Price Impact Of Hidden Orders In The Limit Order Market

In stock market, the hidden order traders are usually liquidity takers. Thecounterparties’ behavior, which means liquidity providers’ order submission behavior,will definitely have non-trivial impact on the trading and the price impact. Of course,this kind of impact is exerted indirectly through the order book profile or order bookdepth. Based on this idea, this paper studies how the price impact of hidden ordersdepend on the limit order placement behavior. And then we provide an unifiedexplanation frame from the perspective of liquidity.First we build an agent-based model with hidden order trading. We also designthree liquidity providing styles, namely limit order uniform placement, limit ordercorrelated placement and limit order power-law placement. And then, we simulatedthe order book properties under these three liquidity providing assumption. We findthat,⑴In cases of limit order uniform placement and limit order correlatedplacement, there will show an increasing average order book along the relative price.In other words, the shares at the best tend to “liquidity hole”, the shares near the bestincrease in the linear way, and the shares far away from the best tend to a stable value.Meanwhile, the average order book is asymmetric. When the market price is in risingstage, sell order book depth increases faster than buy order book depth along therelative price axis. On the contrary, when the market price is in downward stage, buyorder book depth increases faster than sell order book depth along the relative priceaxis. Besides, the shares at the best are power law distributed with the decay exponentequal to-1.5. Furthermore, in case of limit order correlated placement, the sharesdistribution at the best decay slower than the case of limit order uniform placement.⑵In case of limit order power-law placement, the average order book profile istotally different from the other two cases. Under power law assumption, the sharesgrow quickly to a peak value, and then decay in a power law way along the orderbook. This order book profile is more consistent with that in the real market. Besides,the volume distribution at the best is also different with the other two cases. Under theassumption of limit order power-law distribution, the volumes at the best are followedas Gamma distribution. Farther, we simulated the transient price impact of hidden orders and thepermanent price impact of hidden orders. Also, we provide the explanation on thedifferent price impact function, based on above analysis of the average order bookproperties. We find that,⑴In cases of limit order uniform placement and limit ordercorrelated placement, the transient price impact of hidden orders is concave with theirorder sizes, which can be fitted using power law function with exponent δ=0.60.The increasing order book profile can provide a reasonable explanation to thisconcave price impact. Besides, with the increasing of hidden order participant rate, theprice impact curve becomes more concave. This can resort to the fact that the fasterthe hidden order trade, the slower the eaten liquidity on the order book recover. Notethat, the price impact in case of limit order uniform placement is a litter higher thanthe case of limit order correlated placement, when the hidden order sizes are settingsame. This is due to the slower decay of the volume distribution in case of limit ordercorrelated placement, thus this case can be considered as having better liquidity. Forthe temporal structure of hidden order price impact, under the assumption of these twocase, the market price will display a certain amount of reversion after the hidden orderis completed. This can also be understudied from the perspective of liquidity.⑵Incase of limit order power-law placement, there exists an interesting critical transitionsize Q. When Q <Q, the price impact of hidden orders is concave, while whenQ> Q, the price impact of hidden orders is convex. This phenomenon can beexplained by the “first grow then decay” order book profile. Furthermore, we find thatwith the increasing of hidden order participant rate, the price impact becomes moreconcave in the region Q <Q, and becomes more convex in the region Q> Q. Thiscan also be explained by the recovery speed of the eaten liquidity. Besides, we alsofind that, when we set the hidden order participant rate as a certain value, the hiddenorder price impact is nearly linear, which means that during the execution of thehidden order the liquidity has not changed. In this situation, we can predict that themarket price will not revise after the hidden order is completed. We also confirm thispoint by simulation.Finally, we employ the hidden order model to study the relationship between thebias order flow and the implied volatility skew. We find that, the implied volatilitycurve is a horizontal line when there are no hidden orders, while the implied volatilitycurve is downward skewed when there are bias order flows. Furthermore, we find that,with the increasing of the hidden order participant rate, the skew degree becomes lower. We believe that these results can be explained from the perspective of tradingcost.