Liquidity Has a Price: What Immediacy Costs Across Volatility Regimes, and Why Paper Alpha Must Pay It First

Every trade that demands immediacy pays for it. The payment is sometimes visible, as the spread between the price at which the market will buy and the price at which it will sell, and sometimes hidden inside the price itself, as the movement an order causes on its way to being filled. Both are the fee that patient liquidity providers charge impatient liquidity takers, and both rise when the market is under stress, which is precisely when a forecast is most likely to say that something must be done now. A strategy's paper alpha is what its forecasts would have earned had the market stood still while it traded. The market does not stand still.

This piece sets out a simple and explicitly stylized model of that fee, the square-root impact model, and uses it to ask three questions. How does the cost of immediacy change as volatility moves through its regimes? What happens to a strategy's alpha as the capital behind it grows and its orders become a larger share of the volume they trade against? And at what point should a predictive relationship be allowed to call itself alpha at all? Our answer to the last question is the one we hold most firmly: not until it has paid, first in the model and then in practice, for its own execution.

The Square-Root Law

The regularity we build on is well documented in the market-microstructure literature and simple enough to write in a line. The average price impact of an order, measured from the price prevailing when the order begins to the price at which it is completed, grows with the square root of the order's size relative to the volume that normally trades, and in proportion to the volatility of the security.^[1] In symbols, impact ≈ Y · σ · √(Q/V), where Q is the order, V is average daily volume, σ is daily volatility, and Y is a constant that empirical studies place somewhere in the vicinity of one. We use Y = 0.7 throughout and call the ratio Q/V the participation rate.

The square root is the important part. It says that the first slice of an order is the most expensive slice per share, and that doubling an order raises its total cost by a factor of about 2.8 rather than 4, since the cost per unit rises by √2 while the number of units doubles. It also says that the same order costs twice as much in a market that is twice as volatile. Figure 1 draws the model at three volatility levels chosen to stand for a calm market, a typical one, and a stressed one. The vertical axis is the impact cost in basis points of the order's value; the horizontal axis is the order's share of daily volume.

Two things stand out. The curves are steep near the origin and flat far from it, so an execution process gains more from keeping participation low than from any refinement once participation is high. And the gap between the calm and stressed curves is not a shift but a multiplication: at any participation rate the stressed market charges roughly four times what the calm one does for the same order, because volatility enters the model as a factor rather than an addend. A cost model estimated in calm conditions and applied in stressed ones does not merely miss; it misses by a multiple.

Immediacy Across Regimes

The square-root model treats volatility as an input. In practice volatility, the spread, the depth resting at the best quotes, and the volume that trades all move together, and not in the same direction. When volatility rises, liquidity providers widen their quotes because the inventory they take on is riskier to hold, and they post less of it, because a large resting order is a larger free option for anyone who knows something they do not. Volume, meanwhile, rises, because more participants have reasons to trade. The result is a market that trades more and holds less: the flow is heavier and the pool it flows through is shallower.

Figure 2 stylizes those relationships across four volatility regimes. The left panel shows the half-spread, the fee for crossing from bid to ask once, in basis points. The right panel indexes the depth at the best quotes and the volume traded to the calm regime. The specific bars are constructed from simple power laws stated in the note, but the qualitative shape, spreads widening as volatility rises and depth thinning faster than volume grows, is the one we expect anyone who watches order books to recognize.

Put the two panels together with Figure 1 and the cost of immediacy in a stressed regime has three sources rather than one. The spread is wider, so the visible fee is larger. The depth is thinner, so an order of a given size walks further through the book before it is filled. And the impact coefficient is larger, because volatility multiplies it, while the rise in volume offsets only part of that, since the volume gain enters under the square root. In a stylized case where volatility triples and volume doubles, the impact on the same dollar order roughly doubles; add the wider spread and the thinner book, and the total cost of doing the same thing can plausibly be several times what it was a week earlier. That is what it means to say that liquidity has a price and that the price is not constant.

This is why we treat cost estimation as a regime-conditional exercise rather than a fixed schedule. An average cost over a long sample blends calm and stressed periods into a number that describes neither. The forecast a strategy relies on may or may not depend on the regime; the fee it will pay to act on the forecast certainly does. The flash crash of May 2010 showed the extreme case, in which resting depth vanished and the cost of immediacy became effectively unbounded for a few minutes; March 2020 showed the slower version, in which spreads and impact stayed elevated for weeks while volumes were heavy. Neither is the regime a long-sample average describes.

Paper Alpha and the Cost of Scale

The second thing the square-root model does is describe what happens to a strategy as the capital behind it grows. A strategy's paper alpha, the return its forecasts would earn on a market that did not react to its trades, is a rate: so many basis points per unit of capital per year, more or less independent of how much capital there is. Its impact cost is not. Hold the strategy's turnover fixed and let the capital grow by a factor C; every order grows by C, its participation rate grows by C, and the impact per unit traded grows by √C. The cost per unit of capital therefore rises with the square root of scale, and the cost in dollars rises with scale to the power of one and a half.^[2]

Figure 3 follows a hypothetical strategy through that arithmetic. Capital is measured as a multiple of a reference scale at which the strategy's orders are one percent of the daily volume in the securities it trades, and at which, by construction, impact consumes a fifth of gross alpha. Gross alpha is flat. The impact cost rises with the square root of capital. Net alpha, the difference, falls, reaches zero at twenty-five times the reference scale, and goes negative beyond it: past that point the strategy still forecasts correctly and loses money anyway.

The point marked on the net curve matters more than the break-even. The dollars a strategy earns are capital times net alpha, and that product peaks well before net alpha reaches zero, at four-ninths of the break-even scale under this model. At that optimum, net alpha per unit of capital is exactly one-third of gross: two-thirds of the paper alpha has been handed to the market as the price of immediacy, and that is the best the strategy can do. A strategy run at break-even has given all of it away; one run at the optimum has given away most of it. Neither resembles the paper number, and neither should be described using it.^[3]

Figure 4 collects the ratios that fall out of the model. They are worth remembering because they hold whatever the constant Y turns out to be and whatever the calibration at the reference scale; they depend only on the square root.

A Ledger of Stylized Strategies

The argument so far applies to a single strategy. Its sharper form appears when strategies at different horizons are compared, because turnover and paper alpha both tend to rise as the holding horizon shortens, and turnover is the multiplier on cost. Figure 5 runs five stylized strategies, from an intraday reversal signal that turns its capital over daily to a fundamental signal that turns over twice a year, through the same cost model at a common reference scale. Paper alpha is given as an index, with the slowest strategy set to 100; the cost includes a fixed half-spread as well as square-root impact; the last column reports the scale at which tradeable alpha reaches zero.

Read across the rows and the pattern is familiar to anyone who has taken a backtest to an execution desk. On paper the fastest strategy is twenty-five times the slowest. After costs at the reference scale it is still ahead, but the gap has closed sharply, and its break-even scale is a small multiple of the reference, while the slow strategy's is so large that scale is not what limits it. Ranked by paper alpha the strategies run one way; ranked by the dollars they could earn at their own optimal scales, which grow with the break-even column, the order largely reverses. The fast strategy is not worse. It is a different asset: high alpha density, small capacity, and a cost of immediacy that grows with every dollar added.

The table also shows why the phrase “net of costs” is not a footnote to a research result but the result itself. A signal's information coefficient, its hit rate, its risk-adjusted return on paper: these describe the forecast. Only the tradeable column describes what the forecast is worth, and it depends on parameters, turnover, participation, regime, that the forecast itself says nothing about.^[4]

What This Means for How We Work

Three practices follow. The first is sequencing: the cost model runs before the alpha is graded, not after. A candidate signal enters our research process attached to an assumed turnover, an assumed participation rate, and a regime-conditional cost curve, and the number that is reviewed is the net one. A signal that looks strong on paper and weak net of costs is not a strong signal with an execution problem; it is a weak signal, and it is triaged as one.

The second is that scale is a parameter of the strategy, not a consequence of its success. Because net dollar alpha peaks well inside the break-even scale, every strategy carries an intended scale derived from its own cost curve, and adding capital past it is a decision to earn less. The third is that the cost model is itself a research object, re-estimated as regimes change and checked against realized executions, because a cost model that is wrong by a multiple in exactly the conditions that matter is worse than none: it produces confidence at the moment confidence is least warranted.

None of this lowers the price of liquidity. It is what the market charges for doing something now rather than later, and it is charged to everyone. What a systematic process can do is know the price before it agrees to pay it, size its ambitions to the price, and reserve the word alpha for what is left afterward.