2.3.1 Execution Algorithms from Zero

Basic Prerequisites: Market Microstructure & Order Types.

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1. Intuition & Practical Objective

An institutional fund rarely wants to “buy stock” the way you buy a lunch. It wants to buy $1,000,000$ shares - a parent order - and the market’s depth at any moment is only a few thousand shares. This page builds why that single discrepancy forces the entire machinery of execution algorithms, with no prior knowledge needed. The objective is one idea: because a parent order is larger than the market’s instantaneous capacity, someone must decide how and when to break it into tradeable pieces - and that decision has a measurable cost.

Three truths, three “aha”s:

1. A parent order is not a trade; it is a sociotechnical event. If you dump it at once, you sweep the book, walk the price up through every resting limit order, and pay for it. Walk the price up = market impact, paid dollar-for-dollar.
2. The only reason to slice at all is that impact is superlinear in size. Halving the order does not halve the cost - it reduces it by more than half. That is why “go fast = expensive” is not a linear statement; it is the whole economic justification for slicing (the temporary-impact term of Almgren–Chriss).
3. Slicing has a price of its own: time. While you wait to cut impact, the market drifts. So execution is globally a trade-off between impact (go fast risks it) and timing risk (go slow risks drift) - and every algorithm is a different answer to that trade-off.

The one-sentence essence. “You cannot trade a parent order as one thing; you must chop it, and chopping trades impact against timing risk - the schedule and the benchmark are the two dials that define the algorithm’s character.”

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2. Mathematical Ground Truth & Derivations

The impact non-linearity (Almgren–Chriss temporary impact). In the AC model the temporary impact paid on a block traded at rate $v$ is linear:

$$h(v)=\epsilon +\frac{\eta }{\tau }v\text{(per share)},$$

so the total cost of trading $n$ shares in one interval is $n\cdot h(n/\tau )$ - and the key fact is that this grows quadratically in the slice size. The temporary-impact contribution to expected cost is

$$\frac{\tilde{\eta }}{\tau }\sum _{t=1}^N{n}_t^2,\text{so if you split into }N\text{ equal slices, it becomes }\frac{\tilde{\eta }}{\tau }\frac{X^2}{N}.$$

Splitting into $N$ equal parts divides the temporary-impact term by $N$ - not by $\sqrt{N}$, exactly by $N$. That is the cleanest number in the field: why slicing is not optional.

What slicing costs you instead. The risk (variance of shortfall) is

$$V={\sigma }^2\sum _{t=1}^N\tau x_t^2,$$

which shrinks as you trade faster (lower $x_t$ inventory) - the exact reverse of impact. Setting these two against each other is the Almgren–Chriss frontier (02 · The Execution Problem). For this page the message is structural: impact wants you slow and even; risk wants you fast and done; the schedule is your position on that line.

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3. Computational Implementation - sweeping vs. slicing

Buy $X=10^6$ shares; compare the temporary-impact cost of dumping the whole parent in one slice against TWAP slicing over 50 buckets. Stdlib only; reproduced exactly.

Read the number. The fixed $\epsilon X=$ $20,000 cost is identical in both - it is schedule-independent. The rest is the quadratic term, and it collapses by a factor of $50$ because the temporary-impact cost scales as $\sum n_t^2$. Sweeping a 5-day parent order in one shot would cost the desk ~$2500$ bps of pure impact before any price drift; slicing brings it under $100$ bps. This is why the practice exists.

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4. Failure Modes & First-Principles Breakdowns

1. “Slicing always helps” is false - it always helps impact but loads risk. The intermediate slices you didn’t have to trade (had you gone all-at-once) sit exposed to $\sigma \sqrt{\tau }$ drift each period. The sweep’s $\frac{1}{2}\gamma X^2$ permanent impact in AC is schedule-independent, but its temporary and risk terms are not; there is no free lunch, only a dial.
2. The fixed-cost trap. If $\epsilon$ (half-spread + fees) dominates, extra slicing adds more fixed costs than it saves in impact - the optimum is then not “slice as much as possible.” This is precisely why real desks solve for $N^{\ast }$, not $N=\infty$.
3. Impact is not the only cost. A slow, gentle TWAP has the lowest impact and can be the most expensive execution if the market trends against you - which is the seed of adverse selection addressed in 05 · Failure Modes.

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5. References

• Almgren, Robert & Chriss, Neil - “Optimal execution of portfolio transactions,” Journal of Risk 3(2), 5–40 (2000). Where the $\sum n_t^2$ temporary-impact term and the sweep-vs-slice math come from.
• Hasbrouck, Joel - Empirical Market Microstructure (2007)
• Kissell, Glantz & Malamut - Optimal Trading Strategies (2003)

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6. Connected Graph Bridges

• Base: Market Microstructure & Order Types · Market Impact & Depth
• Continue: 02 · TWAP-VWAP-POV · Index Hub
• Deep dive: Optimal Execution · The Execution Problem