Information Share, or, measuring the importance of different markets

Transcription

1 Information Share, or, measuring the importance of different markets The Information Share concerns ways of measuing which market place is most important in price discovery. It is attributed to?. It is an example of the usage of cointegration. A security is traded in two separate markets. The evolution of prices follows p 1,t = p 1,t 1 + w t p 2,t = p 1,t 2 + ε t p 1,t and p 2,t can be both transaction prices or quotes. w t and ε t are zero mean iid disturbances.

2 Key Assumption: Random Walk. Since p 1,t is an asset price, justified in assuming it is a random walk. All else follows from this. The price in the second market follows the price in the first market with a delay. It is thefore also integrated (of order 1). But the difference is not. To see this, take the difference between the two prices: p 1,t p 2,t = p 1,t 1 + w t p 1,t 2 + ε t = (p 1,t 2 + w t 1 ) + w t p 1,t 2 + ε t = w t 1 + w t ε t Since these variables are mean zero noise, the difference between the two prices is stationary. The price difference in the two markets is therefore cointegrated. (A linear combination of integrated variables that is stationary.)

3 Two representations of the model Vector moving average representation p 1,t = w t p 2,t = δp 1,t 2 + ε t ε t 1 = w t 2 + ε t ε t 1 Common trends representation ( t ) p 1,t = p 1,0 + w s s=1 ( t ) p 2,t = p 2,0 + w s + (w t w t 1 + ε t ) s=1 Price series sum of initial value, common random-walk term, and stationary term. Interpretation: random walk component of security price the implicit efficient price.

4 Information Share Information share: attribution of the source of variation in the random-walk component to the innovations in the various markets. To estimate, use estimable representation of the model: Error correction model p 1,t = w t p 2,t = (p 1,t 1 p 2,t 1 ) p 1,t 1 + ε t Promlem: Not an unique representation. General representation (vector form): ( t ) p t = p 0 + ψ e s ı + Ψ (L)e t s=1 e t is a vector of disturbances with covariance matrix Ω.

5 Information Share, ctd Suppose p t represents prices in n different markets. The information share S j : measure the contribution of market j to the total variance of the random walk component. Unfortunately, when the covariance matrix Ω is not diagonal there is not a unique solution to this calculation. Instead,? shows that one can find upper and lower bounds on the information share. Therefore: Report upper and lower bounds on the information share.

6 Information share in R There is an R routine that implements estimation of information shares. pdshare, in the ifrogs library. It is not part of the CRAN library, the following is necessary to install it: install.packages("ifrogs", repos="

7 R documentation pdshare package:ifrogs R Documentation Computes information share & component share weights Description: This function implements the two most commonly used techniques to measure price discovery in multiple markets. These are information share (Hasbrouck (1995)) and the component share approach (Booth et al. (1999), Chu et al. (1999), Harris et al. (2002)) which utilises the Gonzalo and Granger (1995) permanent-transitory decomposition. Usage: pdshare(x, override.lags = NULL, lag.max = 10) Arguments: x: Is a numeric matrix / data frame which has two columns with log prices of two markets. override.lags: Is an integer that specifies user-defined lags to be used for VECM estimation. Uses NULL as default. See Details. lag.max: Is an integer that specifies the maximum lag order to be used for selecting the number of lags in VARselect. Is used when override.lags is NULL. Uses 10 as default.

8 R documentation Details: The function estimates the information share (IS) and component share weights for a two market case. This is done by first estimating the VECM model for two cointegrated price series using the functionality in package urca. The subsequent VAR and VMA representation are obtained using package vars. The number of lags to be used for VECM estimation can be pre-specified by the user in the argument override.lags. The default is NULL. If the argument is kept as NULL, then the number of lags are decided based on the AIC criterion using the function VARselect from package vars. The maximum number of lags to be used in VARselect can be specified using the argument lag.max. To achieve the lower and upper bound of IS using triangularization of covariance matrix, the function first computes IS for the supplied ordering. The resulting IS estimate maximises the share of Market 1 and minimises the share of Market 2. The results are saved in is.original.ordering. Subsequently, the ordering is reversed. The corresponding IS estimate maximises the share of Market 2 and minimises it for Market 1.

9 R documentation Value: A list of the following five elements: is.original.ordering: Information shares of Market 1 & 2 under the supplied ordering is.reversed.ordering: Information shares of Market 2 & 1 under the reversed ordering component.share: Component share weights of Market 1 & 2 var.covar.matrix: Variance covariance matrix of the residuals lags.used: Number of lags used in VECM estimation

10 Example of IS calculation - NHY The stock NHY (Norsk Hydro) is traded on several exchanges. Question: how much of the price discovery happens in the main market (OSE). Compare to e.g. trading at Chi-X. The basis for inference is contemporaneous observations of prices at two market places. Our example: five-second observations of the last transaction price.

11 The first few elements of the data we use: NHY.OL is the code for NHY traded in Oslo, and NHYo.CHI NHY traded at Chi-X. time;nhy.ol-price;nhy.ol-no trans;nhyo.chi-price;nhyo.chi-no_trans 2012-Dec-03 08:00:40;27.3;10;27.3; Dec-03 08:00:45;27.3;0;27.3; Dec-03 08:00:50;27.24;3;27.3; Dec-03 08:00:55;27.27;1;27.3; Dec-03 08:01:00;27.27;0;27.3; Dec-03 08:01:05;27.24;1;27.3; Dec-03 08:01:10;27.24;4;27.3; Dec-03 08:01:15;27.24;0;27.26; Dec-03 08:01:20;27.24;0;27.26; Dec-03 08:01:25;27.25;1;27.26; Dec-03 08:01:30;27.25;0;27.26; Dec-03 08:01:35;27.24;4;27.26; Dec-03 08:01:40;27.24;0;27.26; Dec-03 08:01:45;27.24;0;27.26; Dec-03 08:01:50;27.24;0;27.26; Dec-03 08:01:55;27.24;0;27.26; Dec-03 08:02:00;27.24;0;27.26; Dec-03 08:02:05;27.24;0;27.26; Dec-03 08:02:10;27.24;0;27.26; Dec-03 08:02:15;27.24;3;27.26;1...

12 Calculate IR for the two exchanges trading of NHY on this day. Load library and Prepare the data: > infile <- "/home/bernt/data/2017/reuters_trade_data/2017_01_sampled_prices/fiv > library(ifrogs) > data <- read.table(file=infile,sep=";",header=true) > head(data) time NHY.OL.price NHY.OL.no.trans NHYo.CHI.price Dec-03 08:00: Dec-03 08:00: Dec-03 08:00: Dec-03 08:00: Dec-03 08:01: Dec-03 08:01: NHYo.CHI.no_trans

13 Find Log Prices > data <- na.omit(data) > NHY.OL <- data$nhy.ol.price > NHYo.CHI <- data$nhyo.chi.price > > logprices <- as.matrix(cbind(log(nhy.ol),log(nhyo.chi))) > colnames(logprices) <- c("lnnhy.ol","lnnhyo.chi") > head(logprices) lnnhy.ol lnnhyo.chi [1,] [2,] [3,] [4,] [5,] [6,]

14 Estimating IS without specifying lag structure: > is <- pdshare(logprices) > print(is) $is.original.ordering IS lnnhy.ol lnnhyo.chi $is.reversed.ordering IS lnnhyo.chi lnnhy.ol $component.share CS lnnhy.ol lnnhyo.chi $var.covar.matrix lnnhy.ol lnnhyo.chi lnnhy.ol e e-08 lnnhyo.chi e e-08

15 Want: information share for OSE. $is.original.ordering IS lnnhy.ol lnnhyo.chi $is.reversed.ordering IS lnnhyo.chi lnnhy.ol The IS for trading NHY at the OSE is in the interval [0.71, 0.94]. $component.share CS lnnhy.ol lnnhyo.chi Supported by the CS for the OSE of 0.74.

16 Also show estimation imposing a lag structure with 12 lags: > is <- pdshare(logprices,override.lags=12) > print(is) $is.original.ordering IS lnnhy.ol lnnhyo.chi $is.reversed.ordering IS lnnhyo.chi lnnhy.ol $component.share CS lnnhy.ol lnnhyo.chi $var.covar.matrix lnnhy.ol lnnhyo.chi lnnhy.ol e e-08 lnnhyo.chi e e-08