to trade at market open than at other times. Conversely mean trading intensity is reduced on market open for high trade count stocks (stocks trading 401 trades per day or more) because these stocks only experience 1 trade during the NYSE market opening period whereas these stocks would expect to trade several times during the same period (NYSE market open is approximately 4-5 minutes).

13 4.4 Variance is Scaled by Trade Count

£ Λ(t) ¤ The self-normalized integrated intensity Variance - Var Λ(T ) The variance of the self-normalized integrated intensity is dependent on trade count with higher trade count stocks having lower self-normalized integrated intensity variance. The relationship between trade count and variance is approximately empirically scaled by the inverse square root of trade count (see ﬁgure 4 and equation 5). This idea is developed further in a related paper by the author (McCulloch [17]) where it is shown the scaling of the variances of the self-normalized integrated intensities of stocks with diﬀerent trade counts is closely related to the Hurst exponent (Embrechts and Maejima [10]) of the integrated intensity of the underlying Cox process of trade arrivals. · ¸ Λ(t) 1 Var ∝ √ (5) Λ(T ) K

NYSE RPM Intensity Variance Daily Trade Count

0.04 1 - 5 6 - 10

0.035 11 - 20 21 - 50 51 - 100 0.03 101 - 200 201 - 400 0.025 401 - 5022

0.02

0.015

0.01

0.005

0 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00 15:30 16:00

Figure 3: The variance of the self-normalized integrated intensity for stocks within different trade count bands. Note that trade count has a significant effect on self-normalized integrated intensity variance.

14 NYSE RPM Scaled Intensity Variance Daily Trade Count

0.12 1 - 5 6 - 10 11 - 20 0.1 21 - 50 51 - 100 0.08 101 - 200 201 - 400

0.06 401 - 5022

0.04

0.02

0 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00 15:30 16:00

Figure 4: This graph shows the variance of the Random Probability Measure for diﬀerent trade count√ bands scaled by the inverse of the relationship in equation 5 (scaled by K).

15 5 A Time Scaled Model of the Integrated In- tensity

An examination of the mean and variance of the self-normalized integrated intensity, Λ(t)/Λ(T ), permits an analysis of simple models of the integrated intensity Λ(t) of the underlying Cox process. The integrated intensity is the compensator (Brémaud[4]) of the random counting measure N(t) of the Cox process, so Λ(t) = E[N(t)] and in particular, Λ(T ) = E[N(T )]. Since stocks have different final trade counts the integrated intensity for stock i,Λi(t), can be modelled using a baseline integrated intensity Λ0(t) in two ways.

(i) The integrated intensity for stock i is modelled as a baseline integrated intensity scaled by a stock specific constant, αi, related to final trade count. Λi(t) = αiΛ0(t). (ii) The integrated intensity for stock i is modelled as a baseline integrated intensity where the time parameter is multiplied by a stock specific constant, αi, related to final trade count. Λi(t) = Λ0(αit).

In the ﬁrst case, the self-normalized integrated intensity is the same for all stocks irrespective of ﬁnal trade count and therefore all higher moments of the self-normalized integrated intensity will be the same irrespective of trade count.

Λ (t) α Λ (t) Λ (t) i = i 0 = 0 Λi(T ) αiΛ0(T ) Λ0(T ) However, the variance for the self-normalized integrated intensity is different for different trade counts and therefore the first case is rejected by the evidence in figure 3. This strongly supports case (ii), that stock i can be modelled as a baseline integrated intensity where the time parameter is multiplied by a stock specific constant.

Λi(t) = Λ0(αit), αi = E[Ni(t)]

16 6 Data - Observing Relative Trade Counts

New York Stock Exchange (NYSE) trade data from the TAQ database was used to collect relative trade count data of all stocks that traded from 1 June 2001 to 31 August 2001 (a total of 62 trading days5) for a total of 203,158 relative trade count sample paths for all stocks. The relative trade count data was collected in a 2-D histogram with time in minutes in the x-axis and relative volume in the y-axis. Two histogram sizes were used, a 391 × 253 and 391 × 1261 histogram. These histogram sizes where chosen for the following reasons. In the time x-axis, the NYSE is open from 9:30 to 16:00, a total of 390 minutes. It is natural to collect sample path data each minute at precisely 9:31 to 16:00 giving 390 data points in the time axis. This 390 points is augmented with the initial state of the market at 9:30 when no trades have executed, giving a total of 391 data points in the time x-axis. In the relative volume y axis (proportion of final trading completed) an examination of the properties of the sample path distributions indicated that approximately 250 bins (see page 22, Simonoff [21] for details) was appro- priate. However if 250 bins were used then stocks with n final trades and k intra-day trades at some time during the day would have an intra-day k executed proportional trade count of n . This fraction would fall on the bin boundary of a 250 bin grid if n and 250 have a common divisor. On exact boundaries, a problem arises caused by rounding a floating point y value (proportion executed) to an integer number of bins - the actual integer value generated (which bin) becomes uncertain and is dependent on the hardware/software implementation of the particular computer floating point algorithm. The solution was to use a prime number of bins (no common k divisor), 251, so that the proportional trade count of n never falls on exact bin boundaries. The total number of bins in the y axis is then augmented by including the 2 end conditions of ‘no trades executed’ and ‘no further trades executed’ to give a total of 253 y axis bins. The higher resolution histogram, 391 × 1261 (prime 1259 + 2 end conditions), was used to check for any sampling artifacts in the relative trade count data that the 391 × 253 bin matrix may have introduced - no sampling artifacts were found.

53 July 2001 (half day trading) and 8 June 2001 (NYSE computer malfunction delayed market opening) were excluded from the analysis.

17 3000

2000

1000 200

Volume 100 100

200 Time 300

Figure 5: Binned relative trade count trajectories in a 253 × 391 Histogram. This is an estimation of the unconditional ﬁnite dimensional distributions of R(t) and is a binomial mixture distribution. The binned trajectories were generated from stocks on the NYSE that traded at least 50 times between 9:30 and 16:00 from 1 June 2001 to 31 August 2001.

18 7 Summary

If trade arrival is modelled as a Cox point process, then relative volume is modelled as a Doubly Stochastic Binomial point process. The conditional distribution of this point process is shown to be binomial using an enlargement of filtration on the underlying Cox process where the filtration is enlarged by knowledge of trade final count. The Doubly Stochastic Binomial point process is also shown to be a semi-martingale.

µ ¶ · ¸ · ¸ K Λ(t) aK Λ(t) (1−a)K P ( R(t) = a | Λ(T )) = 1 − aK Λ(T ) Λ(T )

1 K − 1 K ∈ Z+, a ∈ {0, ..., , 1}, t ∈ [0,T ] K K

The unconditional probability distribution of relative volume is a binomial mixture distribution where the mixing distributions are the ﬁnite dimensional distributions of the self-normalized integrated intensity of the underlying Cox process.

µ ¶ Λ(t) Φ (s) = P ≤ s , s ∈ [0, 1] t Λ(T )

Z 1 P ( R(t) = a) = Binomial(aK; K, s)Φt(ds) 0

Relative volume is scaled to the unit interval, thus the empirical finite dimensional distributions of Relative Volume R(t) for the NYSE are simply and readily collected in a 2-D histogram. By modelling R(t) as a binomial mixture distribution, the moments of the self-normalized integrated intensity are calculated from these empirical fi-di distributions. The expectation of the self-normalized integrated intensity is the same as the expectation of relative volume and the deterministic intra-day trading variation, the classic ‘U’ shape found in equity markets. The variances of the self-normalized integrated intensity Λ(t)/Λ(T ) are scaled for stocks with different final trade counts. This implies that the integrated intensity for stock i,Λi(t), can be modelled as a baseline integrated intensity Λ0(t) where the time parameter is multiplied by a stock specific constant αi = E[Ni(t)], Λi(t) = Λ0(αit).

19 References

[1] Anat Admati and Paul Pﬂeiderer, A Theory of Intraday Patterns: Vol- ume and Price Variability, Review of Financial Studies 1 (1988), 3–40.

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