Transaction cost and liquidity¶

Measure what each trade cost the taker, and where that cost went.

For a trade at price :math:p_t with aggressor side :math:D_t (+1 for a buy, -1 for a sell), against the mid-price :math:m_t prevailing when it printed and the mid-price :math:m_{t+\Delta} prevailing one horizon later:

• effective spread :math:= 2 D_t (p_t - m_t) — the round-trip cost of crossing, as the taker experienced it. Doubled so it is comparable with a quoted spread, which also spans both sides of the mid.
• realized spread :math:= 2 D_t (p_t - m_{t+\Delta}) — the part the liquidity provider kept, measured once the trade's information has had horizon to reach the price.
• price impact :math:= 2 D_t (m_{t+\Delta} - m_t) — the rest: how far the trade moved the market. The two parts add back to the effective spread exactly, row by row.

The horizon matters and there is no neutral choice. Too short and the price has not finished reacting, so impact is understated; too long and unrelated moves are counted as this trade's impact. Five minutes is the equity convention; the default here is one minute, because captures of a fast crypto tape are usually measured in minutes rather than hours. Report the horizon alongside the number.

A trade in the last horizon of the quote frame has no future mid to be measured against. Rather than reuse the final quote — which would report a shrinking horizon as though it were the full one — those rows get NaN realized spread and impact, and keep their effective spread.

:math:m_t is the mid of the last quote strictly before the trade. When the quote frame is a depth_summary built from the same event stream, the row stamped at the trade's own instant is the book after the trade took the touch, and measuring against it would charge the taker nothing for a move they caused. Venue timestamps are coarse, so a quote a few events older is the closest honest reference available. Crossed quotes are skipped for the same reason: a book whose best bid is above its best ask has no midpoint, so the last uncrossed quote is used instead.

Every number here inherits the quality of the book it is measured against. On a diff feed a trade can print through resting orders the venue never withdrew, which reads as a negative effective spread — the taker apparently paying less than the mid. That is a finding about the capture, not about the market: run :func:~ob_analytics.analytics.detect_stale_orders (or the audit command) before reading the costs as execution quality.

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Examples:

>>> from ob_analytics import Pipeline, sample_csv_path, transaction_costs
>>> result = Pipeline().run(sample_csv_path())
>>> costs = transaction_costs(result.trades, result.depth_summary, horizon="5s")
>>> costs["effective_spread_bps"].median()
0.1276...

Reduce a :func:transaction_costs frame to one volume-weighted figure each.

Each measure is averaged over the trades that actually carry it, weighted by trade size: the effective spread over every trade with a mid, the realized spread and the impact over the smaller set that also had a mid a horizon later. A measure with no usable trade is NaN rather than an error, so a summary of a capture shorter than its own horizon still returns and says so through :attr:~CostSummary.n_realized.

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Amihud (2002) illiquidity: price move per unit of turnover.

::

amihud = |return| / turnover

Read it as the price response a unit of trading buys. A market where a large turnover barely moves the price is deep, and scores low; one where a small turnover swings it is thin, and scores high. It needs no quotes and no aggressor side, so it is the one liquidity measure available on a tape that carries nothing but prices, sizes and times.

The return is taken within each window, from its first trade price to its last, and turnover is the price times size summed over the same trades. Amihud's original is a daily close-to-close return over that day's volume; a capture is not a series of trading days, and taking each window's own first and last print keeps every window usable — including the first, which a close-to-close definition cannot measure — and makes the whole-session case (window None) the same calculation over one window rather than a different one.

The result carries a unit: one over the turnover unit of the input, which on a pipeline frame is ticks times lots. It is comparable across windows of one run, and across runs of one instrument, but not across instruments without rescaling. Published figures are multiplied by a power of ten to reach readable digits; this returns the raw ratio and leaves that choice to the caller.

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Roll (1984): the spread implied by bid-ask bounce in the price series.

A tape with no quotes still shows the spread, because consecutive trades alternate between hitting the bid and lifting the ask, and that bounce makes successive price changes negatively correlated. Roll turns the size of that correlation back into a spread::

roll_spread = 2 * sqrt(-cov(dp_t, dp_{t-1}))

where dp is the change in trade price.

The model assumes an efficient price plus a bounce of constant half-spread, with order flow that carries no information. Under it the bounce is the only source of price change, so the lag-1 autocorrelation of the changes is exactly -0.5. That is the number to check: autocorrelation is returned beside the estimate, and the further it sits from -0.5, the less of the price movement the bounce explains.

Two things push it away. Real flow carries information, which raises the autocovariance and makes Roll read low against a measured effective spread. More decisively, the efficient price moves between trades, and on a sparse tape in a volatile instrument it moves much further than half a spread — the bounce then accounts for a few per cent of the variance and the autocovariance comes out positive, where the formula has no real root. The estimate is NaN there rather than a number the model does not support, and the two diagnostic columns say why. Sampling more often than the tape trades will not help: the fix is a denser tape or a wider spread, and where quotes exist :func:transaction_costs measures the spread directly instead of inferring it.

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