Market microstructure notes

Interpreting the E as the Intercept of a Regression Model

I can think about D as a regressor, as "the parameter to be estimated". This model is also useful to estimate the bid-ask spread. However, there is an issue with this model: the D indicator may not be available, which means that I don't have the data to indicate if the order is a buy or a sell order. That's why we have another way to estimate the bid-ask spread, the indirect way. This indirect estimate is based on the covariance of the stock returns.

We need to understand what are the possible price changes (slide 222). When I'm on the bid side, the price can go up because there is a buy order, or it stays constant because there is a sell order. The price cannot decrease because in this framework there is no news. If I go from the bid to the ask, the next price change (p) can be equal to 0 or -s. The covariance will be equal to the expected value of the product minus the product.

of the individual expected values. The individual expected values will be equal to 0 (the assumption here is that E=0). I'm interested in the bid-ask spread, so after having found the covariance, I can find it:

This bid-ask spread is computed without knowing the bid and the ask quotes. I need the negative sign because in order to make the square root I need a positive number, and I know the covariance is negative because in this model, by construction, the covariance and the autocorrelation are negative. This is the meaning of the Roll's model.

The second objective of Roll's paper is to check if the model works. To check if the model works, the most direct way is using real actual bid-ask spread. I calculate the estimated value with Roll and the true value with the bid and ask quotes. When we don't have bid and ask, we estimate the serial covariance of the returns on a yearly basis, which means just estimating the covariance. Based on the covariance, Roll estimated the

spread with the formulas we saw. As a way to test for the soundness of this estimator, he ran a regression where the dependent variable is the estimated spread and the regressor was the natural log of the market cap. We assume that has a negative sign. We use the natural log of the regressor because in this way Ib will be smoothing very large values (outliers), I damp the importance of outliers.

Paper: The t-statistic has two properties: - The sign of the t must be the same as the coefficient one; - When the t-statistic is greater than 2, its statistical significance can be assumed.

The puzzling evidence is the fact that there are many negative spread estimates. They are puzzling because we have out of 47.000 observations, 24.000 have negative spreads. Negative spreads are not possible source of concern. Additionally, we have the assumption of the àindependence transactions. In practice transactions are not independent over time.

The easier way to adjust the problems of the Roll’s

The model is to define the Roll's model in two ways. If the covariance is negative, then we can calculate the Roll's model. If the covariance is positive, we will set equal the spread equal to 0.

Transaction costs for large trades (price impact measures)

People in many cases use trading volumes as liquidity measure. But trading volume is just an ex-post measure of liquidity. The two dimensions of liquidity are time and cost. If I observe that the market is very active, I may infer that the price of trading was low or that the time to trade was low. However, when I look at trading volumes, these are indirect proxies because I don't see what is the bid-ask spread that was paid. Trading volume is a measure of market activity. Ex-ante measures are the spread and the price impact. To calculate the spread for small trades I have different measures:

• Bid-ask spread estimated from quotations
• Bid-ask spread estimated from Roll's model
• Trading days with zero returns (=

The price is constant, there is no trading): the larger the number of trading days with zero return, the larger is the number of days without trading and consequently the lower is the liquidity of the stock.

Length of price runs: the run is a succession of price changes in the same direction. The longer is the sequence of prices going up or down, the lower is the liquidity of the stock because this implies that the stock will incur large swings in order to accommodate the equilibrium in demand and supply.

Amihud illiquidity ratio is the ratio between the absolute value stock return and trading volume in monetary terms.

We take the absolute value because we don't need to know if the price goes up or goes down. The idea is that stocks that are more liquid have a lower price change. If I keep constant the trading volume and this will cause a large price swing, this means that the stock is liquid because I'm comparing different stocks and for all the stocks I'm taking.

The ratio between price change and monetary volume. What happens is that dividing the return for the number of shares traded, I end up with some stocks whose price reaction is larger and other stocks whose price reaction is lower. So, stocks with lower price reaction per unit of trading are more liquid because they are able to absorb trading activity without a large swing in price.

20P is the current price (prior to a transaction shock). In time t there is a trade. What is the price impact of this trade? The change in price will be a proxy of how liquid is the market, because if the market price following the trade will stay at the level A, this will tell me that the market is able to absorb the shock in the trading volumes. Otherwise, if keeping constant the €100.000 of trading, the price will go to B and in order to absorb this transaction shock, the transaction price was much larger. That's the idea of the Amihud illiquidity measure. The larger is the Amihud illiquidity measure,

The lower is the liquidity of the stock, especially in terms of market impact. This measure measures the price impact of the order flow: the change in price which is associated with a unit in value of trading volume. What is the big issue of this measure?

According to the framework we saw, prices change only for one reason, because there is disequilibrium between demand and supply (the price goes to B because there is too much demand). But one might say "The price went up not because of a shock in the relationship between demand and supply, but just because there was a positive news and the market went up, or because there was a negative news and the market went down". If we look at the formula of Amihud illiquidity measure, it is not able to capture the difference between the sources of price change, it only takes the price change but we don't know why this price change took place. If it is for disequilibrium between demand and supply, it's a correct proxy. Otherwise,

If it is for news-related reasons, we are observing a biased measure of liquidity.

Lecture 13 (240-267) 4-12-2017

We need to distinguish ex-post liquidity measures, based on trading volume. If trading volume is large, the market is active but it doesn't mean that the market is liquid because trading volume doesn't incorporate trading costs. When we incorporate trading costs, we have a perspective on both ex-ante and ex-post liquidity measures. We distinguish small trades liquidity proxies (based on spread measures) and large trades liquidity proxies (based on price impact measures). Price impact is related to the impact on market price caused by a large order. In the first family of liquidity proxies we have:

• Bid-ask spread (direct estimate) which can be:
• Quoted if it refers to bid and ask quotes that are displayed in the limit order book
• Effective if it also incorporates the transaction price
• Realized if it incorporates the transaction price at a later point in time, not

Necessarily the same time as the tradeBid-ask spread estimated from Roll’s model (indirect estimate)

Trading days with zero return: this is a proxy for liquidity, not a precise measure.

Length of price runs: a run is the number of times the price goes in the same direction.

The quality of the proxy depends on the quality of the data. If we have good quality data, we don’t need to rely on trading days with zero returns or on the number of runs. These measures are used for small trades because we are interested in understanding what is the price to be paid for a number of shares which is consistent with what is displayed in the limit order book. If the quantity we want to trade is lower than the displayed one, than the bid-ask spread is a good proxy. Otherwise, we need to rely on price impact measures. Among price impact measures, Amihud illiquidity measure is widely used. The intuition behind this measure is that if the stock is liquid, for any unit of monetary

trading volume, the lower is the numerator, the lower is the pricechange caused by one unit of trading volume. The only problem with this measure is that theprice movement may be caused by different reasons. We provided three different explanationsthat are:

• Liquidity: when there is an imbalance between demand and supply. This is what we areinterested in: understand if the price jumps up because there is too much demand in themarket with respect to the available supply.
• Information: prices change because of news. Amihud illiquidity measure is not able todistinguish the source of the price changes.
• Noise: noise is when we don't know what is the private information but we believe thatwe are informed.Since Amihud illiquidity measure is not able to distinguish the source of the price changes, wewant to aggregate the daily measure in yearly measure. We just take the average value(summation of the daily Amihud) times 1 over the days of the year.

Amihud measure is quite unstable