Price Discovery: How Information Reaches the Price

The previous lesson established that a trade carries information. Follow that idea to its conclusion and it becomes a theory of how prices get to be right.

Someone who knows an asset is undervalued buys it. Their buying pushes the price up. They keep buying while it remains cheap and stop when it does not. The price ends up near the value they knew about, and it got there without anyone publishing anything.

This is price discovery, and the important part is that trading is not merely how you act on information. It is the only channel through which private information becomes price. An analyst who is certain and never trades moves nothing.

The consequence is uncomfortable for anyone trading in size. If buying is how information enters the price, then your buying looks exactly like information entering the price, whether or not you have any.

The standard treatment is Albert Kyle's, in "Continuous Auctions and Insider Trading", Econometrica, volume 53, issue 6, 1985, pages 1315 to 1335.

The setup has three participants. A single risk-neutral insider knows the asset's eventual value. Noise traders submit orders for reasons unrelated to value. Competitive risk-neutral market makers see only the combined order flow, not who sent what, and set a price at which they expect to break even.

That last clause is the whole model. The market maker cannot separate the insider's order from the noise it is mixed into, so it must price the aggregate, treating every unit of net buying as partial evidence of good news.

The insider, knowing this, faces a trade-off. Buying more captures more of a known mispricing, and also moves the price further against themselves. They are not a price taker, and the model's content is how they resolve that.

The equilibrium has a strikingly simple shape. The price the market maker sets is the prior expectation plus a constant times the net order flow:

price  =  prior value  +  lambda * (net order flow)

Lambda is the price impact coefficient, and it is the single number that summarises how expensive a market is to trade in size. Its reciprocal, 1/lambda, is the market's depth: how much net buying it absorbs per unit of price movement.

Two comparative statics follow, and both match intuition once stated. Lambda rises with the amount of private information in the market, because each unit of flow is stronger evidence. Lambda falls as noise trading volume rises, because a given order is easier to mistake for noise.

So depth is not a property of the asset. It is a property of the mixture of participants trading it, and it can change within a day as that mixture changes.

The model's most quoted result concerns behaviour rather than pricing. The insider does not buy everything at once.

Dumping the full position immediately produces a huge order flow imbalance, which the market maker prices immediately, so most of the mispricing is destroyed before the position is built. Instead the insider trades gradually, in quantities small enough to be plausibly noise.

Kyle's term for what makes this work is camouflage: the noise traders' volume is what conceals the insider's activity, and the insider's optimal aggressiveness is calibrated to how much of it there is.

Taking the interval between auctions to zero gives a continuous-trading limit with three notable properties: prices follow Brownian motion, the depth of the market is constant over time, and all of the insider's private information is incorporated into prices by the end of trading.

That last property is the punchline. The information gets in regardless. What the insider controls is how much of its value they keep on the way.

Impact splits into two pieces with completely different remedies, and separating them is the most useful diagnostic a trader has.

Temporary impact is the cost of demanding immediacy. You consumed resting orders faster than they were replenished, so the price moved and then came back once the book refilled. Trading more slowly reduces it, because replenishment gets time to happen.

Permanent impact is the market's revised opinion. Your order flow was evidence, the market maker updated, and the price stayed moved. Trading more slowly does not remove it, because the same total quantity still reveals the same total evidence.

The practical test is to look at the price some interval after your order is finished. What reverted was liquidity. What did not is information, and it is yours whether you had any or not.

Lambda is not an abstraction; it is a regression coefficient you can compute for any instrument you trade.

def estimate_lambda(buckets):
    """buckets: (signed_volume, price_change) per interval.
    Signed volume is buys minus sells, classified by which side
    of the mid each trade printed on."""
    n = len(buckets)
    mean_q = sum(q for q, _ in buckets) / n
    mean_d = sum(d for _, d in buckets) / n
    cov = sum((q - mean_q) * (d - mean_d) for q, d in buckets)
    var = sum((q - mean_q) ** 2 for q, _ in buckets)
    return cov / var if var else float("nan")

# price change per unit of net buying, in price units per share
lam = estimate_lambda(five_minute_buckets)
cost_of_100k = lam * 100_000       # rough permanent impact of that size

Run it per instrument and the numbers vary by orders of magnitude, which is the quantitative form of "this one is liquid and that one is not".

Two cautions. Trade classification is imperfect, and errors bias lambda downward. And a linear model is a local approximation: the empirical relationship between size and impact is closer to a square root, which the lesson Costs, Capacity, and a Protocol You Can Trust develops.

Read the relationship in the other direction and it becomes a forecast rather than a cost.

If price change is a function of net order flow, then observing net order flow tells you something about the price change that is coming. Order flow imbalance, measured over short horizons from the book and the tape, is one of the most robust short-term predictors in equity markets, and much of high-frequency market making and short-horizon trading rests on it.

This is not a contradiction of market efficiency. The information is public and the horizons are short, so the predictability is available mainly to participants able to act within it, and acting on it is what removes it.

It also explains why hiding order flow is worth money. Techniques that fragment a large order across venues, sizes and time all exist to prevent exactly this inference. The counterpart is that detecting such fragmentation is itself a strategy, which is the arms race the next lesson is about.

Everything above collides in one decision: how fast to trade a position you have decided to take.

The first row is the cost of haste; the third is the cost of patience. Trading slowly means the price may move away from you for reasons entirely unrelated to your order, and that risk grows with the time you take.

There is therefore an interior optimum rather than a rule, and it depends on how urgent your information is. A signal that decays in minutes must be traded aggressively even at high impact. A valuation view good for months should be worked patiently.

Notice what the second row says: the permanent component is invariant to schedule. That is the part you cannot optimise away, only decide whether to incur.

Kyle's model is a idealisation and it is worth being precise about where it stops.

It assumes one insider. Real markets have many, competing with each other, which accelerates how fast information enters the price and reduces what any one of them keeps. It assumes noise trading volume is exogenous, when in practice uninformed participants withdraw exactly when adverse selection is worst. And its linear impact function is a local approximation.

What survives all of that is the structural claim, and it survives because it does not depend on the functional form: the market cannot distinguish your order from an informed one, so it must price all order flow as though it might be informed.

That single sentence explains the spread from the previous lesson, impact from this one, and the capacity ceiling on any strategy. It is also why execution is a discipline rather than a clerical step, and why market design turns out to be contested. Both are the subject of the final lesson.