Kyle's lambda

The model in one line

In Kyle's model, published in Econometrica in 1985, market makers see only the total net order flow and set the price as a linear function of it: Δp = λ × net flow. Lambda is therefore the inverse of market depth. The same paper framed liquidity in three dimensions, tightness, depth and resiliency, and lambda is the depth dimension in number form.

Estimating it from bars

Over a window of bars, take x as the bar's net signed volume, its delta, and y as the change in the close. The regression through the origin gives λ = Σxy / Σx². A version with an intercept absorbs the window's drift. Estimating it bar by bar in this way follows Hasbrouck's 1991 approach. Alternatives replace the delta with its signed square root, reflecting the square-root law of impact, or with the difference between the number of buy and sell trades.

A worked example

Over 120 bars, the sum of delta times close change is 1,500 and the sum of squared deltas is 2,000,000, with prices in index points. Then λ = 1,500 / 2,000,000 = 0.00075 points per contract of net flow. Per 1,000 contracts of net buying, the close moved on average 0.75 points, three ticks on a contract with a 0.25 tick. If the same calculation an hour later gives 0.00150, net flow is moving price twice as much: the book has become thinner or the flow more informative.

In Senzoukria

Kyle Lambda (price impact) is a pane indicator in the Quantitative group. Its 'Window (bars)' is 120 by default, from 10 to 2,000. 'Order flow' chooses 'Net signed volume (delta)', the default, 'Signed square root of delta' or 'Trade count imbalance'. 'Fit an intercept' is off by default, since in the model zero net flow has no reason to move price. 'Unit' is 'Ticks per 1 000 units of flow' by default, or price per unit of flow, or basis points per 1,000 units. A window with missing data, or without any net flow, produces no value rather than a zero. The indicator's own description states its limits: it is an average contemporaneous impact that does not separate permanent from temporary impact, does not predict the cost of the next order, depends on the quality of the source's aggressor classification and changes with the timeframe.

Common mistakes

• Comparing lambdas across timeframes or instruments without converting units.
• Reading a rising lambda as a directional signal; it describes how sensitive price is to flow, not which way flow will go.
• Trusting lambda computed on a feed whose aggressor side is inferred rather than reported.