Order flow means buyer- or seller-initiated transactions at electronic exchanges. Order flow consumes liquidity provided by market makers and drives a wedge between transacted market price and equilibrium price, even if the flow is based on information advantage. Flow distorts market prices for two reasons. First, the need for imminent transaction carries a convenience charge. Second, the prevalence of informed flow justifies a charge for market risk on the part of the market maker. Standard models suggest that the price impact is increasing in the square root of the order flow, i.e. increases with the order size, but not linearly so. New theoretical work suggests that the price impact function may be “S-shaped”, i.e. increases more than proportionately in the smaller size range and less than proportionately for large sizes. The price effects of order flow are relevant for the design of algorithmic trading strategies, both as signal and execution parameter.

Kijima, Masaaki and Christopher Ting (2019), “Market Price of Trading Liquidity Risk and Market Depth”

The below are excerpts from the paper. Headings and text brackets have been added.

The post ties in with this site’s summaries on price distortions and implicit subsidies.

### What is order flow?

“Order flow is transaction volume that is signed. __A trade at the ask price is said to be buyer-initiated (positive) and that at the bid price, seller-initiated (negative)__. Market orders and marketable limit orders are means by which transactions are initiated and executed. With immediacy, these trades consume liquidity provided by dealers and market makers. On electronic exchanges, liquidity is said to be provided by traders who submit limit orders, which are displayed on an electronic limit order book.”

“When the electronic market is open for trading, buy and sell orders are flowing into the exchange matching engine for execution. __Aggregating over a trading period, the order flow (buyer initiated trades less seller-initiated trades) provides an ex post measure of net demand for the asset__, which translates into a market price. This relationship between the order flow and price…is a superior candidate for modelling the variation in foreign exchange rate compared to arguments along the lines of macroeconomics.”

“Market depth is intrinsically related to the amount of liquidity in the (electronic) market, i.e., the numbers of contracts waiting to buy and to sell.”

### The two channels of price impact of order flow

“__Whatever be the information content… transactions induce the market to move in the direction that reflects the price pressure over a time horizon__…An institutional investor seeking to acquire or liquidate a substantial amount of a security is likely to cause the market price to deviate from the reference price, which is the market price prevailing at the time an investment decision has just been made. This deviation from the reference price is a reaction of the market to the institutional investor’s imminent need to trade.”

“Another cost of trading is known as the price impact cost or the market impact cost. It is the compensation the liquidity providers receive for rendering the service (of accommodation of mismatched order flow), and the market impact cost paid by liquidity demanders…__The price impact is the result of the correlation between an incoming market order and the subsequent price change, in such a manner that the second buy trade is on average more expensive than the first__ because of its impact on the market price (and vice versa for sells). In other words, price impact is a post-trade explanation for the potential price disparity when a market order or marketable limit order is executed… [For example,] for major currency pairs, using data sampled at the one-minute frequency, publicly announced macroeconomic information not only causes exchange rates to move, but also causes order flow to significantly change in a direction consistent with the exchange rate movement.”

“Even before trades occur, market participants are mindful that their market orders may impact prices; traders will take into account the market impact of their trades. In other words, they know that there is going to be a disparity between the price at which their market order is executed and the price at which they wish their orders can be executed immediately. More often than not, the price is disadvantageous to the traders. For that matter, price impact is created…for all traders who consume liquidity.”

### Industry models of price impact

“Many prime broker-dealers have their own proprietary methods to measure the price impact…Price impact models used in the industry include Bloomberg’s model [and] JP Morgan’s model.”

“In these phenomenological models σ2 is the volatility, and V the volume. Respectively, EDV and EPV denote the expected daily volume and the expected period volume.”

“The first terms in the models of Bloomberg and JP Morgan correspond to the temporary price impact, and the second term is meant to estimate the permanent impact. Temporary price impacts include the bid-ask spread. They affect prices temporarily by disturbing the supply and demand equilibrium. These __transient disturbances are typically immediate at the point of transaction__. On the other hand, __permanent price impacts are caused by material information that will prompt investors to urgently update their belief__ about the value of the asset being traded. They have a relatively long-lasting effect on the subsequent prices.”

“The industry models, while consistent with the insights gleaned from the market microstructure literature, are heuristic in nature. Generally, the __price impact is assumed to be a square root of the trade size__.”

### New proposed price impact function

“Traded prices are dependent on the order size and another price referred to as the marginal price or the true price. The __true price…is unobservable, as it is the price corresponding to the absence of trades__, i.e., zero order flow.”

“In making the true price process to be under the risk-neutral measure so as to reflect no risk-free arbitrage opportunity, __we introduce the market price of (market) risk and also the market price of liquidity risk__. We solve the Bernoulli differential equation and obtain a closed-form solution for the price impact function and the market price of liquidity risk. ..It turns out that the parameters of the market price of liquidity risk are encapsulated in a measure that can be interpreted as the market depth of the electronic market. The larger or deeper the market depth is, the market can be said to be more liquid.”

“In the more general case, __we find that the price impact function is S-shape for all possible values of order flows__. Empirically, we find that our model provides a better fit [than prevailing industry models]. Our price impact function may be a better model to estimate the potential price impact in pre-trade and post-trade analyses…Empirically, it is well known that when the size of order flow is small, the price impact is approximately linear. But when the trade size is very large, nonlinear effects become pronounced.”

“Interestingly, our empirical analysis shows that __the inflection point of the S-shape function can be interpreted as the market depth__…By construction, the inflection point is in the unit of order flow (number of contracts). For our S-shape function, the unique inflection point is the order flow at which the gradient of the function is at its maximum, which means that the price impact from the order flow is at its highest. Thus, __if the inflection point is far away from the origin, it means that a larger volume of order flow is needed to create the maximum impact__. Put differently, the market is deep enough to ameliorate the price impact from small- to medium-sized order flows…The inflection point is dependent on parameters of the market price of liquidity risk, quantities of the order flow, and the volatility of the true price return through its correlation r with the order flow.”

“As a possible application of the S-shape price impact function, we show that the notion of __market depth can be implemented to capture the level of liquidity risk in trading, much like volatility is a measure of price uncertainty or risk__…Our model of order flow yields…a plausible cause of why volatility and correlation are stochastic in nature… order flow as a manifestation of trading will give rise to additional volatility that is otherwise absent. This excess volatility is stochastic because the order flow is stochastic.”