Do Bitcoin Markets have a Conversion Tort Problem?

In many jurisdictions, a buyer of stolen goods does not acquire their legal title. This principle, “nemo dat“, often applies regardless of whether the buyer knows the goods were stolen. The rightful owner can sue the current holder and take possession.

Bitcoins (and other cryptocurrencies) are well-known to be stolen in large quantities. Bitcoin thieves are usually hard to identify or outside the reach of developed markets’ courts, so victims believe they have no recourse. But the Bitcoin blockchain allows coins to be traced with high precision, and it seems possible for stolen coins to be returned once they re-enter the regulated domain. I’m not a lawyer and don’t understand the relevant intricacies, so this post is just intended to start a discussion. I may be wrong, but buying coins of unknown provenance could be a major risk for Bitcoin holders and intermediaries.

Pro-rata Tracking

All transfers between Bitcoin addresses are recorded and publicly accessible. In principle, this audit trail should make it easy for victims to track stolen coins and reclaim them if they enter the custody of an identifiable entity. There are, however, a few complications. Bitcoin transactions can have multiple inputs and outputs, so if only some inputs are stolen, it may be difficult to determine the portions of the outputs that should be returned.

Bitcoin is frequently called “digital gold,” and perhaps that analogy is useful for tracking ownership across multi-party transactions. As a toy example, say that Tom steals 1 bar of gold from Val. Tom then melts Val’s gold, combines it with another bar of gold, and sells the two resulting bars to unsuspecting buyers, Bob and Barbara (who buy 1 bar each). How should we determine how much of Bob’s and Barbara’s purchases belong to Val? I think the intuitive choice is that each of them should return half a bar. This pro-rata allocation seems fair even though we don’t know which of the individual gold atoms were previously Val’s, or how thoroughly Tom stirred the melted gold.

In some circumstances there could be documentation that suggests an alternate allocation. Suppose Tom steals a bar from Val, then entrusts it to a gold broker, who melts it with other bars. The broker then sells a bar of the blended gold to Bob on Tom’s behalf, recording a trade between them, and giving Tom the cash. Even though “Bob’s” bar doesn’t contain the same gold atoms as Val’s did, perhaps a sensible outcome is for Bob to return the entire bar to Val?

Applying the same intuition to Bitcoin, we could determine which current holders should return which fractions of their holdings, to which victims. The default flow of ownership would be pro-rata, unless there’s documentation proving otherwise — such as trades facilitated by an exchange with client coins in its custody. [1]

Legal Venue

Every jurisdiction deals with this liability differently. Many don’t follow “nemo dat” at all, instead using versions of the “market overt” rule. [2] Other venues only return stolen property if it’s tangible. [3] The statute of limitations can vary dramatically between venues. Another question is to what extent a “merchant exemption” applies to misappropriated coins. [4]

So, could lawsuits proceed in a victim-friendly venue like New York? You can argue that Bitcoin is inherently pan-jurisdictional. Bitcoins’ value derives from the consensus of participants on the network, who download the blockchain to confirm that coins are authentic and not part of a fork. If some coins were left out of the version downloaded by NY residents, those coins would be worth much less. Thus, every coin’s value is dependent on NY activity. [5]

Potential Consequences for Market Participants

If the above reasoning applies, then victims have a good chance of recovering their stolen coins. It’d be logistically complex, but I can imagine a group of victims demanding that major exchanges, custodians, and intermediaries return the appropriate amounts, if coins with certain address-chains come into their control.

If enforced by a powerful court, demands like this could have serious effects on the Bitcoin market. Many people would have coins — that they innocently bought — seized. These people might then sue their counterparties and exchanges. Depending on the number of stolen Bitcoins circulating, some exchanges could go bankrupt (if they have assets within the reach of the relevant court). OTC market makers, which may account for over half of volume [6], could have massive liabilities to their customers.

Bitcoins would have different values depending on their transaction history. [7] Coins considered “suspicious” [8] or sold without a guarantee from a well-capitalized counterparty [9], might be worth much less than freshly mined ones. In an extreme case, the market could cease to function until the provenance of enough Bitcoins is publicly known. And perhaps future transactions would have to be fingerprinted and authenticated though an ancillary blockchain.

National Stolen Property Act

Is it possible that it’s a crime to knowingly sell (or transmit) stolen Bitcoins? I have no understanding of this, but if you’re worried you possess stolen coins and want to sell them, you may want to consult a lawyer first. The National Stolen Property Act could apply to intangible goods, and I don’t want to hear about any readers of this blog going to jail.

Bringing Transparency to Bitcoin

Established institutions are trying to make Bitcoin a more trustworthy asset. That requires knowing that marketable coins have similar values and are legitimately owned, as well as reducing the incentives of crime. There’d be some pain involved in reversing past thefts, but it may be better for the Bitcoin community to deal with these issues sooner rather than later.

[1] Some Bitcoin exchanges may not keep good records of counterparties to trades they facilitated. Perhaps the allocation method should be pro-rata in these cases, i.e. anyone with Bitcoin in the custody of such an exchange has their coins partly tainted when stolen coins are deposited at the same exchange address. This may feel unfair to the innocent depositors at the exchange, but it does seem a lot like a shady gold broker combining legitimate with stolen gold, and returning the blended product to innocent customers. Customers may feel bilked when a victim of theft reclaims gold that the customers believed was theirs, but that’s the fault of the broker, who should compensate them.

This method makes particular sense in the extreme case of a Bitcoin exchange that intentionally loses records, which is arguably similar to a “mixer.” In the gold analogy, a “mixer” would melt together gold from many sources, and return the same quantities to the sources. You can certainly question the motivations of mixer-users, but even those acting in good faith probably have an idea that they might be acquiring tainted coins.

[2] “Market overt” generally grants the title to stolen goods to an innocent buyer. See, e.g., Schwartz’s footnote 15 and Appendix for a brief overview of jurisdictions that follow “market overt.”

[3] Bitcoins are generally considered electronic and intangible, but some people print them out on sheets of paper.

[4] In the US, if a merchant sells goods entrusted to it in the “ordinary course of business” — whether authorized or not — the purchaser acquires the title. So perhaps if a Bitcoin exchange sold client coins to an innocent buyer on the exchange, without transferring the proceeds to the client, the buyer would acquire the title to the coins (in the US). But, if an exchange absconded with client coins and sold them later, outside the normal course of business, the victim would still own the title. Likewise for coins stolen from an exchange by a third party.

This treatment could have important consequences for the missing Mt. Gox coins, worth several billion USD at current prices. We still don’t know the full story behind those coins, but it seems likely that the victims own the title in the eyes of many jurisdictions.

[5] Similar arguments have allowed lawsuits concerning allegedly misappropriated art to proceed in NY.

[6] Bobby Cho of Cumberland, a DRW subsidiary and large OTC market maker, says that:

On any given day, exchanges are reporting anywhere from $200 million to $250 million traded in a 24-hour period. OTC trading is not reported anywhere, but I’d imagine that the OTC market is larger than that number.

[7] JP Koning discusses the notion of fungibility as it relates to “nemo dat.” That is, if “nemo dat” applied to legal tender, identical units of currency would have different (and hard to ascertain) values depending on their history. If only one jurisdiction applies “nemo dat” to Bitcoin, then its value could become unstandardized everywhere.

[8] Especially high-risk coins might include the Silk Road coins auctioned by the US Marshals. The US Government said that Ross Ulbricht “is the only person or entity reasonably believed to be a potential claimant to the Computer Hardware Bitcoins.” I don’t know how civil forfeiture works, so maybe this doesn’t matter at all, but isn’t there at least a chance some of those coins didn’t belong to him? Ulbricht was convicted of charges related to drug-dealing, hacking, and money-laundering. I’m not sure it’d be a surprise if something in his possession had been stolen.

GBTC, a Bitcoin-holding trust listed on OTC Markets, bought 48,000 BTC from this source. Cumberland reportedly bought 27,000 BTC. Tim Draper bought around 30,000 of “these bitcoins with a tainted past,” and deployed them at market maker and exchange Mirror. Exchange itBit, which will form part of the settlement price for CME Bitcoin futures, bought 13,000 BTC.

[9] I don’t know if OTC market makers keep large cash reserves, but perhaps some clients patronize them because they offer more reliable protections in case they accidentally sell illegitimate Bitcoins. I can imagine a situation where clients purchasing Bitcoin require market makers to post margin as an additional protection; and conversely market makers could require margin when buying coins from clients.

A Curious Feature of IEX Auctions

IEX is busy setting up its listing business and has designed an auction mechanism. If I understand IEX’s documentation properly — and I might not [1] — their auctions may have some pretty weird functionality. [2]

IEX’s auction documentation says:

For the Opening/Closing Auction, non-displayed buy (sell) orders on the Continuous Book with a resting price within the Reference Price Range will be priced at the Protected NBB (NBO) for the purpose of determining the clearing price, but will be ranked and eligible for execution in the Opening/Closing Auction match at the order’s resting price.

Now, I’m not easily surprised by complex exchange logic, but this got my attention. Again, I may have misunderstood something, but if I haven’t, this logic could be controversial with long-term traders once they familiarize themselves with IEX auctions.

Take IEX’s “Example 1” from their “CLEARING PRICE EXAMPLES”:

The Closing Auction Book includes the following orders:
 LOC order to buy 1,500 shares with a limit price of $10.10; and
 LOC order to sell 1,000 shares with a limit price of $10.10.
Shares are maximized at $10.10; therefore
 1,000 shares would execute at the IEX Official Closing Price of $10.10.

My understanding is that a hidden order submitted after these LOC orders could gain priority without providing a better price. If the NBBO were 10.09/10.13, and someone submitted a hidden bid at $10.11 100ms before the auction, I believe IEX’s auction logic would still price the auction at $10.10 [3] — but the hidden order would be executed in place of the LOC bid. This may feel “unfair” because the hidden order was submitted long after the LOC order and had virtually zero risk of moving the matching price (because the hidden bid is priced at the $10.09 “for the purpose of determining the clearing price”). And unlike LOC (or MOC) orders, the hidden order does not have a “no-cancel” period, so could be canceled even 1ms before the auction.

“Example 2” in “PRIORITY OF EXECUTION EXAMPLES” provides a slightly more complex illustration of this behavior. [4]

Opaque, Non-Competitive Auctions

Under certain conditions, this functionality gives traders the opportunity to simultaneously gain priority while offering a less competitive price, a rare confluence in normal markets. One reason why traders submit LOC/MOC orders (and displayed orders) is because they want to announce their intentions to attract liquidity. IEX’s hidden order logic shouldn’t change that. But other auction traders only announce their intentions because they want priority. This latter group may find the advantages of hidden orders hard to resist.

Here is a potentially problematic example:

  1. After the final opportunity to submit LOC/MOC orders, a closing auction has a big sell imbalance: there is 1 LOC buy at $10.00 for 1,000 shares, and 1 LOC sell at $10.00 for 10,000 shares.
  2. In the remaining 10 seconds of the day, traders will compete to trade with the sell imbalance. On IEX, the only way they can do so is by using displayed and undisplayed orders.
  3. Maybe 10ms before the close, the NBBO is $10.05/$10.07 (with all displayed quantity on IEX).
  4. The bidder at $10.05 realizes they can get a better price and better priority in the auction by deleting their displayed order and resubmitting a hidden buy order at $10.06. They do so, and the NBBO is now $10.04/$10.07.
  5. The bidder at $10.04 does the same thing. Likewise those at $10.03, $10.02, and $10.01. The NBBO is now $10.00/$10.07.
  6. The auction occurs at $10.00. The hidden orders are filled completely, but the LOC bid at $10.00 doesn’t receive any execution.[*]

I can’t imagine that either LOC side would be happy in this circumstance. The LOC sell order was filled at a non-competitive price. And the LOC buy order was skipped-over in favor of hidden orders that were submitted after it, but effectively at the same price. This process could also negatively affect market stability by causing the NBBO to widen at the most important and volatile part of the day. [5]

Now, I don’t think this sort of outcome will happen most of the time. But it certainly could happen some of the time. And even when it doesn’t, traders in the final second of the day may play a guessing game, trying to determine whether it’s worth submitting a displayed order on top of suspected hidden orders. (In our example, a trader might want priority over the hidden orders and submit a displayed bid at $10.06.)

I’m also not sure this functionality was an oversight. IEX says it exists “to protect the anonymity of resting non-displayed interest.” And “IEX Auctions were strategically designed after extensive research and informal discussion with various market participants.” But IEX did not find it appropriate to employ this functionality for auctions occuring after volatility pauses or halts, perhaps because those auctions can be chaotic and have a particular need for transparency. The thing is, some days the open and close can also be volatile, and those are the days when it’s most important for exchanges to run a smooth auction. [6]

To be clear, I wouldn’t say that this feature is scandalous. The behavior (I think) is fully disclosed, and professional traders should always read the manual before using any exchange. But issuers are probably not equipped to do that. And if IEX wants its auction to be “simple,” I’m not sure this is the right approach.


[1] It’s hard for me to completely understand IEX’s documentation. It has some clear errors in it — such as the “CLEARING PRICE EXAMPLES,” which say “Each example below assumes the Protected NBBO is $10.09 by $10.11 at the time of the Closing Auction,” but “Example 3” has displayed limit orders on IEX at $10.12 (buy) and $10.13 (sell) — so something is not quite right. Even aside from errors in the documentation, I may have made errors interpreting it.

[2] I don’t mean to write so much about IEX, but functionality like this has not gotten the attention it deserves.

[3] I think the auction would actually occur at $10.09 in this case, were it not for the rule:

If more than one price maximizes the number of shares that will execute, resulting in an auction price range, the Reference Price is set to the price at or within such range that is not lower (higher) than the most aggressive unexecuted buy (sell) order.

Which I think guarantees that even if the LOC bid at $10.10 is skipped-over and doesn’t receive an execution, it will still affect the auction price. I suppose this is better than performing the auction at $10.09, but it will still frustrate the LOC-sender who missed their execution and moved the closing price.


The Regular Market Continuous Book Contains the following orders:
 Midpoint Peg order to buy 2,500 shares with a resting price of $20.20.
The Closing Auction Book includes the following orders:
 LOC order to buy 500 shares with a limit price of $20.19; and
 LOC order to sell 2,000 shares with a limit price of 20.18.
For purposes of determining the clearing price, the Midpoint Peg order is priced to the Protected NBB ($20.19), but remains ranked and eligible to execute at its resting price;
Accordingly, shares are maximized between $20.18 and $20.19 resulting in an auction price range, and $20.19 is the only price within such range that is not below the price of the most aggressive unexecuted buy order; therefore
 2,000 shares would execute at the IEX Official Closing Price of $20.19;
 The Midpoint Peg buy order would receive an execution of 2,000 shares;
 The LOC sell order would receive an execution of 2,000 shares; and
 The LOC buy order would not receive an execution, because the LOC sell order is fully filled after matching with the Midpoint Peg buy order with superior priority.

[5] It also seems potentially risky to have the auction price depend other exchanges’ order books. If somebody submits a 100-share order on another exchange, they could potentially move the IEX open or close by multiple ticks. Even ignoring the possibility of manipulation, it might not be the best idea for an important cross to be sensitive to small orders that aren’t eligible to participate in that cross.

[6] NYSE learned this the hard way after Rule 48 probably contributed to the market disruption on Aug. 24, 2015.

[*] I have edited this example because I made an error in the original version. The LOC bid in the original version was at $10.10. In that circumstance, the LOC bid at $10.10 *would* get fully filled at $10.00 (because it’s more aggressively priced than the hidden orders), in contrast to what happens to the LOC bid at $10.00 in our example.


Liquidity Disincentives and IEX

Exchange fee-schedules are probably the most boring topic in market structure. But there are occasional exceptions that are almost interesting, like IEX’s new fee. In essence, the new fee will charge liquidity takers an extra 30 mils if the “Crumbling Quote Indicator” is on at the time IEX completes an execution. [1][2] Most exchanges segment order flow with incentives; IEX is attempting to carve out a niche by using disincentives.

With the fee, IEX seems to be deliberately targetting traders they view as harming their market. John McCrank relays the following quote from IEX’s Eric Stockland:

If we can stop HFT (high-frequency trading) market makers from getting picked off by predatory HFT strategies, the byproduct should be more liquidity and a better experience for folks interacting on IEX

And IEX’s filing makes clear that the fee targets “Only 13 Members… likely to be engaging in a deliberate strategy to target resting orders at soon to be stale prices.”

We have discussed the peculiarility of an exchange penalizing trades at prices that are “soon to be stale,” a phrase that borders on oxymoron. Volume and price-changes are inseparable in a healthy market: traders buy when they think the price is about to increase, and prices tend to increase after buyers initiate trades. These processes are natural features of markets, and ordinarily can be prevented only via transaction-blocking price controls. So I find IEX’s goal of “[p]reventing someone from anticipating an NBBO change” a very odd one.

Part of me suspects that what IEX really hoped to do was extend the last-look feature of its peg orders to displayed liquidity, but doing so might’ve been too problematic even for them. Brad Katsuyama told Matt Levine:

“People have asked us why not make D-peg a displayed order. That becomes very challenging… within the context of displayed liquidity — seeing something and having it fade on you — we’ve been very cognizant that’s what started this whole journey for us. We don’t want to contribute to that.”

So, instead of allowing displayed orders to fade “during periods of quote instability,” IEX is opting to charge takers the legal maximum.

Now, IEX certainly knows the identity of the 13 members that would pay this fee. Stockland says that the fee is targeted at “predatory” HFT strategies [3], and designed to help other HFT strategies. An exchange is free to make a good-faith judgment about what types of HFT they want to encourage or discourage, but IEX continues to use emotionally-charged language. I hope that their feelings were not a factor in their determination.

In any case, if the fee succeeds and market-makers quote more displayed liquidity on IEX, then the mix of traders paying the fee will likely change. I strongly suspect that the Thor router, as described in “Flash Boys,” would pay this fee on most of its eligible orders. If Thor swept Nasdaq’s book, it’s very likely that IEX would determine that the quote is “crumbling” by the time Thor’s order exited the IEX speedbump. Thor seems to be designed for click-traders, so it could delay sending its orders to Nasdaq and Bats by an additional 350us, preventing IEX from realizing the quote is about to “crumble” until after its execution. But many smart order routers and execution algos trade during periods of market activity, and cannot delay their orders without experiencing slippage. [4]

Displayed Liquidity and Listings

IEX desperately needs to increase displayed liquidity on their market. In principle they may not mind being a predominantly dark venue, but if they want to attract listings they will need to convince issuers that they can realistically facilitate price discovery. To be frank, the current level of pre-trade transparency on IEX is an embarrassment and any issuer choosing to list there is taking a serious risk. I suspect IEX knows this and the new fee schedule is an genuine attempt to incentivize displayed quoting.

That doesn’t mean it will work though. IEX believes that their refusal to pay rebates is the reason their exchange is mostly dark. [5] Another possible purpose of the new fee is to increase support for a reduced access fee cap. The fee and any copycats may annoy market participants that are currently indifferent to the access fee cap. Reducing the current cap from 30 mils would cut the rebates exchanges can afford to pay. The result, if IEX’s claim is true, would be to improve their displayed market share. I’m a bit skeptical: I suspect that IEX struggles with displayed liquidity because their last-look-functionality allows dark orders to preferentially trade with low-alpha aggressors, leaving abnormally high-alpha flow to hit their lit quotes.

Charging Traders For Ex-post Market Moves

The new fee depends on the state of the market after an order is sent. If the access fee cap were higher, this would enable pricing that is economically equivalent to last look. But even with the current cap, there are some weird possibilities. If this fee is allowed, would there be anything stopping an exchange from levying a fee that depends on the market a millisecond in the future? What about a second? Or a day? An exchange could, for instance, charge 30mils extra if a trade is followed by a favorable price movement larger than 0.1%. With the revenue generated, it could pay a huge rebate to resting orders that are blown-through (by say, 1%) during volatile moments, compensating market makers for their losses. This could make the exchange’s order book thicker during high-volume periods, resulting in market share gains. It’s easy to see how such pricing could backfire, with the exchange having to make larger-than-expected payments if a tail-event occurs on the thickened book. And there is something uncomfortable about paying traders for being wrong. But it’s an intriguing possibility that issuers might like.

An exchange could also try the opposite, paying high-alpha traders for correct predictions and penalizing noise-traders. Most likely, noise traders would move to other venues, leaving only orders that are sent by confident participants. This would add another tier to the venue segmentation spectrum. Under the current regime, low-alpha flow is disproportionately directed to internalizers and wholesalers, forcing high-alpha traders to interact with each other on lit exchanges, the centers of price discovery. Perhaps a few ultra-high-alpha traders would be attracted to this new type of venue of last resort, particularly when prices are in transition. I doubt it’d be wise for a major exchange to introduce this type of pseudo-sin-tax, but perhaps a smaller venue like PSX or EdgeA will try it out.

I’m usually skeptical of complex exchange logic, but it’ll take time to see the consequences of this type of pricing. IEX’s proposal shows that they are starting to think like their competitors do, and they’ve spun it in a way that no other exchange could. It may not be to the liking of long-term traders, but there’s no doubt that IEX is gradually becoming more sophisticated. [6]

[1] The filing says the fee will be applied when “Taking Liquidity During Periods of Quote Instability, as defined in IEX Rule 11.190(g)” if the volume of such executions exceeds “5% of the sum of a Member’s total monthly executions on IEX [and] at least 1,000,000 shares during the calendar month, measured on an MPID basis.”

[2] Matt Hurd is not a fan of the new pricing. For more details on the “Crumbling Quote Indicator,” see his analysis and update on the logistic regression IEX uses to assess whether a quote is “crumbling.” The regression uses non-delayed market data, which arrives at IEX 350us later than traders’ orders.

[3] IEX discloses that the largest such trader would have paid about $120k (footnote 17) under the new fee schedule for all of June. Assuming, generously, that these frowned-upon trades made 30mils, then their profit would be about $120k/month.

[4] IEX appears to think that anyone hit by the fee, presumably including these SORs, “degrade the qualtiy of the market.” And “ideally, the fee will be applied to no one, because participants will adjust their trading activity to account for the pricing change.” I could be wrong, but I suspect many brokers disagree.

[5] IEX executives have repeatedly referred to rebates as “kickbacks.” Matt Hurd argues that if a payment is a kickback, then so is providing a service for free:

IEX fail to recognise that zero pricing for lit at IEX would meet their own description of a kickback as a kickback is a remunerative, not monetary, exchange. Giving something away for free, such as an order execution, might also be considered a kickback; as may something for a discount. So if rebates are kickbacks, then IEX is also offering kickbacks. It becomes a matter of degree.


[6] It’s worth reflecting on “Flash Boys,” which opined:

Creating fairness was remarkably simple… They would pay no kickbacks to brokers or banks that sent orders; instead they’d charge both sides of a trade the same amount: nine one-hundredths of a cent per share (known as 9 “mils”). They’d allow just three order types: market, limit, and Mid-Point Peg

IEX has come a long way since then.

The Shape of Supply and Demand Curves in Rapidly Clearing Markets

A central challenge in economics is understanding how price affects the quantity of supply and demand, a relationship often assumed to be approximately linear. But there are markets where this notion of linearity, sometimes called “elasticity,” may not hold. In a paper that deserves more attention, Donier and Bouchaud show that supply/demand curves of rapidly clearing markets (with a Brownian price process) have an average shape that is locally quadratic, with no linear term.

Here is a classical illustration of how total supply and demand tend to vary with price:

Total demand decreases — and total supply increases — monotonically with price. The curves intersect at p*, which is the volume-maximizing, market price. [1] Near the market price, supply and demand vary linearly with changes in price.

If we “clear” the market, so that the supply on offer at p^* trades with the demand at p^*, then the curves will look like:

Which is the same as the first plot, just shifted downwards by the quantity traded. The curves are still linear near p*. And if we zoom in, the average supply/demand curves will look like [2]:

But, Donier and Bouchaud [3] show that markets with certain features will actually have *average* supply/demand curves that are locally quadratic. I.e.:

In this regime, a market order will have price impact that scales with the square-root of its size, on average.

Their result also raises the question whether some markets operate at saddle points in their production and consumption curves, which may look like:

Where W_{saddle} is the rough width of the “saddle zone,” the region where the curves are predominantly quadratic.

This strikes me as qualitatively different from the classical economic picture. It also makes sense intuitively: when an asset is volatile, it’s difficult to know the exact price where supply and demand balance. Donier and Bouchaud don’t speculate on the size of W_{saddle}, but there is at least the possibility that their results apply to a wider range of prices than expected. If any real markets have a large W_{saddle}, estimating their price elasticities would be difficult or impossible. It could also explain the ennui of financial markets — where headline-generating price moves have little effect on real-world supply and demand. [4]

The Donier-Bouchaud Model

Donier and Bouchaud (and previous co-authors) use a reaction-diffusion model to obtain this result. Briefly and roughly:

  1. New buy orders are created (in time interval dt) with probability \omega_+(y), where y = p - p^* is the difference between the price of the new order (p) and the market price (p^*). Sell orders are created with probability \omega_-(y).
  2. Existing orders are canceled with probability \nu_{\pm}(y).
  3. The market clears with periodic time-interval \tau — when buy and sell orders with crossing prices are matched, and removed from the market.
  4. The underlying process of the price p^* is Brownian.

When \tau is small, they show that supply and demand curves are locally quadratic, for any reasonable \omega_{\pm}(y) and \nu_{\pm}(y).

Of course, real-world markets do not instantly clear crossed meta-orders, even when markets trade in continuous time. For example: a trader might intend to buy $100M of stock at any price below $100/share, while another trader intends to sell $100M of stock at any price above $90/share. The two traders might dribble out their order-flow over weeks, instead of instantly trading with each other at a price between $90 and $100 per share. [5]

Nonetheless, it is conceivable that some markets behave as if they’re in the small \tau limit. In the Bitcoin market, traders may be less inclined to hide their intentions than in traditional markets, and the visible order book might represent the true levels of supply and demand near the market price. The authors present the average displayed supply and demand for Bitcoin in Figure 6, which is very close to a quadratic function for prices within 2% of the clearing price (where cumulative supply/demand is typically ~400k BTC). So electronic markets’ “saddle zones” may be about as wide as their daily volatility, which doesn’t seem surprising; few oil producers are going to increase drilling because the price went up by 1%.

Latent Liquidity as First-Passage Time

Donier and Bouchaud’s result seems to be a general feature of Brownian price processes, and doesn’t depend much on the specifics of the model. The spirit of their model raises the question whether there’s a connection between the marginal supply/demand at a given price, and the time required for the market to move through that price. That is, perhaps latent liquidity has properties similar to first-passage time statistics.

A quadratic supply/demand curve is equivalent to marginal supply/demand varying linearly with price. By definition, cumulative supply (S(y)) at a price y away from the “true” price is just the sum of marginal supply (\rho_{S}) available up to that price: S(y) = \int_{0}^{y} \rho_{S}(y') dy'. [6] \rho_{S}(y) may also be called the volume of latent sell orders available at price y.

One way to reach Donier and Bouchaud’s result is by assuming that \rho_{S}(p) builds with time, after the market price moves through p. [7] To be clear, the following model is different and less sophisticated than what Donier and Bouchaud did, but I think it’s a good way to capture the intuition.

As an illustration, consider what happens to the latent order book after the clearing price drops instantly from p_0 to p_1. At first, the supply on offer between p_0 and p_1 will be zero:

Afterwards, inside this new gap, latent sell orders will start to build. Let’s assume that \rho_{S} grows as a function only of the time t(p) since the price dropped through p: \rho_{S}(t(p)). [8] The precise form of the function \rho_{S}(t) doesn’t really matter.

If the market doesn’t move after that initial price drop, time t_1 later, the latent order book will have replenished. Between p_0 and p_1, there will be quantity \rho_{S}(t_1) on offer:

To calculate the expected value of \rho_{S}(t(y)), we need the probability density of the time since the price last passed y away from the present price: \mathbf{p}_{y}^{LPT}[t]. For a time-reversible process, this distribution is the same as that of the first-passage time, \mathbf{p}_{y}^{FPT}[t].

For continuous-time Brownian motion, the first-passage time distribution is well-known:

\mathbf{p}_{y}^{FPT}[t] = \frac{y}{\sqrt{2\pi \sigma^2 t^3}} e^{-y^2 / (2 \sigma^2 t)}

For y \ll \sigma \sqrt{t}, this is linear in y:

\mathbf{p}_{y}^{FPT}[t] \approx \frac{y}{\sqrt{2\pi \sigma^2 t^3}}

Which gives an average marginal supply curve that’s linear in y:

\mathbf{E}_{t}[\rho_{S}(t)] = \int_{0}^{T} \rho_{S}(t) \mathbf{p}_{y}^{FPT}[t] dt \propto y

Where T is the total time that the market has been operating.

Thus, the expected cumulative supply is quadratic in y: \mathbf{E}_{t}[S(y)] = \int_{0}^{y} \mathbf{E}_{t}[\rho_{S}(y')] dy' \propto y^2.

The model should fail at sufficiently large t, when it has been a long time since the price last reached its current level. E.g., if the price of oil rises past its high for the year to $60/bbl, then we’d expect the level of marginal supply near $60 to reflect the real-world economics of oil extraction. So clearly, the model shouldn’t work for t=1yr in the oil market. But if oil breached only its high-of-the-day, perhaps the marginal supply would just be a mechanical function of that duration. We could argue that the model will start to fail when t is long enough for businesses to react to new highs/lows, which should be about the typical time between business decisions (t_{InterDecision}). [9] In that case W_{saddle} \sim \sigma \sqrt{t_{InterDecision}}, which could be quite large for illiquid markets.

Discretizing the Process

If \tau > 0, the market clears every period \tau in a batch auction, and the price process becomes a discrete-time random walk. An infinitely long \tau should recover the uncleared, classical supply/demand curves at the top of this post. So, as \tau increases, we expect a transition from the quadratic supply of Brownian motion to a linear regime.

To get the average supply curve for discrete markets, we need the first-passage time distribution. When a random walk has price steps that are independently drawn from a symmetric, continuous probability distribution with finite second moment, its first-passage time PDF is asymptotically: [10]

\mathbf{p}_{y}^{FPT}[n] \approx  (\frac{1}{2 \sqrt{\pi n^3}} + \frac{y}{\sqrt{2\pi \sigma_{step}^{2} n^3}}) e^{-y^2 / (2 \sigma_{step}^2 n)}

Where n is the number of steps the random walk has taken, and \sigma_{step}^2 is the variance of each step’s price movement. The approximation is valid in the limit n \to \inf with \frac{y}{\sqrt{n}} finite. The step count is related to continuous time via n = \frac{t}{\tau}. And if the underlying process is Brownian, \sigma_{step} = \sqrt{\tau}\sigma.

When the price process has a typical step size (\sigma_{step}) that’s small compared to the distance from the market price (y), then the second term dominates and \mathbf{E}_{t}[\rho_{S}(t)] is identical to the continuous Brownian case. That is, cumulative supply varies quadratically with price when \tau \ll \frac{y^2}{\sigma^2}.

When the process is heavily discretized, y is small compared to \sigma_{step} and the first term dominates, which will be approximately constant in y. Thus the marginal supply will be constant, and the cumulative supply linear in y.

This result is the same as Donier and Bouchaud’s. In fact, if we expand \mathbf{p}_{y}^{FPT}[n] to first order in \frac{y}{\sigma_{step}\sqrt{n}} (near the market price), then we get:

\mathbf{E}_{n}[\rho_{S}(n)] \approx L(y + u_0 \sigma \sqrt{\tau})

Where L is some constant obtained from integrating out t, identified as a measure of liquidity by Donier and Bouchaud. And u_0=\sqrt{\frac{1}{2}} \approx 0.71 is a constant not terribly far from the one obtained by Donier and Bouchaud (u_0 \approx 0.82). [11][12]

Thus, slowly-clearing markets — which are heavily discretized — may not have a saddle zone. [13]

Supply/Demand Curves when the Price Process is a Lévy Flight

The above asymptotics apply to a broad class of random walks if their variance is finite. But markets can have price fluctuations with fatter tails, particularly on shorter timescales. A Lévy flight of index 0 < \alpha < 2 has price increments (x_{t} = p_{t} - p_{t-\Delta t}) with divergent variance and a power-law tail: \mathbf{p}[x] \sim \frac{1}{|x|^{\alpha + 1}}.

The first passage times of a Lévy flight have asymptotic PDF: [14]

\mathbf{p}_{y}^{FPT}[t] \sim \frac{y^{\alpha / 2}}{t^{3/2}} for long t. [15]

This distribution gives, on average, a cumulative supply curve S(y) \sim y^{(2+ \alpha) / 2}. And a market order will have price impact \mathcal{I}(q)=S^{-1}(q) \sim q^{2 / (2+\alpha)}. As an example, \alpha=1.5 would correspond to a rather “jumpy” market, and would have S(y) \sim y^{1.75} and \mathcal{I}(q) \sim q^{0.57}. [16]

Supply/Demand Curves for a Sub-Diffusive Price Process

The volatility of a sub-diffusion increases with timescale more slowly than the volatility of an ordinary Brownian motion. For example: \sigma_{\Delta t}^2 \sim \Delta t^{\gamma} . When \gamma = 1, the volatility scales in the usual way for Brownian motion: linearly with the timescale. When 0 < \gamma < 1, the process is a sub-diffusion. A sub-diffusive market is mean-reverting in the sense that a price fluctuation is likely to be reversed in the future. Because sub-diffusive markets have “memory,” they’re considered “inefficient.” [17]

The first passage time distribution of a sub-diffusion is asymptotically: [18]

\mathbf{p}_{y}^{FPT}[t] \sim \frac{y}{t^{1 + \gamma / 2}}

This has linear price-dependence like ordinary Brownian motion. So the cumulative supply is again quadratic, and market impact is again square-root.

Certain types of “efficiency” can lead to square-root price impact. But if this model is approximately accurate, then “inefficient” markets like sub-diffusions can also have square-root impact.

Update: Benzaquen and Bouchaud just examined a reaction-diffusion model for sub-diffusions. They show that the latent order book is locally linear (eq. 10), like in the crude first-passage analysis here. For quickly executed meta-orders, they show \mathcal{I}(q) \sim \sqrt{q}. But for slow meta-orders that give latent orders more time to react mid-execution, they get \mathcal{I}(q) \sim q^{1-\gamma / 2}.

Order-of-Magnitude Scaling of Impact in Diffusions and Sub-Diffusions

I find this result interesting because it appears, at first glance, to contradict the simplest, order-of-magnitude “derivation” of the square-root impact law. But on closer inspection, I think order-of-magnitude logic is consistent with sub-diffusions and ordinary diffusions having similar impact scaling.

If a market is Brownian, its price changes will scale like \sigma \sqrt{\Delta t}. One view of price discovery is that a fraction (of order 1) of those price changes come from traders’ impact. Thus, a meta-order’s impact will roughly scale with the square root of its duration: \mathcal{I} \sim \sigma \sqrt{t_{OrderDuration}}. Market-wide volume roughly accumulates at a constant rate in time (V), so a meta-order of size q will last for a duration t_{OrderDuration} \sim \frac{q}{V}. This gives \mathcal{I}(q) \sim \sigma \sqrt{\frac{q}{V}}.

Now, if a market is sub-diffusive then its price changes scale more slowly with time: \Delta p \sim \Delta t^{\gamma / 2} . If everything else in the above argument were the same, then we’d get \mathcal{I}(q) \sim q^{\gamma / 2} . But there’s no reason to expect that volume in a sub-diffusive market should occur at a constant rate in clock-time.

Sub-diffusions can be reformulated as processes with independent increments, where the time between steps is variable with a long tail. [19] In this formulation, steps may correspond to trading activity — with each step roughly having volume V_{step}. A meta-order would then last for N steps, with N \sim \frac{q}{V_{step}}. The step lengths of this sub-diffusion have finite variance \sigma_{step}^2, so we can follow the same argument as in the Brownian case, with the step count replacing time: \mathcal{I}(q) \sim \sigma_{step}\sqrt{N} \sim \sigma_{step}\sqrt{\frac{q}{V_{step}}} — which has the same square-root scaling.


Market-behavior is crowd-behavior, and a crowd can be more predictable than the individuals inside it. Donier and Bouchaud — and the loose interpretation in this post — show that that when markets are mechanical and boring, they will have cumulative supply and demand curves that are roughly quadratic, on average. Now, lots of us do not think markets are boring! Even if many markets do operate at saddle points, they probably do not have very wide “saddle zones.” But if any do, a small price change might have virtually zero effect on production and consumption, and a large price change might have a giant effect. That’s different from the standard intuition.

[1] This picture is obviously heuristic. Supply/demand curves are well known to have many shapes, may be non-monotonic, and may have more than one price that locally maximizes the exchanged volume.

[2] The supply and demand curves near p* should (probably) be symmetric around p*, if we average over some ensemble of situations in a given market.

[3] Borrowing from their earlier work with Bonart, Deremble, de Lataillade, Lempérière, Kockelkoren, Mastromatteo, and Tóth. (e.g. 2014 and 2011)

[4] Even when large price changes barely affect supply and demand, they could still serve an important economic function. The opinions and information of traders are incorporated into prices, which — when markets are transparent — provide useful signals for long-term business decisions.

[5] Table 1 and Figure 13 of Waelbroeck and Gomes indicate that the bulk of institutional equity meta-orders are shorter than a few days.

[6] Demand is similar. For ease, let’s just discuss the supply side.

[7] Latent liquidity isn’t usually observable, so we can’t easily test the hypothesis that latent liquidity at a given price builds as a function of time since the market was at that price. But latent liquidity could be measured via the order book — if a market is heavily financialized, transparent, and dominated by traders who don’t hide their intentions. Arguably, Bitcoin is (or used to be) such a market.

[8] We could add an i.i.d noise term to \rho_{S} and it wouldn’t affect the results. The assumption that \rho_{S} has no explicit price-dependence is similar to Donier and Bouchaud’s example of constant \omega and \nu. This assumption is obviously wrong, but sufficiently close to the market price, it may be reasonable enough.

[9] The reaction time of businesses could be significantly longer than the human decision-making timescale. Especially when the market in question isn’t transparent or mature. E.g., if a farmer switches from growing olives to almonds, it might take her grove several years to become productive again. So the price of almonds may need to reach a multi-year high in order for her to be confident enough to switch crops. But, if the farmer could hedge her future production, she might quickly decide to switch crops after it becomes economical to do so. Perhaps a well-developed futures market reduces the valid range of this model from t \lesssim 1yr to t \lesssim 1day.

[10] See eq. 38 of Majumdar and citations for details. I changed the CDF of eq. 38 into a PDF, and dropped a non-leading term.

[11] Eq. 22

[12] Sparre-Andersen scaling applies to any Markov process with a continuous and symmetric distribution of price movements: For large t, first passage time distributions have to decay like t^{-3 / 2}. So, in order for the expectation over t to converge, we must have \rho_{S}(t) < \sqrt{t} for large t (assuming \rho_{S}(t) is monotonically increasing). That is, the marginal supply at a given price must grow more slowly than the square root of the time since the market was at that price. The connection between latent liquidity and first passage times won’t hold for very long t, but this may provide a loose bound on how quickly latent liquidity can replenish.

[13] According to the model, slowly-clearing supply/demand curves are still dominated by the quadratic term when price movements are sufficiently large (and thus last-passage times long). But the model should fail at very long timescales, when businesses are able to react to price moves. Though, as discussed in [9], the model’s valid timescale could be longer if a market is slow and opaque.

[14] See, e.g., eq. 10 in Koren, et al.

[15] Figure 2 shows pretty rapid convergence to this asymptotic result after a handful of time steps.

[16] Perhaps it shouldn’t be surprising that impact is steeper than in the Brownian case. Heuristically, the Lévy tails make the market more “momentum-driven.” In a Lévy-type market, a trader who initiates a price movement could find that the market quickly moves away from her. The Lévy process used here has independent increments, but we can imagine the independence breaking down in continuous-time, “mid-increment.” I.e., it’s conceivable that momentum-traders could trade in the middle of a timestep.

On a related note, Koren et al. show that the mean leapover length diverges for Lévy flights. The leapover length could be interpreted as the profit made by stop orders, *if* they execute in the middle of a price jump and are able to hit liquidity during tail events. Those are big “ifs,” but the potential to make near-infinite profit may partly explain the popularity of stop orders and short-term momentum strategies. It could also explain why traders are reluctant to post much liquidity far away from the current price.

[17] In a sub-diffusive market with low transaction costs, betting on mean-reversion is a profitable strategy. If you know of any electronic markets like this, let me know.

Even if sub-diffusions are rare in electronic markets, linear combinations of different assets could still be sub-diffusive. The stereotypical “pairs trade” involves a mean-reverting spread between prices of two assets. Also, sub-diffusions may be more common in the broader, off-exchange economy than we’d naively expect.

[18] See the first non-constant term in the series expansion of eq. 30 in Metzler and Klafter (“Boundary value problems for fractional diffusion equations”). Eq. 30 is the survival probability (CDF of the first passage time), the PDF is the time derivative. The approximation is valid for y \ll \sqrt{ K_{\gamma}t^{\gamma} }. K_{\gamma} is the “fractional diffusion coefficient,” analogous to \sigma^2 for \gamma=1.

Note that sub-diffusions are not Markov processes, so Sparre-Andersen scaling [12] doesn’t apply. Liquidity must replenish more slowly than \rho_{S}(t) < t^{\gamma / 2} for large t.

[19] Correlated waiting times may also lead to sub-diffusions in a process with independent increments.

CHX and Four Types of Speed Bump

Speed bumps, the latest fad in market structure, are proliferating. The Chicago Stock Exchange (CHX) recently proposed a new type of speed bump in the US equities market, called the “Liquidity Taking Access Delay” (LTAD). [1] The SEC’s decision whether to approve the LTAD could have a big impact on market structure, possibly more so than the IEX decision.

To understand the consequences of approving LTAD, I think it’s helpful to consider four types of asymmetric access delay: [2][3]

  1. An exchange applies a delay to orders that are accessing resting liquidity. Resting orders themselves may be modified or cancelled without delay. This is the LTAD.
  2. The same as #1, except only resting orders that are non-displayed avoid the delay.
  3. The delay applies to all client messages to the exchange, including cancellations and marketable orders. But the exchange operates algorithmic order-types which adjust their attributes without delay. In particular, a displayed order can be pegged to undelayed price data, while its potential counterparties are subject to the delay.
  4. The same as #3, except only non-displayed algorithmic order-types avoid the delay. This is the IEX speed bump. [4]

Here’s a table of the speed bump combinations, split by which types of resting orders have functionality similar to last look:

Last look Can apply to displayed quotes Can apply to non-displayed quotes only
At discretion of resting order-sender #1 (LTAD) #2
At discretion of exchange algo #3 #4 (IEX)

In my opinion, all four types do more harm than good to market end-users (in the context of Reg NMS) — each allows resting liquidity to fade in some way. But many end-users disagree, and asked the SEC to approve #4. So, now that speed bumps exist and non-displayed peg orders may elide them, which of the other three asymmetric delays should be allowed?

One school of thought is that exchanges have a duty to protect their peg orders to the maximum extent possible. Combining this principle with the new “de minimis” interpretation may imply that speed bumps of types #3 and #4 should be allowed. Healthy Markets takes this view. [5] I think it’s a bad idea for exchange algorithms to have a time-advantage that isn’t offered to traders. Exchanges have neither the incentive nor the expertise to optimize their pricing algorithms. [6] The traditional role of the exchange is to provide a meeting place for buyers and sellers, allowing them to transact at prices of their own choosing. Exchanges offer algorithmic order-types (like pegs) mainly for convenience, and I don’t think there should be any illusion that these order-types are as good as the techniques used by sophisticated brokers and traders. Providing exchange algos with a time-advantage will mean that they out-compete traders, who are otherwise superior in terms of their pricing accuracy, stability under stress, and diversity. Traders and their algorithms do make mistakes, but it’s hard to believe that an exchange algo monoculture can do better. Subsidizing inferior business methods reduces an industry’s productivity, and giving exchange algorithms a structural advantage would be no different.

In my view then, if #4 is allowed, then #2 should be allowed. And, if #3 is allowed, then #1 should be allowed. Since #4 is already approved, the only thing to consider is whether it should be permissible for delays to have asymmetries in how they apply to accessing displayed quotes. The comments mostly do a good job explaining why asymmetries are problematic for displayed market structure. [7] In short, if displayed orders are given extra time to decide whether to consummate a trade, a large number of quotes will be practically inaccessible, even though it’d be prohibited to lock, cross, or trade-through those quotes. [8] Even in markets without these regulations, such as FX, last look causes difficulty. [9] The consequences of combining last look with order protection are unpredictable, but they seem unlikely to be good for long-term traders.

So, if we don’t want Reg NMS order protection to apply to quotes eligible for last look, only options #2 and #4 should be permitted. There is a tradeoff, though. Giving only hidden orders a time-advantage will incentivize dark trading. I doubt that the major exchanges would become as dark as IEX, but the equities market would probably become less transparent.

IEX’s approval is already inspiring a dramatic increase in complexity. CHX might not be the most important exchange, but like IEX, any precedent it sets applies to other exchanges. [10] Restricting speed-bump-asymmetries to hidden orders has drawbacks, but it might be the only way Reg NMS can keep limping on.

[1] Technically, it’s not really new. The LTAD is similar to an old proposal from Nasdaq’s PSX, which was rejected by the SEC.

[2] In order to highlight the essential elements, I’m leaving out some details.

[3] There are other speed bump types of course. For example, an exchange could delay traders from cancelling resting orders, but allow incoming marketable orders to execute without delay. That sort of delay could be used to address complaints about “fading liquidity“, and is similar in some respects to Nasdaq’s tentative “Extended Life Order” and EBS’s “Minimum Quote Life“.

Delays in market data are speed bumps of an entirely different class. These are prohibited (I think) in US equities markets, which require quotes and trades to be published to the SIP without delay. Though perhaps this requirement doesn’t apply to “de minimis” delays?

[4] Algorithmic order-types may also use their undelayed data feeds to trade aggressively with resting orders which can’t be cancelled without going through a delay. IEX’s DPEG does this via its “book recheck” mechanism.

[5] From their comment letter:

Time delays should not apply to an exchange’s ability to price orders on behalf of all participants (i.e. Pegging).

Dave Lauer clarified on Twitter that this principle also applies to displayed peg orders.

[6] CHX’s justification for the LTAD evokes a useful thought experiment. CHX argues that its ETF quotes are victims of “latency arbitrage,” and that the LTAD will prevent this. If the SEC rejects the LTAD, CHX might propose another type of speed bump, where CHX manages displayed orders in ETFs (e.g. SPY, QQQ, etc) by pegging them to undelayed CME market data. Will CHX understand the relationship that these ETFs have to futures markets as well as professional traders do? And these ETFs are just the simple cases — imagine what kind of pegs an exchange might come up with to prevent market makers from being “picked off” on XOM when the price of crude or CVX moves. I’m sure exchanges can think of peg algos that market makers would find very useful, but is that really what we want exchanges doing?

[7] No discussion of the LTAD would be complete without mentioning CHX’s market data revenue sharing program. This post, though, isn’t intended to be a complete discussion. Many of the comment letters (and Matt Hurd’s playful summaries) address this issue.

I found it interesting that Virtu publicly supported the LTAD as soon as it was announced, presumably before they had time to review the filings. Was the LTAD proposed at Virtu’s request? Could Virtu’s CHX profitability rely on market data revenue sharing?

[8] Without using ISOs, that is. If enough displayed orders were granted a last look, then the market would be clogged with inaccessible quotes. Only traders with the legal infrastructure to submit ISOs would be able to navigate the equity market.

[9] This opinion isn’t unanimous. Some long-term traders appreciate last look, like Norway’s SWF, NBIM.

[10] James Angel’s letter argues that CHX should be given the benefit of the doubt just like IEX was. I’m not a lawyer, and I think the LTAD could hurt long-term traders, but some of his points are persuasive.

High-speed Trading Networks and Societal Value

The romanticized trader works in a caravan, braving the elements to move property and information between cities. Over millenia, humans have gone from trading physical goods with wagons and ships to trading symbolic contracts with undersea cables and specialized radio networks. Markets’ adoption of communications technology has undoubtedly benefited society. But has it gone too far? Could the benefits of further speed be outweighed by the costs? A lot of people think so. And high-speed networks aren’t cheap; a new cable between Tokyo and London may cost $850M. But while I’ve heard lots of complaints about the cost of improving network infrastructure, I haven’t seen any estimates of the benefit. I will attempt to provide one here.

The “Arms Race”

Budish, Cramton, and Shim call further investment in speed “socially wasteful.” They also describe the dynamic between trading firms as a “prisoner’s dilemma,” where firms could increase their profits if they all avoid expensive technology investments. But if one firm “deviates” and improves its speed, it can temporarily increase its profit at the expense of its rivals — until they make their own similar investments. The result is that firms are continually investing in speed, but these expenditures don’t increase industry revenue. Many HFTs, being the supposed prisoners in this dilemma, agree that this process is a waste — for example Mani Mahjouri of Tradeworx:

[W]e went from piecing together existing fiber routes to digging trenches across the country to lay straight fiber to now sending signals through microwave towers and laying new trans-Atlantic cables, doing these very, very expensive technological investments… But if you look at it from the perspective of society, that’s a tremendous amount of capital that’s being invested… In my view, maybe that would be better spent making a more competitive price, as opposed to spending it on speed

Now, capitalism is supposed to force competitors to invest in order to provide a better service. If an industry complains that high investment costs are squeezing its margins, that’s a sign that competition is working the way it should. The concern is normally the opposite, that competitors are colluding to avoid “defections” from the investment “prisoner’s dilemma.” The question is whether there’s something unique about trading networks that makes the investments worthless to society. I don’t see why there should be.

The Speed of Information and Social Welfare

It is obvious that social welfare suffers when communications between markets are sufficiently slow. A revealing example is the experience of a Mediterranean trader (c.a. 1066), who needed to know the price of silk in an away market before he could proceed with his business. He sent the following letter, where he reports that his time awaiting this information will be wasted:

The price in Ramle of the Cyprus silk, which I carry with me, is 2 dinars per little pound. Please inform me of its price and advise me whether I should sell it here or carry it with me to you in Misr (Fustat), in case it is fetching a good price there. By God, answer me quickly, I have no other business here in Ramle except awaiting answers to my letters… I need not stress the urgency of a reply concerning the price of silk

Clearly, there would have been an economic benefit if the trader had faster communications. In general, how can we estimate the size of this benefit? When prices aren’t synchronized, wealth is arbitrarily transferred between traders — some transactions receive a slightly better price and others a slightly worse price, purely through luck. These wealth transfers are zero on average, so it’d be wrong to say their cost to society is their entire amount. But it’d also be wrong to say that such transfers are completely innocuous because they’re “zero-sum.” Which means that a reasonable estimate of the societal cost of unsynchronized markets is: the volume (V_{transfer}) of wealth transfers due to price discrepancies between venues, multiplied by a factor between 0 and 1, representing the economic cost per unit wealth transfer (C_{transfer}).

Let’s start by estimating C_{transfer}. Here are three examples where markets assign a value to the cost associated with random wealth transfers:

  1. Insurance: Where customers transfer the risk of bad luck to insurance companies, paying significantly more than expected losses to offload risk. Loss ratios in the insurance industry are generally around 80%. [1] So the insurance market has a C_{transfer} of about 0.2. [2]
  2. The equity risk premium: Aswath Damodaran calculates the historical excess return of equities over short-term government debt to be 4.4% globally, and the standard deviation of the difference in returns to be 17.1% (Table 6). This means, roughly, that investors typically demand an additional 4.4% return in exchange for risking 17.1% of their capital — translating to a C_{transfer}=\frac{0.044}{0.171} \approx 0.25. Excluding the US, this ratio is 0.2, and in some countries isn’t far above 0.1 (e.g. Belgium or Norway).
  3. Jensen’s analysis of synchronization between fish markets in Kerala: After fishermen obtained mobile phones, they were able to find the nearby market with the highest price for fish, while they were still on the water. This information made it much easier for them to smooth out local fluctuations in supply and demand (see, e.g., Figure IV). Jensen estimates that mobile phones increased fishermen’s profits by 8% and decreased the average cost of fish by 4%. [3] These numbers suggest that the Kerala fish market has a C_{transfer} of at least 0.1. [4]

It’s striking that these disparate methods all give approximately the same value. For our calculation, we’ll take C_{transfer}=0.1.

Now, we just need to estimate the reduction in random wealth transfers, \Delta V_{transfer}, for a famous HFT network — The NY-Chicago microwave route seems like a good example. We’ll just make a simple, order-of-magnitude estimate. It’s reasonable to approximate \Delta V_{transfer} with the total executed volume (V) for products with multiple important trading centers connected by the route, multiplied by the typical price dispersion (\sigma_{\Delta}) prevented by a reduction in inter-market latency. The CME traded over a quadrillion dollars notional in 2015. Some of that volume is “inflated” by contracts with a high notional value (e.g. options, Eurodollars) or may not be tightly coupled with away markets — so let’s cut it by 90% and say that V \approx \$10^{14} yr^{-1} is the volume in Chicago that’s highly-correlated with important markets in New York or Europe. [5][6] We can estimate \sigma_{\Delta} by making the usual assumption that price movements are Brownian, so that volatility scales with the square root of time. Let’s just roughly assume that the relevant CME contracts have an annual volatility of 10%. This makes it easy to calculate how much their prices typically move in the ~2.5ms that wireless networks save over fiber (one-way): \sigma_{\Delta} = 0.1 yr^{-\frac{1}{2}} \sqrt{2.5 ms \cdot \frac{1.6 \times 10^{-10} yr}{ms}} \approx 2 \times 10^{-6} [7]

So we have that \Delta V_{transfer} \approx V \cdot \sigma_{\Delta} \approx \$10^{14} yr^{-1} \cdot 2 \times 10^{-6}. In other words, upgrading the Chicago-NY route from fiber to microwave prevents around $200M of arbitrarily transferred wealth per year. Multiplying this by our estimated economic cost per unit transfer, C_{transfer}=0.1, gives the societal value of this microwave route: about $20M per year.

Comparing a Network’s Value to its Cost

Laughlin, Aguirre, and Grundfest estimate that Chicago-NY microwave networks required around $140M in capital expenditures, with $20M per year in operating expenses. Our estimate of the route’s value is very rough, but still surprisingly close to their (also rough) numbers. And I imagine that some fraction of the costs, which include things like salaries and radio licensing fees, are not a complete loss to the economy. So, it’s quite plausible that these microwave networks are “worth it” to society.

There’s also the possibility that the networking technology developed for HFTs will have beneficial applications in other industries, many of which are latency-sensitive. Finance has a history of subsidizing important innovations. According to Rocky Kolb, the first application of the telescope was spotting cargo ships and using that information to trade. Galileo himself described how impressed Venetian leaders were with this application, who rewarded him:

And word having reached Venice that I had made one [a spyglass], it is six days since I was called by the Signoria, to which I had to show it together with the entire Senate, to the infinite amazement of all; and there have been numerous gentlemen and senators who, though old, have more than once climbed the stairs of the highest campaniles in Venice to observe at sea sails and vessels so far away that, coming under full sail to port, two hours and more were required before they could be seen without my spyglass.

If — in my armchair — I were put in charge of the economy and somebody came to me with a $200M project to cut inter-market latency by 3ms, I really wouldn’t know whether it was a good idea. Sure, I could do a more thorough calculation than we did here. But I bet the best decision for the economy would be based on whether people would actually pay to use the network. That’s what telecoms do already. Capital markets sometimes invest in dead-end projects, but identifying these mistakes, in advance, is rarely easy.

Reducing latency from 2 weeks to 1 second has obvious benefits to society. Going from 7ms to 4ms is more subtle. [8] But just because progress is incremental, doesn’t mean we should dismiss its value. The economy is big, and our markets process tremendous volumes. A small improvement in price discovery can make a meaningful difference.

[1] Typical loss ratios for property and casualty insurance in 2015 were about 69% (Table 1). For accident and health insurance, they were about 80% (Figures 10 and 11).

[2] Of course, buyers of insurance may be especially risk-averse and willing to pay high premiums to avoid catastrophes. But hopefully competition and regulation keeps exploitation to a minimum.

[3] Jensen estimates that consumer surplus (the welfare benefit to fish-buyers) increased by 6%, a bit bigger than the decline in fish prices.

[4] Also keep in mind that prices were still not perfectly synchronized after the introduction of mobile phones. Perhaps profits would increase and prices decrease even more if fishing boats had automated, low-latency routing. They almost certainly would if new technology made boats faster.

[5] It isn’t just S&P 500 futures in Chicago that are highly-correlated with markets in NY and Europe. You can make obvious arguments to include FX and fixed income futures in this category. Energy pricing is critically important for a wide range of asset markets. Agricultural products and metals also trade around the world, and can provide important trading signals for some equities.

[6] I don’t know whether the current lowest latency route between Chicago and Europe goes through New York. It probably depends on whether Hibernia allows customers to connect at their Halifax landing, and use their own microwave networks to shuttle data between Halifax and Chicago. I’m far from an expert on such things, but Alexandre Laumonier suggests that Hibernia may have restricted connections at a different landing:

Different informants in the industry (and one journalist) told me that Hibernia will not allow (at least for now) dishes at the Brean landing station. I tried to know more about that but the only answers I got were some “neither confirm nor deny” responses. Huh! People know but don’t talk. I wrote an email to Hibernia but I got zero answer (obviously). Then other informants told me Hibernia may finally allow dishes…

It’d be pretty funny if Hibernia let customers connect at Halifax instead of New York or Chicago, but charged extra for the ostensibly inferior service (like airlines do).

In any case, even if there are proprietary wireless networks from Chicago to Halifax, they probably share some towers with the Chicago-NY route. So maybe it’s fair to include Chicago-Europe traffic in our estimate of the economic value of Chicago-NY microwave networks.

[7] Assuming there are ~250 trading days per year, and each trading day is ~7 hours — so there are about 1.6 \cdot 10^{-10} trading years per millisecond.

[8] Matt Levine describes (as many do) “a market as a giant distributed computer for balancing supply and demand; each person’s preferences are data, and their interaction is the algorithm that creates prices and quantities.” This analogy may be helpful for understanding high-speed trading. A supercomputer’s performance can depend heavily on its interconnect. If the market is a giant supercomputer, reducing its interconnect latency from 7ms to 4ms could dramatically increase its processing power — for some tasks by 40%. For such tasks, we’d expect a large increase in inter-node traffic when latency is improved. Perhaps we are seeing this increase in modern financial markets, which have far higher trade and message volumes than in the past.

Price Impact in Efficient Markets

Market prices generally respond to an increase in supply or demand. This phenomenon, called “price impact,” is of central importance in financial markets. Price impact provides feedback between supply and demand, an essential component of the price discovery mechanism. Price impact also accounts for the vast majority of large traders’ execution costs — costs which regulators may seek to reduce by tweaking market structure.

Price impact is a concave function of meta-order [1] size — approximately proportional to the square-root of meta-order size — across every well-measured financial market (e.g. European and U.S. equities, futures, and bitcoin). There are some nice models that help explain this universality, most of which require fine-grained assumptions about market dynamics. [2] But perhaps various financial markets, regardless of their idiosyncrasies, share emergent properties that could explain empirical impact data. In this post, I try to predict price impact using only conjectures about a market’s large-scale statistical properties. In particular, we can translate intuitive market principles into integral equations. Some principles, based on efficiency arguments, imply systems of equations that behave like real markets.

In part I, we’ll start with the simplest principles, which we’ll only assume to hold on average: the “fair pricing condition”, and that market prices efficiently anticipate the total quantity of a meta-order based on its quantity already-executed. In part II, we’ll replace the fair pricing condition with an assumption that traders use price targets successfully, on average. In part III, we’ll return to fair pricing, but remove some efficiency from meta-order anticipation — by assuming that execution information percolates slowly into the marketplace. In part IV, we’ll emulate front-running, by doing the opposite of part III: leaking meta-orders’ short-term execution plans into the market. In parts V and VI, we’ll discuss adding the notion of urgency into meta-orders.

Definitions and Information-Driven Impact

We can motivate price impact from a supply and demand perspective. During the execution of a large buyer’s meta-order, her order flow usually changes the balance between supply and demand, inducing prices to rise by an amount called the “temporary price impact.” After the buyer is finished, she keeps her newly-acquired assets off the market, until she decides to sell. This semi-permanent reduction in supply causes the price to settle at a new level, which is higher than the asset’s initial price by an amount called the “permanent price impact.” Changes in available inventory cause permanent impact, and changes in flow (as well as inventory) cause temporary impact. [3]

Another view is that informed trading causes permanent impact, and that uncertainty about informedness causes temporary impact. When a trader submits a meta-order, its permanent impact should correspond in some fashion to her information. And its temporary impact should correspond to the market-estimate of permanent impact. In an “efficient” market, the informational view and the supply/demand view should be equivalent.

Before we proceed, we need some more definitions. Define \alpha(q) as the typical permanent price impact associated with a meta-order of quantity q. By “typical”, I mean that \alpha(q) is the expectation value of permanent impacts, \alpha_{s}(q), associated with a situation, s, in the set of all possible situations and meta-orders, S. Specifically, \alpha(q) = \mathbf{E}_{s \in S}[\alpha_{s}(q)]. It’s reasonable to associate the colloquial term “alpha” — which describes how well a given trade (s) predicts the price — with \alpha_{s}.

Also define \mathcal{I}(q) as the typical temporary price impact after a quantity q has been executed. Again, “typical” means \mathcal{I}(q) = \mathbf{E}_{s \in S}[\mathcal{I}_{s}(q)].

These expectations can be passed through the integrals discussed below, so we don’t need to pay attention to them. In the rest of this post, “permanent impact” will refer to the expectation \alpha(q)=E_{s \in S}[\alpha_{s}(q)] unless otherwise specified (and likewise for “temporary impact” and \mathcal{I}(q)).

I. A Bare-Bones Model

Starting from two assumptions of market efficiency, we can determine the typical price-trajectory of meta-orders. The two conditions are:

I.1) The “fair pricing condition,” which equates traders’ alpha with their execution costs from market-impact (on average):

\alpha(q) = \frac{1}{q} \int_{0}^{q} \mathcal{I}(q') dq'

The integral denotes the quantity-averaged temporary impact “paid” over the course of an entire meta-order. “Fair pricing” means that, in aggregate, meta-orders of a given size do not earn excess returns or below-benchmark returns.

Temporary price impact (black line) over the course of a meta-order of size q. After the execution is finished, the price impact decays (dashed line) to \alpha(q) (red), the quantity-weighted average of the meta-order’s temporary impact trajectory

I.2) Efficient linkage between temporary and permanent price impact:

\mathcal{I}(q') = \mathbf{E}_{q}[\alpha(q)|q \geq q'] = \int_{q'}^{\infty} \alpha(q)p[q|q \geq q']dq


Here, p[q] is the PDF of meta-order sizes and P[q] is the CDF. And p[q|q \geq q'] is the truncated probability density of meta-order sizes, \frac{p[q]}{1 - P[q']} — which represents the probability distribution of q given that quantity q' from the meta-order has already executed. This condition means that, on average, “the market” observes how much quantity an anonymous trader has executed, and uses that to estimate the distribution of her meta-order’s total quantity. “The market” then calculates an expectation value of the meta-order’s alpha, which sets the current clearing price (i.e. temporary impact). [4] [5] To emphasize, only the average temporary impact is determined this way; a single meta-order could have much different behavior. Here’s a heuristic example:
A. A trader is buying a lot of IBM stock and has so far bought a million shares.
B. The rest of the market sees signs (like higher price and volume) of that purchase and knows roughly that somebody has bought a million shares.
C. Once a trader has bought a million shares, there is a 50% chance that she’ll buy 5 million in total, and a 50% chance that she’ll buy 10 million. “The market” knows these probabilities.
D. For 5 million share meta-orders, the typical permanent price impact is 1%, and for 10 million share meta-orders it’s 2%. So “the market” expects our trader’s meta-order to have permanent impact of 1.5%. The *typical* temporary impact is determined by this expectation value. This particular meta-order may have temporary impact smaller or larger than 1.5%, but meta-orders sent under similar circumstances will have temporary impact of 1.5% on average.



An illustration of this linkage. The temporary price impact trajectory is the black line. At a given value of q', \mathcal{I}(q') (blue) is equal to the expected value (blue) of the permanent price impact given that the meta-order has size q' or bigger. The probability density of the final meta-order size, p[q|q \geq q'], is shown in shaded red. The permanent impact associated with those meta-order sizes is shown in green.

Relationship with Efficiency

The fair pricing condition could emerge when the capital management industry is sufficiently competitive. If a money manager uses a trading strategy that’s profitable after impact costs, other managers could copy it and make money. The strategy would continue to attract additional capital, until impact expenses balanced its alpha. (Some managers are protective of their methods, but most strategies probably get replicated eventually.) If a strategy ever became overused, and impact expenses overwhelmed its alpha, then managers would probably scale back or see clients pull their money due to poor performance. Of course these processes take time, so some strategies will earn excess returns post-impact and some strategies may underperform — fair pricing would hold so long as they average out to a wash.

A strictly stronger condition than I.2) should hold in a market where meta-orders are assigned an anonymous ID, and every trade is instantly reported to the public with its meta-order IDs disclosed. Farmer, Gerig, Lillo, and Waelbroeck call a similar market structure the “colored print” model. Under this disclosure regime, if intermediary profits are zero, the expected alpha would determine the temporary impact path of individual meta-orders, not just the average \mathcal{I}(q') as in I.2). All meta-orders would have the same impact path: \mathcal{I}_{s}(q') = \mathbf{E}_{q}[\alpha(q)|q \geq q'] = \int_{q'}^{\infty} \alpha(q)p[q|q \geq q']dq for any s. [6] Now, the colored print model doesn’t seem very realistic; most markets don’t have anywhere near that level of transparency. Nonetheless, Farmer et al. show preliminary measurements that partly support it. [7]

Even without colored prints, the linkage property I.2) could be due to momentum and mean-reversion traders competing away their profits. As discussed by Bouchaud, Farmer, and Lillo, most price movement is probably caused by changes in supply and demand. That is, if prices move on increased volume, it’s likely that someone activated a large meta-order, especially if there hasn’t been any news. So, if average impact overshot I.2) significantly, a mean-reversion trader could plausibly watch for these signs and profitably trade opposite large meta-orders. Likewise, if average impact undershot I.2), momentum traders might profit by following the price trend.

Solving the System of Equations

We can combine I.1) and I.2) to get an ODE [8]:

\alpha''(q) + (\frac{2}{q} - \frac{p[q]}{1-P[q]})\alpha'(q) = 0

This ODE lets us compute \alpha(q) and \mathcal{I}(q) for a given meta-order size distribution, p[q].

It’s common to approximate p[q] as a Pareto[q_{min},\beta] distribution (p[q] = \frac{\beta q_{min}^{\beta}}{q^{\beta+1}}). If we do so, then \frac{p[q]}{1-P[q]} = \frac{\beta}{q}, and the ODE has solution \alpha(q) = c_1 q^{\beta-1}+c_2. Equation I.1) implies \mathcal{I}(q) = \alpha(q)+q\alpha'(q), so we have that \mathcal{I}(q) = c_1 \beta q^{\beta-1} + c_2. Impact should nearly vanish for small q, so we can say that c_2 \approx 0. The post-execution decay in impact is then given by \frac{\alpha(q)}{\mathcal{I}(q)} = \frac{1}{\beta}

If we choose \beta = \frac{3}{2} (roughly in-line with empirical data), we get the familiar square-root law: \mathcal{I}(q) \propto \sqrt{q}. We also get an impact ratio of \frac{\alpha(q)}{\mathcal{I}(q)} = \frac{2}{3}, very close to real-world values.

A similar method from Farmer, Gerig, Lillo, and Waelbroeck gives the same result. They use the fair pricing condition, but combine it with a competitive model of market microstructure. [9] Here, instead of having a specific model of a market, we’re making a broad assumption about efficiency with property I.2). There may be a large class of competitive market structures that have this efficiency property.

Distribution of Order Sizes Implied by a Given Impact Curve

Under this model, knowing an asset’s price elasticity (\mathcal{I}(q)) is equivalent to knowing its equilibrium meta-order size distribution(p[q]). [10] If a market impact function \mathcal{I}(q) is assumed, we can calculate the meta-order size distribution. [11] For instance, Zarinelli, Treccani, Farmer, and Lillo are able to better fit their dataset with an impact function of the form \mathcal{I}(q) = a Log_{10}(1+bq) (p17). This impact curve implies a p[q] that’s similar to a power-law, but with a slight bend such that its tail decays slower than its bulk:


Meta-order size distribution implied by the impact curve \mathcal{I}(q) = 0.03 Log_{10}(1+470q), which Zarinelli, Treccani, Farmer, and Lillo fit to their dataset of single-day meta-orders. In this case, q would be analogous to their chosen measure of size, the daily volume fraction \eta. The impact function’s fit might be invalid for very large meta-orders (q \approx 1), so the lack of a sharp cutoff near q \approx 1 in the implied size distribution isn’t problematic.

II. A Replacement Principle for Fair Pricing: Traders’ Effective Use of Price Targets

The two integral equations in part I can be modified to accommodate other market structure principles. There’s some evidence that our markets obey the fair pricing condition, but it’s fun to consider alternatives. One possibility is that traders have price targets, and cease execution of their meta-orders when prices approach those targets. We can try replacing the fair pricing of I.1) with something that embodies this intuition:

II.1) \alpha(q) = a\mathcal{I}(q) + d

Where a and d are constants. This principle should be true when traders follow price-target rules, and their targets accurately predict the long-term price (on average). If d=0 and a=\frac{5}{4}, then traders typically stop executing when the price has moved \frac{4}{5} of the way from its starting value to its long-term value. If a=1 and d=0.01, then traders stop executing when the price is within 1% of its long-term value.

If we keep I.2), this gives the ODE:

\alpha'(q) + \frac{p[q](a-1)}{1 - P[q]}\alpha(q) + \frac{p[q]d}{1 - P[q]}=0

It’s readily solved. [12] In particular, if q \sim Pareto[q_{min},\beta] and a \neq 1 :

\alpha(q) = c q^{\beta (1-a)}+\frac{d}{1-a} and \mathcal{I}(q) = \frac{c q^{\beta (1-a)}+\frac{d}{1-a}-d}{a}.

For typical values of \beta \approx 1.5, we can get the usual square root-law by setting a \approx \frac{2}{3}. We need 0 < a < 1 in order for impact to be a concave, increasing function of order size, in agreement with empirical data. This suggests that perhaps traders do employ price targets, only instead of being conservative, their targets are overly aggressive. In other words, this model gives a realistic concave impact function if traders are overconfident and think their information is worth more than it is. [13] More generally, the partial reversion of impact after meta-orders’ completion could be explained with overconfidence. And when the “average” trader is overconfident just enough to balance out her alpha, the market will obey the fair pricing condition. I think there’s more to fair pricing than overconfidence, but this link between human irrationality and market efficiency is intriguing.

III. A Replacement Principle for Efficient Linkage, with Delayed Dissemination of Information

We can also think about alternatives for I.2). In I.2), “the market” could immediately observe the already-executed quantity of a typical meta-order. But markets don’t instantly process new information, so perhaps the market estimate of meta-orders’ already-executed quantity is delayed:

III.2) \mathcal{I}\left(q'\right) = \mathbf{E}_{q}[\alpha(q)|q \geq (q'-q_d)^+] = \frac{\int_{(q'-q_d)^+}^{\infty } p[q] \alpha (q) \, dq}{1-P[(q'-q_d)^+]}

Where q_d is a constant and (q'-q_d)^+ is the positive part of (q'-q_d): max(0,(q'-q_d)).
This condition should be true when the market (on average) is able to observe how much quantity an anonymous trader executed in the past, when her executed quantity was q_d less than it is in the present. This information can be used to estimate the distribution of her meta-order’s total size, and thus an expectation value of its final alpha. The temporary impact is set by this expectation value.

Intuitively, small meta-orders may blend in with background activity, but large ones are too conspicuous. If someone sends two 100-share orders to buy AAPL, other traders won’t know (or care) whether those orders came from one trader or two. But if a large buyer is responsible for a third of the day’s volume, other traders will notice and have a decent estimate of the buyer’s already-executed quantity, even if they don’t know whether the buyer was involved in the most recent trades on the tape. So, it’s very plausible for market participants to have a quantity-lagged, anonymized view of each other’s trading activity.

Combining III.2) with fair pricing I.1) gives the delay differential equation [14]:

\begin{cases} q \alpha ''(q) + \alpha '(q) \left(2-\frac{q p[q-q_d]}{1-P[q-q_d]}\right)-\left(\alpha (q)-\alpha (q-q_d)\right)\frac{p[q-q_d]}{1-P[q-q_d]}=0, & \mbox{if } q \geq q_d \\ \mathcal{I}(q)=\alpha(q)=constant, & \mbox{if } q < q_d \end{cases}.

We can solve it numerically [15]:


\mathcal{I}(q) and \alpha(q) when q is Pareto-distributed, for several values of q_d. The general behavior for q \gg q_d is similar to that of q_d=0, as in I.


The impact ratio \frac{\alpha(q)}{\mathcal{I}(q)} for several values of q_d. This ratio is 1 when the price does not revert at all post-execution, and 0 when the price completely reverts.

I gather that fundamental traders don’t like it when the price reverts on them, so some may want this impact ratio to be close to 1. Delayed information dissemination helps accomplish this goal when meta-orders are smaller than what can be executed within the delay period. But traders experience bigger than usual reversions if their meta-orders are larger than q_d. This behavior is intuitive: if a meta-order has executed a quantity less than q_d, other traders will have zero information about it and can’t react. But as soon as its executed quantity reaches q_d, the market is made aware that somebody is working an unusually big meta-order, and so the price moves considerably.

Some bond traders are pushing for a longer delay in trade reporting. One rationale is that asset managers could execute meta-orders during the delay period, before other traders react and move the market. The idea feels superficially like condition III.2), but isn’t a perfect analogy, because counterparties still receive trade confirmations without delay. And counterparties do use this information to trade. [16] So, delaying prints may not significantly slow the percolation of traders’ information into the marketplace, it just concentrates that information into the hands of their counterparties. Counterparties might provide tighter quotes because of this informational advantage, but only if liquidity provision is sufficiently competitive. [17]

In theory, it’s possible for market structure to explicitly alter q_d. [18] An exchange could delay both prints and trade confirmations, while operating, on behalf of customers, execution algorithms which do not experience a delay. This was the idea behind IEX’s defunct router, which would have been able to execute aggressive orders against its hidden order book and route out the remainder before informing either counterparty about the trades. The router would’ve increased the equity market’s q_d by the resting size on IEX’s hidden order book, which (I’m guessing) is very rarely above $100k notional — an amount that doesn’t really move the needle for large fundamental traders, especially since orders larger than q_d experience significant price reversion. Regardless, it’s interesting to think about more creative ways of giving exchange execution algorithms an informational advantage. The general problem with such schemes is that they are anti-competitive; brokers would have to use the advantaged exchange algos, which could command exorbitant fees and suffer from a lack of innovation. [19]

IV. A Replacement Principle for Efficient Linkage, with Information Leakage from Sloppy Trading or Front-Running

In III., we altered condition I.2) so that market prices responded to meta-orders’ executions in a lagged fashion. We can try the same idea in reverse to see what happens if market prices adjust to meta-orders’ future executed quantity:


\mathcal{I}\left(q_{tot},q_{executed}\right) = \begin{cases} \mathbf{E}_{q}[\alpha(q)|q \geq q_{executed}+q_{FR}] = \frac{\int_{q_{executed}+q_{FR}}^{\infty } p[q] \alpha (q) \, dq}{1-P[q_{executed}+q_{FR}]}, & \mbox{if } q_{executed}<q_{tot}-q_{FR} \\ \mathbf{E}_{q}[\alpha(q)|q=q_{tot}] = \alpha \left(q_{tot}\right), & \mbox{if } q_{executed}\geq q_{tot}-q_{FR} \end{cases}

Where \mathcal{I}\left(q_{tot},q_{executed}\right) is the temporary impact associated with a meta-order that has an already-executed quantity of q_{executed} and a total quantity of q_{tot}. q_{FR} is a constant. On average, a meta-order’s intentions are partly revealed to the market, which “knows” not only the meta-order’s already-executed quantity, but also whether it will execute an additional quantity q_{FR} in the future. If a meta-order will execute, in total, less than q_{executed}+q_{FR}, the market knows its total quantity exactly. “The market” uses this quantity information to calculate the meta-order’s expected alpha, which determines the typical temporary impact.

This condition may be an appropriate approximation for several market structure issues:

A. The sloppy execution methods described in “Flash Boys”: If a sub-par router sends orders to multiple exchanges without timing them to splash-down simultaneously, then “the market” may effectively “know” that some of the later orders are in-flight, before they arrive. If most fundamental traders use these sloppy routing methods (as “Flash Boys” claims), then we might be able to describe the market’s behavior with a q_{FR} approximately equal to the typical top-of-book depth.
B. Actual front-running: E.g., if fundamental traders split up their meta-orders into $10M pieces, and front-running brokers handle those pieces, the market will have a q_{FR} \approx \$ 10M. Though, brokers know their customers’ identities, so they may be able to predict a customer’s permanent impact with better precision than this model allows.
C. Last look: During the last-look-period, a fundamental trader’s counterparty can wait before finalizing the trade. If the fundamental trader sends orders to other exchanges during this period, her counterparty can take those into account when deciding to complete the trade. This is similar to A., except traders can’t avoid the information leakage by synchronizing their orders.

We can examine the solutions of this version of condition 2). Combining it with the fair pricing condition I.1) gives, for meta-orders with q_{tot}>q_{FR}: [20]

\alpha '(q_{tot}) \left(2-\frac{(q_{tot}-q_{FR}) p[q_{tot}]}{1-P[q_{tot}]}\right)+(q_{tot}-q_{FR}) \alpha ''(q_{tot})=0

If q_{tot} \sim Pareto[q_{min},\beta], this has solution:

\alpha (q_{tot}) = c_1 + c_2 (q_{tot}-q_{FR}){}^{\beta-1} \, _2F_1(1-\beta,-\beta;2-\beta;\frac{q_{FR}}{q_{FR}-q_{tot}})

For q_{tot} \gg q_{FR}: the _2F_1(...) \approx 1, so \alpha (q_{tot}) \approx c_1 + c_2 q_{tot}^{\beta-1}, which is the same behavior we saw in the base model I.

If we look at the solution’s behavior for q_{tot} \gtrsim q_{FR}, the story is quite different:


Permanent Impact and Peak-Temporary Impact when q_{tot} is slightly above q_{FR} = 10^{-4}, with constants c_1=0 and c_2=1. The temporary impact for a meta-order of size q_{tot} reaches its peak just before the meta-order’s end becomes known to the market, at q_{executed}=q_{tot}-q_{FR}. Peak-temporary impact goes negative when q_{tot} is sufficiently close to q_{FR}, but it’s possible to choose constants so that it stays positive (except at q_{tot}=q_{FR}, where it’s complex-valued). \alpha(q_{tot}), on the other hand, has a regular singular point at q_{tot}=q_{FR} and it is not possible to choose non-trivial constants such that \alpha(q_{tot}) is always positive. Temporary impact is calculated numerically via equation IV.2).

Under this model, meta-orders slightly larger than q_{FR} necessarily have negative long-term alpha. It’s possible that traders would adapt to this situation by never submitting meta-orders of that size, altering the Pareto-distribution of meta-order sizes so that no commonly-used q_{tot} is associated with negative alpha. But, it’s also possible that some traders would continue submitting orders that lose money in expectation. Market participants have diverse priorities, and long-term alpha is not always one of them.

V. Adding Time-Dependence

The model template above gets some general behavior right, but glosses over important phenomena in our markets. It makes no explicit mention of time, ignoring important factors like the urgency and execution rate of a meta-order. It’s not obvious how we could include these using only general arguments about efficiency, but we can imagine possible market principles and see where they lead.

For the sake of argument, say that every informed trading opportunity has a certain urgency, u, defined as the amount of time before its information’s value expires. For example, an informed trader may have a proprietary meteorological model which makes predictions 30 minutes before public forecasts are published. If her model predicts abnormal rainfall and she expects an effect on the price of wheat, she’d have 30 minutes to trade before her information becomes suddenly worthless. Of course, in real life she’d have competitors and her information would decay in value gradually over the 30 minutes, perhaps even retaining some value after it’s fully public. But let’s just assume that u is a constant for a given trading opportunity and see where it leads us.

If we try following a strict analogy with the time-independent model, we might write down these equations:

V.1) A “universal-urgency fair pricing condition,” that applies to meta-orders at every level of urgency:

\alpha(q,u) = \frac{1}{q} \int_{0}^{q} \mathcal{I}(q',u) dq'


This is a much stronger statement than ordinary fair pricing. It says that market-impact expenses equal alpha, on average, for meta-orders grouped by *any* given urgency. There are good reasons to expect this to be a bad approximation of reality — e.g. high-frequency traders probably constitute most short-urgency volume [21] and have large numbers of trades to analyze, so they can successfully tune their order sizes such that their profits are maximized (and positive). Perhaps some traders with long-urgency information submit orders that are larger than the capacity of their strategies, but I doubt HFTs do.

V.2) Efficient linkage between temporary and permanent price impact:

\mathcal{I}(q',u') = \mathbf{E}_{q,u}[\alpha(q,u)|q \geq q', u \geq u'] =\int_{u'}^{\infty}\int_{q'}^{\infty} \alpha(q,u)p[q,u|q \geq q', u \geq u']dqdu

Where p[q,u] is the PDF of meta-order sizes and urgencies, and P[q,u] is the CDF. p[q,u|q \geq q',u \geq u'] is the truncated probability distribution of meta-order sizes and urgencies, \frac{p[q,u]}{1 - P[q',\infty] - P[\infty,u'] + P[q',u']} — which represents the probability distribution of q and u given the knowledge that quantity q' from the meta-order has already executed in time u'. This is similar to the time-independent efficient linkage condition I.2). For example, a trader splits her meta-order into chunks, executing 1,000 shares per minute starting at 9:45. If she is still trading at 10:00, “the market,” having observed her order-flow imbalance, will “know” that her meta-order is at least 15,000 shares and has an urgency of at least 15 minutes. “The market” then calculates the expected alpha of the meta-order given these two pieces of information, which determines the average temporary impact.

We can combine these two equations to get a rather unenticing PDE. [22] As far as I can tell, its solutions are unrealistic. [23] Most solutions have temporary price impact that barely changes with varying levels of urgency. But in the real world, temporary impact should be greater for more urgent orders. The universal-urgency fair pricing here is too strong of a constraint on trader behavior. This condition means that markets don’t discriminate based on information urgency. Its failure suggests that markets do discriminate — and that informed traders, when they specialize in a particular time-sensitivity, face either a headwind or tailwind in their profitability.

VI. A Weaker Constraint

If we want to replace the universal-urgency of V.1) with something still compatible with ordinary fair pricing, perhaps the weakest constraint would be the following:

VI.1) \mathbf{E}_{u|q}[\alpha(q,u)] = \mathbf{E}_{u|q}[\frac{1}{q} \int_{0}^{q} \mathcal{I}(q',u) dq']

Which says that, for a given q, fair pricing holds on average across all u.

Requiring this, along with V.2), gives a large class of solutions. Many solutions have q -behavior similar to the time-independent model I, with u -behavior that looks something like this:


Stylized plot of permanent (\alpha) and temporary (\mathcal{I}) price impact vs urgency. Meta-orders of some urgencies pay more (on average) in temporary impact than they make in permanent impact, while meta-orders of other urgencies pay less than they make.

This weaker constraint leaves a great deal of flexibility in the shape of the market impact surface \mathcal{I}(q,u). Some of the solutions seem reasonable, e.g. for large u, \mathcal{I} could decay as a power of u. But there are plenty of unreasonable solutions too, so perhaps real markets obey a stronger form of fair pricing.


Price impact has characteristics that are universal across asset classes. This universality suggests that financial markets possess emergent properties that don’t depend too strongly upon their underlying market structure. Here, we consider some possible properties and their connection with impact.

The general approach is to think about a market structure principle, and write down a corresponding equation. Some of these equations, stemming from notions of efficiency, form systems which have behavior evocative of our markets. The simple system in part I combines the “fair pricing condition” with a linkage between expected short-term and long-term price impact. It predicts both impact’s size-dependence and post-execution decay with surprising accuracy. Fair pricing appears to agree with empirical equities data. The linkage condition is also testable. And, as discussed in part III, its form may weakly depend on how much and how quickly a market disseminates data. If we measure this dependence, we might further understand the effects of price-transparency on fundamental traders, and give regulators a better toolbox to evaluate the evolution of markets.

[1] A “meta-order” refers to a collection of orders stemming from a single trading decision. For example, a trader wanting to buy 10,000 lots of crude oil might split this meta-order into 1,000 child orders of 10 lots.

[2] There’s a good review and empirical study by Zarinelli et al. It has a brief overview of several models that can predict concave impact, including the Almgen-Chriss model, the propagator model of Bouchaud et al. and of Lillo and Farmer, the latent order book approach of Toth et al. and its extension by Donier et al., and the fair pricing and martingale approach of Farmer et al.

[3] Recall the “flow versus stock” (“stock” meaning available inventory) debate from the Fed’s Quantitative Easing programs, when people agonized over which of the two had a bigger impact on prices. E.g., Bernanke in 2013:

We do believe — although, you know, there’s room for debate — we do believe that the primary effect of our purchases is through the stock that we hold, because that stock has been withdrawn from markets, and the prices of those assets have to adjust to balance supply and demand. And we’ve taken out some of the supply, and so the prices go up, the yields go down.

For ordinary transactions, the “stock effect” is typically responsible for about two thirds of total impact (see, e.g., Figure 12). Central banks, though, are not ordinary market participants. But there are hints that their impact characteristics may not be so exceptional. Payne and Vitale studied FX interventions by the SNB. Their measurements show that the SNB’s price impact was a concave function of intervention size (Figure 2). The impact of SNB trades also appears to have partially reverted within 15-30 minutes, perhaps by about one third (Figures 1 and 2, Table 2). Though, unlike QE, these interventions were sterilised, so longer-term there shouldn’t have been much of a “stock effect” — and other participants may have known that.

[4] We can assume without loss of generality that the traders in question are buying (i.e. the meta-order sizes are positive). Sell meta-orders would have negative q, and the same arguments would apply, but with “\geq” replaced by “\leq“. Though, the meta-order size distribution for sell orders might not be symmetric to the distribution for buy orders (i.e. p[q] \neq p[-q]). Note that this model assumes that traders don’t submit sell orders when their intention is really to buy. There’s some debate over whether doing so would constitute market manipulation and I doubt it happens all that much, but that’s a discussion for another time.

[5] I’m being a little loose with words here. Say a meta-order in situation s has an already-executed quantity of q_{executed,s}, and the market-estimate of q_{executed,s} is \hat{q}_s. I.2) is not the same as saying that \mathbf{E}_{s \in S}[\hat{q}_s] = \mathbf{E}_{s \in S}[q_{executed,s}]. The market-estimate \hat{q}_s could be biased and I.2) might still hold. And I.2) could be wrong even if \hat{q}_s is unbiased.

[6] I’m being imprecise here. Intermediaries could differentiate some market situations from others, so we really should have: \mathcal{I}_{s_p}(q') = \mathbf{E}_{q}[\alpha_{S_p}|q \geq q'] = \int_{q'}^{\infty} \alpha_{S_p}(q)p[q|q \geq q']dq, where \alpha_{S_p} = \mathbf{E}_{s_p \in S_p}[\alpha_{s_p}(q)] is the average alpha for possible situations s_p given observable market conditions. E.g. average alpha increases when volatility doubles, and other traders know it — so they adjust their estimates of temporary impact accordingly. In this case, S_p is the set of meta-orders that could be sent when volatility is doubled. For this reason, and because impact is not the only cause of price fluctuations, the stronger “colored print” constraint wouldn’t eliminate empirically measured \mathbf{Var}_{s}[\mathcal{I}_{s}] — though it should dramatically reduce it.

[7] The draft presents some fascinating evidence in support of the colored print hypothesis. Using broker-tagged execution data from the LSE and an estimation method, the authors group trades into meta-orders. They then look at the marginal temporary impact of each successive child order from a given meta-order (call this meta-order M_{1}). In keeping with a concave impact-function, they find that M_{1}‘s child orders have lower impact if they’re sent later in M_{1}‘s execution. However, if another meta-order (M_{2}) is concurrently executing on the same side as M_{1}, M_{2}‘s child orders have nearly the same temporary impact, regardless of whether they occur early or late in the execution of M_{1} (p39-40). This means that “the market” is able to differentiate M_{1}‘s executions from M_{2}‘s!

I.2) might seem like a sensible approximation for real markets, but I’d have expected it to be pretty inaccurate when multiple large traders are simultaneously (and independently) active. There should be price movement and excess volume if two traders have bought a million shares each, but how could “the market” differentiate this situation from one where a single trader bought two million shares? It’s surprising, but the (draft) paper offers evidence that this differentiation happens. I don’t know what LSE market structure was like during the relevant period (2000-2002) — maybe it allowed information to leak — but it’s also possible that large meta-orders just aren’t very well camouflaged. A large trader’s orders might be poorly camouflaged, for example, if she has a favorite order size, or submits orders at regular time-intervals. In any case, if a meta-order is sufficiently large, its prints should effectively be “colored” — because it’s unlikely that an independent trading strategy would submit another meta-order of similar size at the same time.

A. Take a \frac{d}{dq} of I.1): \mathcal{I}(q)=q \alpha '(q)+\alpha (q)
B. Set A. equal to the definition of \mathcal{I}(q') in I.2): q' \alpha '(q')+\alpha (q')=\frac{\int_{q'}^{\infty } p[q] \alpha (q) \, dq}{1-P[q']}
C. Take a \frac{d}{dq'} of B.: q' \alpha ''(q')+2 \alpha '(q')=\frac{P'[q'] (\int_{q'}^{\infty } p[q] \alpha (q) \, dq)}{(1-P[q'])^2}-\frac{p[q'] \alpha (q')}{1-P[q']}
D. Plug B. into C. to eliminate the integral: q' \alpha ''(q')+2 \alpha '(q')=\frac{P'[q'] (q' \alpha '(q')+\alpha (q'))}{1-P[q']}-\frac{p[q'] \alpha (q')}{1-P[q']}
E. Use P'[q']=p[q']: \alpha '(q') (2-\frac{q' p(q')}{1-P(q')})+q' \alpha ''(q')=0
F. And for clarity, we can change variables from q' \rightarrow q, and divide by q (since we’re not interested in the ODE when q=0).

[9] There’s a helpful graphic on p20 of this presentation.

[10] This equivalence comes from ODE uniqueness and applies more generally than the model here. Latent liquidity models have a similar feature. In latent liquidity models, traders submit orders when the market approaches a price that appeals to them. In addition to their intuitive appeal, latent liquidity models predict square-root impact under a fairly wide variety of circumstances.

It’s helpful to visualize how price movements change the balance of buy and sell meta-orders. Let’s call N_{s}(q) the number of meta-orders, of size q, live in the market at a given situation s (a negative q indicates a sell meta-order). When supply and demand are in balance, we have \sum_{q=-\infty}^{\infty} qN_{s}(q) = 0 (buy volume equals sell volume).

Say a new meta-order of size q' enters the market and disrupts the equilibrium. This changes the price by \delta_{s}(q'), and morphs N_{s}(q) into a new function N_{s}(q, \delta_{s}(q')), with \sum_{q=-\infty}^{\infty} qN_{s}(q, \delta_{s}(q')) = -q'. I.e., a new buy meta-order will fully execute only if the right volume of new sell meta-orders appear and/or buy meta-orders disappear. Here is a stylized illustration:


Pre-impact (blue) and post-impact (orange) distributions of meta-order sizes live in the market, at an arbitrary situation s. Before a new buy meta-order (red) enters the market, the volume between buy and sell meta-orders is balanced. After the new meta-order begins trading, the distribution shifts to accommodate it. This shift is facilitated by a change in price, which incentivizes selling and disincentivizes buying.

By definition, \mathcal{I}(q) = \mathbf{E}_{s \in S}[\delta_{s}(q)], where the expectation is over all situations when a meta-order of size q might be submitted. Also by definition, N_{s}(q) — if we assume that meta-orders are i.i.d. (which would preclude correlated trading behavior like herding) — is the empirical distribution function of meta-order sizes. So N_{s}(q) and p[q] have the same shape if there are a large number of meta-orders live.

Donier, Bonart, Mastromatteo, and Bouchaud show that a broad class of latent liquidity models predict similar impact functions. Fitting their impact function to empirical data would give a latent liquidity model’s essential parameters, which describe the equilibrium (or “stationary”) p[q], as well as how it gets warped into p[q,\delta] when the price changes by \delta.

[11] From the ODE: \frac{p[q]}{1 - P[q]} = \frac{\alpha''(q)}{\alpha'(q)} + \frac{2}{q}. We can use I.1) to get \alpha(q) from \mathcal{I}(q), and thus find p[q] (for a continuous probability distribution, p[q] \propto \frac{p[q]}{1 - P[q]} e^{-\int\frac{p[q]}{1 - P[q]}dq}).

[12] That is, if a \neq 1 : \alpha(q) = \frac{d}{1-a}+K \exp \left(\int_0^q \frac{(1-a) p[q']}{1-P[q']} \, dq'\right). And in the case that a=1 : \alpha(q) = d \int_0^q \frac{p[q']}{P[q']-1} \, dq'+K.

[13] If fund managers knowingly let their AUM grow beyond the capacity of their strategies, then “overconfidence” might not be the right word. Then again, maybe it is. Clients presumably have confidence that their money managers will not overload their strategies.

A. Take a \frac{d}{dq} of the fair pricing condition I.1): \mathcal{I}(q)=q \alpha '(q)+\alpha (q)
B. Set equal to III.2): q' \alpha '\left(q'\right)+\alpha \left(q'\right)=\frac{\int_{q'-q_d}^{\infty } p[q] \alpha (q) \, dq}{1-P[q'-q_d]}
C. Take a \frac{d}{dq'} : q' \alpha ''\left(q'\right)+2 \alpha '\left(q'\right)=\frac{P'[q'-q_d] \left(\int_{q'-q_d}^{\infty } p[q] \alpha (q) \, dq\right)}{\left(1-P[q'-q_d]\right){}^2}-\frac{p[q'-q_d] \alpha \left(q'-q_d\right)}{1-P[q'-q_d]}
D. Substitute B. into C. to eliminate the integral: q' \alpha ''\left(q'\right)+2 \alpha '\left(q'\right)=\frac{\left(q' \alpha '\left(q'\right)+\alpha \left(q'\right)\right) P'[q'-q_d]}{1-P[q'-q_d]}-\frac{p[q'-q_d] \alpha \left(q'-q_d\right)}{1-P[q'-q_d]}
E. And use P'[q'-q_d]=p[q'-q_d] to get q \alpha ''(q) + \alpha '(q) \left(2-\frac{q p[q-q_d]}{1-P[q-q_d]}\right)-\left(\alpha (q)-\alpha (q-q_d)\right)\frac{p[q-q_d]}{1-P[q-q_d]}=0

[15] The solutions were generated with the following assumptions:

q \sim Pareto[q_{min}=10^{-7},\beta=\frac{3}{2}]
Initial conditions for q_d=0 : \alpha(q_{min})=10^{-5}, \alpha'(q_{min})=10^{3}
Initial conditions for q_d=10^{-5} : \alpha(q_{min})=1.1 \times 10^{-4}, \alpha'(q_{min})=10^{-2}
Initial conditions for q_d=10^{-2} : \alpha(q_{min})=1.1 \times 10^{-4}, \alpha'(q_{min})=10^{-3}
The q_d=0 solution was generated from the ODE of I.1).

[16] Here’s Robin Wigglesworth on one reason bank market-makers like trade reporting delays:

These days, bank traders are loath or unable to sit on big positions due to regulatory restrictions. Even if an asset manager is willing to offload his position to a dealer at a deep discount, the price they agree will swiftly go out to the entire market through Trace, hamstringing the trader’s ability to offload it quickly. [Emphasis added]

[17] I don’t know whether bond liquidity provision is sufficiently competitive, but it has notoriously high barriers to entry.

Even for exchange-traded products, subsidizing market-makers with an informational advantage requires great care. E.g., for products that are 1-tick wide with thick order books, it’s possible that market-makers monetize most of the benefit of delayed trade reporting. On these products, market-makers may submit small orders at strategic places in the queue to receive advance warning of large trades. Matt Hurd calls these orders “canaries.” If only a handful of HFTs use canaries, a large aggressor won’t receive meaningful size-improvement, but the HFTs will have a brief window where they can advantageously trade correlated products. To be clear, canaries don’t hurt the aggressor at all (unless she simultaneously and sloppily trades these correlated products), but they don’t help much either. Here’s a hypothetical example:

1. Canary orders make up 5% of the queue for S&P 500 futures (ES).
2. A fundamental trader sweeps ES, and the canaries give her a 5% larger fill.
3. The canary traders learn about the sweep before the broader market, and use that info to trade correlated products (e.g. FX, rates, energy, cash equities).

Most likely, the fundamental trader had no interest in trading those products, so she received 5% size-improvement for free. But, if more HFTs had been using canaries, their profits would’ve been lower and maybe she could’ve received 10% size-improvement. The question is whether the number of HFTs competing over these strategies is large enough to maximize the size-improvement for our fundamental trader. You could argue that 5% size-improvement is better than zero, but delaying public market data does have costs, such as reduced certainty and wider spreads.

[18] If q_d were intentionally changed by altering market structure, there’d probably be corresponding changes in the distribution of q and the initial conditions. These changes could counteract the anticipated effects.

[19] A more competition-friendly version might be for exchange latency-structure to allow canaries. But the loss of transparency from delaying market data may itself be anti-competitive. E.g., if ES immediately transmitted execution reports, and delayed market data by 10ms, then market-makers would only be able to quote competing products (like SPY) when they have canary orders live in ES. Requiring traders on competing venues to also trade on your venue doesn’t sound very competition-friendly.

A. Since \mathcal{I} is piecewise, split the fair pricing integral I.1) into the relevant two regions: \alpha(q_{tot}) = \frac{1}{q_{tot}} \left( \int_0^{q_{tot}-q_{FR}} \mathcal{I}(q_{tot},q_{executed}) dq_{executed} + \int_{q_{tot}-q_{FR}}^{q_{tot}} \mathcal{I}(q_{tot},q_{executed}) dq_{executed} \right)
B. Plugging in IV.2) to A.:
q_{tot}\alpha(q_{tot}) = \int_0^{q_{tot}-q_{FR}} \frac{\int_{q_{executed}+q_{FR}}^{\infty } p[q] \alpha (q) \, dq}{1-P[q_{executed}+q_{FR}]} \, dq_{executed}+q_{FR} \alpha(q_{tot})
C. Take a \frac{\partial}{\partial q_{tot}} : q_{FR} \alpha '(q_{tot})+\frac{\int_{q_{tot}}^{\infty } p[q] \alpha (q) \, dq}{1-P[q_{tot}]}=q_{tot} \alpha '(q_{tot})+\alpha (q_{tot})
D. Take another \frac{\partial}{\partial q_{tot}} : q_{FR} \alpha ''(q_{tot})+\frac{P'[q_{tot}] (\int_{q_{tot}}^{\infty } p[q] \alpha (q) \, dq)}{(1-P[q_{tot}])^2}-\frac{p[q_{tot}] \alpha(q_{tot})}{1-P[q_{tot}]}=q_{tot} \alpha ''(q_{tot})+2 \alpha '(q_{tot})
E. Subsitute C. into D. to eliminate the integral, and use P'[q_{tot}] = p[q_{tot}] : \alpha '\left(q_{\text{tot}}\right) \left(2-\frac{\left(q_{\text{tot}}-q_{\text{FR}}\right) p[q_{\text{tot}}]}{1-P[q_{\text{tot}}]}\right)+\left(q_{\text{tot}}-q_{\text{FR}}\right) \alpha ''\left(q_{\text{tot}}\right)=0

[21] The value of HFTs’ information will decay in a complex manner over the span of their predicted time period. An HFT might predict 30-second returns and submit orders within 100us of a change in its prediction. If that prediction maintained its value for the entire 30 seconds (becoming valueless at 31 seconds), then the HFT wouldn’t need to react so quickly. High-frequency traders, almost by definition, are characterized by having to compete for profit from their signals. From the instant they obtain their information, it starts decaying in value.

[22] Thanks to Mathematica.
q g(q,u)^2 \alpha ^{(2,1)}(q,u) = g(q,u) \left(g^{(1,0)}(q,u) \alpha ^{(0,1)}(q,u)+q g^{(1,0)}(q,u) \alpha ^{(1,1)}(q,u) + q g^{(1,1)}(q,u) \alpha ^{(1,0)}(q,u)+g^{(0,1)}(q,u) \left(2 \alpha ^{(1,0)}(q,u)+q \alpha ^{(2,0)}(q,u)\right)+g^{(1,1)}(q,u) \alpha (q,u)\right) - 2 g^{(0,1)}(q,u) g^{(1,0)}(q,u) \left(q \alpha ^{(1,0)}(q,u)+\alpha (q,u)\right)+g(q,u)^3 p(q,u) \alpha (q,u)-2 g(q,u)^2 \alpha ^{(1,1)}(q,u)

With g\left(q,u\right) = \frac{1}{1 - P[q,\infty] - P[\infty,u] + P[q,u]}

The procedure is to plug V.2) into V.1) and take 2 q partial derivatives and 1 u partial:

A. Inserting V.2) into V.1): \alpha \left(q,u'\right)=\frac{\int_0^q g\left(q',u'\right) \int_{u'}^{\infty} \left(\int_{q'}^{\infty} p(q'',u) \alpha (q'',u) \, dq''\right) \, du \, dq'}{q}
B. Take a \frac{\partial}{\partial q} : \alpha ^{(1,0)}\left(q,u'\right)=\frac{g\left(q,u'\right) \int_{u'}^{\infty} \left(\int_q^{\infty} p(q'',u) \alpha (q'',u) \, dq''\right) \, du}{q}-\frac{\int_0^q g\left(q',u'\right) \int_{u'}^{\infty} \left(\int_{q'}^{\infty} p(q'',u) \alpha (q'',u) \, dq''\right) \, du \, dq'}{q^2}
C. Substitute A. into B. to eliminate integrals where applicable: \alpha ^{(1,0)}\left(q,u'\right)=\frac{g\left(q,u'\right) \int_{u'}^{\infty} \left(\int_q^{\infty} p(q'',u) \alpha (q'',u) \, dq''\right) \, du}{q}-\frac{\alpha \left(q,u'\right)}{q}
D. Take another \frac{\partial}{\partial q} : \alpha ^{(2,0)}\left(q,u'\right)=\frac{g^{(1,0)}\left(q,u'\right) \int_{u'}^{\infty} \left(\int_q^{\infty} p(q'',u) \alpha (q'',u) \, dq''\right) \, du}{q}-\frac{g\left(q,u'\right) \int_{u'}^{\infty} \left(\int_q^{\infty} p(q'',u) \alpha (q'',u) \, dq''\right) \, du}{q^2}-\frac{g\left(q,u'\right) \int_{u'}^{\infty} p(q,u) \alpha (q,u) \, du}{q}+\frac{\alpha \left(q,u'\right)}{q^2}-\frac{\alpha ^{(1,0)}\left(q,u'\right)}{q}
E. Substitute C. into D to eliminate integrals where applicable: \alpha ^{(2,0)}\left(q,u'\right)=\frac{g^{(1,0)}\left(q,u'\right) \left(\alpha ^{(1,0)}\left(q,u'\right)+\frac{\alpha \left(q,u'\right)}{q}\right)}{g\left(q,u'\right)}-\frac{g\left(q,u'\right) \int_{u'}^{\infty} p(q,u) \alpha (q,u) \, du}{q}-\frac{2 \alpha ^{(1,0)}\left(q,u'\right)}{q}
F. Take a \frac{\partial}{\partial u'} : \alpha ^{(2,1)}\left(q,u'\right)=\frac{-g^{(0,1)}\left(q,u'\right) \left(\int_{u'}^{\infty} p(q,u) \alpha (q,u) \, du\right)}{q}+\frac{g^{(1,1)}\left(q,u'\right) \left(\alpha ^{(1,0)}\left(q,u'\right)+\frac{\alpha \left(q,u'\right)}{q}\right)}{g\left(q,u'\right)}+\frac{g^{(1,0)}\left(q,u'\right) \left(\frac{\alpha ^{(0,1)}\left(q,u'\right)}{q}+\alpha ^{(1,1)}\left(q,u'\right)\right)}{g\left(q,u'\right)}-\frac{g^{(0,1)}\left(q,u'\right) g^{(1,0)}\left(q,u'\right) \left(\alpha ^{(1,0)}\left(q,u'\right)+\frac{\alpha \left(q,u'\right)}{q}\right)}{g\left(q,u'\right)^2}+\frac{g\left(q,u'\right) p\left(q,u'\right) \alpha \left(q,u'\right)}{q}-\frac{2 \alpha ^{(1,1)}\left(q,u'\right)}{q}
G. To get the result, substitute E. into F. to eliminate integrals where applicable.

[23] I could be wrong, and it’s hard to define what “reasonable” solutions look like. But I checked this three ways:

1. I tried numerically solving for \alpha (and thus \mathcal{I}) assuming various joint probability distributions p[q,u] — where q and u are dependent and generated by functions of Weibull, Pareto, Log-Normal, or Stable random variables. I didn’t see any solutions where \alpha and \mathcal{I} had significant u -dependence without simultaneously having some other ridiculous feature (e.g. an infinity at small q).
2. I tried assuming \alpha(q,u) had a few reasonable forms (e.g. \alpha(q,u) \propto q^x u^{-y}) and solving numerically for p[q,u]. All the solutions I saw were not probability distributions (e.g. had negative probabilities).
3. It’s possible to solve the two integral equations directly if we assume that p and q are independent (p[q,u]=p_q[q]p_u[u]) and the solutions are separable (\alpha(q,u)=\alpha_q(q)\alpha_u(u) and \mathcal{I}(q,u)=\mathcal{I}_q(q)\mathcal{I}_u(u)). In this case, \mathcal{I}_q(q) and \alpha_q(q) obey the same ODE as the original time-independent system in part I. And \alpha_u(u)=\mathcal{I}_u(u)= constant, which isn’t realistic.