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Suppose someone buys $4bn of a particular stock over the period of a few weeks. Depending on how much that stock is being traded, you would expect that the price goes up in a visible way compared to if the purchase hadn't been made. I think there is no exact way to calculate the stock price after such a transaction unless you know what all actors in the market are doing (how many people have limits near the current share price or will sell during the buying time). But is there a good approximate way to estimate the new stock price after this purchase?

I am a physicist by profession so I don't know if this is something standard or not. Thanks for the help!

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    $\begingroup$ A big, complicated and controversial question... I suppose what you are looking for is called a Market Impact Model. $\endgroup$
    – nbbo2
    Commented Jun 9, 2020 at 14:06
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    $\begingroup$ It's actually a very difficult question. There is motivation to publish many models that claim to show that if the stock is bough using someone's commercial product or service, then the market impact is affected (is less). often they are not based on some theory but implementations of someone's expert judgment or empirical observations. But it's impossible to test how the price would have changed if this trade had not happened or happened differently. $\endgroup$
    – Dimitri Vulis
    Commented Jun 9, 2020 at 14:53
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    $\begingroup$ "Hello Dr. DoubleFelix, this is Golden Sacks, we are very good at this, if you give us the job we will buy 4 billion and the price will practically not increase at all (according to our proprietary model)." ;) $\endgroup$
    – nbbo2
    Commented Jun 9, 2020 at 16:58
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    $\begingroup$ I programmed a primitive MIM some 30 years ago. I was told by the experts to assume that the impact of trading n shares of stock s would be $c_s e^{|n|}$ (i.e. some stock-specific constant, and exponential in the number of shares in either direction). I heard these days some experts assume conversely that any impact on the mid price plateaus quickly, and only bid-ask spread continues to widen. To answer the question, there doesn't seem to be any standard creditble model. $\endgroup$
    – Dimitri Vulis
    Commented Jun 10, 2020 at 13:28
  • $\begingroup$ Interesting. The answer could be seen in "experiment" then. If one knew all the limits/purchases/sales of a particular stock over some time, one could see how a given buy would change the price over the same period. Is there data like this available to the public for any particular stock over some time period? $\endgroup$
    – don't train ai on me
    Commented Jun 10, 2020 at 17:29

4 Answers 4

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There are a number of price impact models which seek to predict the bias induced on prices by trading. There are also issues with some of these models (which I will mention later).

Models

Probably the earliest and most-known model is that by Torre and Ferrari (1997) which estimates the impact to be a multiple of the square root of trade size over average daily volume and a multiple of the typical bid-ask spread. This model later appeared in Chacko, Jurek, and Stafford (2008) which did not cite Torre and Ferrari (1997). This model may date as far back as 1991 when work at Salomon Brothers referenced a square root model form.

Permanent and Temporary Impact

Almgren and Chriss (2000) proposed a model with two types of price impact: permanent, which changes the price for subsequent trading and conveys information, and temporary, which only affects a given trade. Their permanent impact model is linear in trade size $x$ while their temporary term includes a fixed fee and a term for the speed of trading $\frac{x}{T}$ (where $T$ is the length of the trading period).

Almgren, Thum, Hauptmann, and Li (2005) proposed a model with permanent impact involving the volatility $\sigma$, trading period $T$, a ratio of trade size to average daily volume (raised to the power $\alpha$), and inverse average daily turnover (raised to the power $\delta$). The temporary term involves the volatility $\sigma$ and the ratio of trade size to average daily volume (raised to the power $\beta$). They also find that the permanent impact is linear in trade size ($\hat\alpha=1$).

Decaying Impact

Finally, the most attractive model is that of Obizhaeva and Wang (2013). Their model features a third type of price impact, decaying impact which may affect subsequent trades but decreases as time goes on. This is meant to reflect that trading takes from the order book and the book needs time to refill. Their permanent term is, again, linear in trade size; the temporary impact only includes a fixed fee (no term for the speed of trading); and, the decaying term involves the trade size and an exponential decay with a decay parameter to reflect how quickly the order book replenishes.

Issues and Quasi-arbitrage

One of the biggest problems with some of these models is that they may allow quasi-arbitrage, the construction of a sequence of trades returning to a flat position yet having a positive expected return. Huberman and Stanzl (2004) discuss this problem and note that permanent impact must be linear in trade size to avoid quasi-arbitrage. Therefore, the Torre and Ferrari (1997, aka "square root") model allows quasi-arbitrage.

Allowing quasi-arbitrage is not merely an academic point. If you use a model that allows for quasi-arbitrage and embed that in a trade scheduler, the scheduler will sometimes take the opposite side of the market to knock prices back in your favor. SEC lawyers have made clear to market participants that this would be seen as market manipulation. Therefore, trading engines need to prevent their trade schedulers from trading opposite the direction of their overall order.

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Let me try to answer: I have seen how equity trades are executed at the order book level. Let's say the price of the stock is 100 (last traded price). Let's say the order book is as follows:

Bids: Bid1 = 99 (size = 10,000), Bid2 = 98 (size = 20,000), Bid3 = 97 (size = 25,000), Bid4 = 96 (size = 30,000), Bid5 = 95 (size = 40,000): total size = 125,000 stocks.

Offers: Offer1 = 101 (size = 10,000), Offer2 = 102 (size = 20,000), Offer3 = 103 (size = 25,000), Offer4 = 104 (size = 30,000), Offer5 = 105 (size = 40,000): total size = 125,000 stocks.

So the bids and offers are symmetrical in this example and the price is in perfect equilibrium. Imagine two scenarios:

Scenario 1: an aggressive buyer comes in and puts in a buy order with a price limit of 104 for 100,000 stocks (which is more than $10 million notional). If the execution of this order is "stupid", it will instantly hit all the offers until price = 104 and suck out all the liquidity up to that price. The buyer will only fill 85,000 of his 100,000 order, the price instantly moves to 104 and most likely the offer at 105 will disappear and move to 106 or higher.

Scenario 2: same buyer, but smart execution: the buyer uses an Iceberg order (this is an order where only a partial size appears in the order book and when it's fully hit, the size keeps "reloading" until the entire buy order is filled). The iceberg order is bid at 95.5 in size 10,000. It will sit there for a while until it gets hit by an offer. After a few hits (maybe 25,000 total pieces) the offers might "freak out" and re-quote best offer at 101.5 or even 102, at that point the Iceberg disappears and comes back later when the price action has calmed down.

Scenario 2 can go one for an entire week, until the buyer is filled. Unless the buyer is unlucky and the entire market is rallying at the same time (so the price action goes against the buyer), it is likely that a very large size can be acquired over time without the price moving too much: that's what smart execution is all about: satisfying buyer or sellers, without moving the price too much.

Obviously, as explained already, when the entire market moves too much based on fundamentals (i.e. recent action), it is difficult to carry out smart execution and the execution algos have to get more aggressive, moving the price more.

In conclusion, large orders can move the price a little (difficult to say by how much): but usually large orders are carried out via smart execution and in fact the execution algorithm gets remunerated for carrying out the order in proportion to how much the algorithm moves the price whilst actively carrying out the order (so it is in the execution market-maker's interest NOT to move the price when filling an order).

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  • $\begingroup$ Interesting... Is this how large companies do stock buybacks and how big institutional investors do their purchases? That is, over a period of few trading days/weeks? $\endgroup$
    – Tan Yong Boon
    Commented Jun 10, 2020 at 0:24
  • $\begingroup$ Newbie here, but if I was a large company it seems that I might want to purchase all shares relatively quickly, thus raising the price as much as possible, since the purpose of share buybacks is to return money to investors. $\endgroup$
    – don't train ai on me
    Commented Jun 10, 2020 at 17:26
  • $\begingroup$ @doublefelix: I think there are strict regulations related to buy-backs: the buy-back cannot "manipulate" the price aggressively. Albeit it sure would appear to be beneficial in the short term to shareholders if the buying company hit it "full gas". $\endgroup$
    – Jan Stuller
    Commented Jun 10, 2020 at 17:55
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    $\begingroup$ It's in general not allowed. Aggressively hitting the ask in a buy-back, distorts the supply/demand dynamics and cause a temporarily market impact, which is not in the investors best interest, as this mainly benefits traders with orders resting away from the market, arbitragers or soon to expire options/warrent holders (management?). According to SEC Rule 10b–18, among many rules it states that repurchase at a price higher than last traded or independent bid is not allowed (yielding hitting impossible in the US) $\endgroup$
    – chjortlund
    Commented Jun 11, 2020 at 7:30
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    $\begingroup$ Some firms have regulations on buybacks, but some firms do exactly as @doublefelix suggests. This tends to happen more when the CEO has compensation tied to the stock price on a particular day, may be granted options struck at a certain day's close (so no buybacks that day) or when the CEO is more exposed to the stock and (in some cases) sells at the same time or shortly afterward. $\endgroup$
    – kurtosis
    Commented Jul 24, 2020 at 18:49
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I’m by no means an “expert”, though I’ve spent a fair amount of time studying this and writing quant software.

There are three important starting places to study this question, in this order:

1 dark pools ( see https://squeezemetrics.com/monitor/dix )

40% to 60% of large trades are now done in dark pools.

2 the “closing auction” at 4pm

3 the “on balance volume” technical chart indicator. This indicator doesn’t “work” to help generate alpha.

Disclaimers: There is no formulaic or algorithmic answer to this question. I’ve proven to myself many many times that any algorithm that “works” on historical data is likely to work against me in the future.

More than 60% of the time, even relatively small “iceberg” type orders from a small time trader like me have effects on price. The more thinly traded the ticker, the bigger the effect. Market makers are there to take your money.

Maybe there are times technical indicators have kept me from taking really dumb trades, however.

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  • $\begingroup$ That is very interesting, thank you. Especially the existence of dark pools. It does seem that the effect of a trade on the price can be extremely variable, especially in the short-term. For the long-term (say, at least a year), do you know of any estimates for the effect on the price which appear semi-reasonable? Or is the answer simply that it seems hopeless to even approximate? $\endgroup$
    – don't train ai on me
    Commented Jun 19, 2020 at 12:16
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Though there is no standard solution to your question, empirical studies have consistently shown that the market impact of a metaorder is a non-linear concave function of its size. The square root law of market impact is a quite simple and popular model for price impact estimation: $$ \Delta p = Y\sigma\sqrt{Q/V} $$

where: $\Delta p$ is the price impact, $Y$ is a constant (needs to be calibrated), $\sigma$ is the annualized daily volatility of the returns and Q is daily trading volume.

Some papers around this model are Gomes and Walbroeck 2015, Zarinelli et al. 2015.

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  • $\begingroup$ The term "empirical" comes from the ancient phlosopher Sextus Empiricus. In contrast, Aristotle (who first wrote about Thales trading options in Politics, incidentally), stated that women have fewer teeth than men, an assertion that probably would not have been supported by empirical studies. $\endgroup$
    – Dimitri Vulis
    Commented Apr 9, 2021 at 12:43
  • $\begingroup$ I assume $Y$ is recalibrated for each individual stock and not a constant in general? $\endgroup$
    – don't train ai on me
    Commented Apr 10, 2021 at 13:18
  • $\begingroup$ In general yes it is, to be calibrated in order to capture the idiosyncrasies of each asset, though it can be assumed that certain stocks/sectors/asset classes are expected to have a very similar value (note that Y is of order of unity). $\endgroup$
    – MaPy
    Commented Apr 12, 2021 at 8:51

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