Federico M. Bandi and Jeffrey R. Russell University of Chicago, Graduate School of Business.
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Transcript of Federico M. Bandi and Jeffrey R. Russell University of Chicago, Graduate School of Business.
Federico M. Bandi andJeffrey R. Russell
University of Chicago, Graduate School of Business
Introduction
• In a rational expectations setting with asymmetric information two prices can be defined (i.e. Glosten and Milgrom (1985), Easley and O’Hara (1987,1992)).– The equilibrium price that would exist if all agents possessed all
public and private information, henceforth, the full information price.
– The equilibrium price that exists in equilibrium given all public information, henceforth, the efficient price.
• Of course, market frictions also induce departures of transaction prices from these equilibrium values.
Measuring transaction costs
• Ideally, estimates of transaction cost should measure deviations of transaction prices from the full information price.
• Most estimates of transaction cost, however, measure deviations from the efficient price. – Bid/Ask spread– Effective spread– Realized spread– Roll’s measure
• This is due to the fact that while both the full information and the efficient price are unobservable the efficient price can, under certain assumptions, be approximated by the midpoint of the bid and ask price.
Our contribution
• We propose a simple and robust non-parametric estimate of the cost of trade defined as deviations from the full information price.
• The estimator is simple because it relies only on second moments of the observed transaction price returns.
• We are robust because we relax all the assumptions imposed by the above mentioned estimators.
Notation
• We consider a fixed time period (a trading day, for instance)
• Let t1, t2, …ti denote a sequence of arrival times where ti denotes the arrival time of the ith trade.
• Let N( ) denote the counting function. That is, the number of transaction that have occurred over the time period 0 to .
h
h
h
The ith observed transaction price at time ti is given by:
Where denotes the full information price and denotes deviations of the transaction price from the full information price.
ii t ip p
The setup
itp
Taking logs and differencing yields:
1
1 1 1
1
ln ln ln ln
where ln
or
where ln ln and i i
i i i i i i
i i
i i i
i t t i i i
p p p p
r r
r p p
Assumption 1: the price process
(1) The log price process is a continuous semimartingale:
Where is a continuous finite variation component and
(2) The spot volatility is a cadlag process.
0
t
t s sM dw
ln t t tp A M
t
tA
Assumption 2: the microstructure noise
(1) is a mean zero covariance stationary process.
(2)
(3)
i
for j k
0 for j>k
j
i i jE
for all , , ,j l m nE j l m n
Lemma 1
Under Assumptions 1 and 2 we obtain:
1
2 2
0
11
2
k
i i k ss
kE E s E
Theorem 1Assume assumptions 1 and 2 are satisfied.
Conditional on a sequence of transaction arrival times such that
we obtain 1max , 1,..., 1 0 as i it t i N h N h
21
1 1
0
1ˆ 1
2
N h N h
i i i k s pki i k s
s N h
r r rk
sN h N h
Technical intuition
• The estimator relies on the different orders of magnitude of the two components of the observed returns.
• The full information returns are .• The noise returns are• In the limit (as the intervals go to zero) the observed
returns are dominated by the noise returns. Therefore, as the trading rate increases, sample second moments of observed returns provide consistent estimates of the corresponding moments of the noise returns.
pO dt
1pO
1pO
Economic intuition
Ask
Bid
Effic. Price
Full Information Price
• • •
• •
• • •
t1 t5t4t3t2 t7t6 t8
Assumption 3
Where Qi denotes signed order flow. Specifically, Qi=1 denotes trades that occur above the full information price and Qi=-1 denotes trades that occur below the full information price.
denotes a constant distance that trades occur from the full information price.
Furthermore, Pr(Qi=1)=Pr(Qi=-1)=.5
1,...,2i i
sQ i N h
2s
Corollary to Theorem 1
Under the assumptions of theorem 1 and assumption 3 we obtain
( )
ˆ2
p
N h
s
Using this interpretation we refer to as anestimate of what we affectionately call the Full Information Transaction Cost (FITC).
ˆ
• Lemma 3 is the standard approach implemented by Roll. Our measure is similar to Roll’s.
• We emphasize that our method greatly relaxes the assumptions necessary to derive Roll’s estimator.
• Specifically, we allow for:– Correlation between full information and noise
returns.– Arbitrary temporal dependence in the noise.– Predictability of the full information return process.
Data
• We obtain transactions data from TAQ for all S&P100 stocks over the month of February 2002.
• Data are filtered to remove outliers.
Distribution of average number of seconds
between trades for S&P100 stocks
706050403020100
40
30
20
10
0
Intertrade Duration
Fre
quen
cy
Histograms of t-stats for 1,2,3,5,10,and 15 autocorrelations.
0-100-200-300-400
60
50
40
30
20
10
0
t1
Fre
quen
cy
0-10-20-30
30
20
10
0
t2
Fre
quen
cy
86420-2-4-6-8-10
30
20
10
0
t3
Fre
quen
cy76543210-1-2
15
10
5
0
t5
Fre
que
ncy
43210-1-2
15
10
5
0
t10
Fre
quen
cy
2.52.01.51.00.50.0-0.5-1.0-1.5-2.0-2.5
20
10
0
t15
Fre
quen
cy
Distribution of FITCs
0.440.400.360.320.280.240.200.160.120.080.04
50
40
30
20
10
0
cost %
Fre
quen
cy
0.080.070.060.050.040.030.020.010.00
20
10
0
cost $
Fre
quen
cy
Some specific FITCs
Symbol Duration Avg Price FITC $ FITCEff.
Spread Ann SD Turnover
GE 4.68 37.50 0.06% $0.0215 0.05% 32.52% 0.03%
IBM 6.51 102.82 0.06% $0.0583 0.04% 26.65% 0.10%
NXTL* 1.10 5.07 0.18% $0.0093 0.56% 218.34% 0.16%
Average100 stocks 14.22 40.70 0.09% $0.0307 0.06% 37.30% 0.25%
How good is the asymptotic approximation?
• Our asymptotic theory suggests that we should sample as frequently as possible.
• If trade-to-trade sampling is sufficient, then an estimate based on sampling every other transaction price should yield similar results.
• We next compare two estimates that use all and every other trade.
Plot of estimates of the FITCs using all and every other trade
00.0005
0.0010.0015
0.0020.0025
0.0030.0035
0.004
0.00450.005
0 0.001 0.002 0.003 0.004 0.005
FITC all data
FIT
C s
kip
1
Estimates based on taking every 30th transaction don’t look so good
(nor should we expect them to).
00.0005
0.0010.0015
0.0020.0025
0.0030.0035
0.004
0.00450.005
0 0.001 0.002 0.003 0.004 0.005
FITC all data
FIT
C s
kip
30
Cross section regressions
• Proposition 1: “Operating Costs” theory suggests stocks with higher liquidity should display smaller transaction costs.
• Proposition 2: “Asymmetric information” theory says that stocks with more private information should have larger transaction costs.
• Proposition 3: Both “operating cost” and “asymmetric information” theory suggest stocks with higher volatility should have larger transaction costs.
Liquidity measures $vol=average dollar volume per tradetrades=average number of trades per day
Asymmetric Information measureturn=(Shares transacted)/(shares outstanding)
Volatility measure
Other variables includedNasdaq=dummy variable for Nasdaq stocksSpread=average log(ask)-log(bid)Price=sample average price level (to capture price discreteness effects).
ˆ daily standard deviation (estimated by realized volatility)
Taking logs of all variables we run the following cross sectional regression:
0 1 2 3
4 5 6
ˆ ˆln ln ln ln $
ln # Price
true priceTurn vol
trades Nasdaq
Regression Results 1
Coef StDev t Prob
Intercept -3.4684 0.6968 4.977 0.000
lturn 0.2186 0.0544 4.017 0.000
lsize -0.1846 0.0834 2.212 0.029
lsdprice 0.2177 0.0756 2.878 0.005
ltrades 0.1184 0.1116 1.061 0.291
price -0.0067 0.0016 4.182 0.000
nasdaq -0.7859 0.3123 2.516 0.0136
Regression Results 2
Coef StDev t P
Intercept 3.2426 0.6640 -4.882 0.000
lturn 0.1812 0.0414 4.369 0.000
lsize 0.1296 0.0654 -1.980 0.050
lsdprice 0.2599 0.0643 4.041 0.000
price 0.0071 0.0015 -4.520 0.000
nasdaq 0.5126 0.1768 -2.898 0.0047
Regression Results 3
Coef StDev t Prob
Intercept 1.1169 0.6609 -1.689 0.094
lturn 0.0917 0.0380 2.410 0.017
lsize 0.040 0.0573 -0.708 0.480
lsdprice 0.1144 0.0594 1.924 0.057
price 0.0005 0.0018 0.282 0.778
nasdaq 0.1746 0.1597 -1.093 0.277
lspread 0.6528 0.1062 6.143 0.000
Difference regression
0 1ln ln . FITC Eff Spread lturn
Coef StDev t Prob
Intercept 1.5166 0.2197 6.901 0.000
lturn 0.1834 0.0342 5.354 0.000
Conclusions
• This paper proposes a new estimator for the cost of trade as measured by the expected deviation of transaction prices from the full information price.
• The estimate is consistent under weak assumptions.
• The proposed estimator is trivial to implement involving nothing more than calculating the second moments of the observed transaction prices.
• Our empirical work demonstrates that we obtain sensible estimates for the S&P100 stocks.
• Skip sampling demonstrates the accuracy of our asymptotic approximations.
• We find support for the operating cost and asymmetric information theory of transaction cost.
• We find that the difference between FITCs and effective spreads can be attributed to asymmetric information measures thereby providing support for our economic interpretation of the proposed measure.
• We examine market microstruture hypothesis about the determinants of the cost of trade.
• More liquid stocks have smaller effective spreads.
• Stocks with a higher proportion of informed traders have larger effective spreads.
• Stocks with higher volatility have larger effective spreads.
• We are currently constructing finite sample MSEs for our estimator.