2.5.3 SOR Logic

Basic Prerequisites: 02 - Fragmentation & the NBBO and 02 - Order Types.

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1. Intuition & Practical Objective

A smart order router is, at its core, an optimizer with a very specific objective: fill the parent order at the lowest possible all-in effective price. This page builds that objective from first principles and implements the routing engine that minimizes it.

The router’s decision hierarchy - the order in which the terms matter - is:

1. Best price. Among venues showing size, prefer the lowest ask (highest bid). This is the dominant term: a half-cent of price beats a fraction-of-a-cent fee almost always.
2. Then size. The best price usually has little size; fill what is shown, then move to the next-best price. The allocation is water-filling down the merged book.
3. Then fees. Replace the raw quote with the all-in price $p_i+f_i$. A venue that displays the best price but charges the most is a trap - the router must rank on the all-in number.
4. Then latency and toxicity. Add the expected adverse-move penalty $\sigma \sqrt{L_i}$ for arrival delay (page 06) and a toxicity penalty for meeting informed flow.

The one-sentence essence. “SOR minimizes the total all-in cost ${\sum }_i{q}_i(p_i+f_i+\varphi (L_i)+k{\theta }_i)$ over how much to send to each venue, which is a linear program solved by greedily filling the merged, effective-price-sorted book.”

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2. Mathematical Ground Truth & Derivations

The routing objective. For a marketable parent order of $Q$ shares, choose allocations $q_i\ge 0$ to

$$\boxed{\underset{\{q_i\}}{\min }\sum _i{q}_i{\hat{p}}_i,{\hat{p}}_i=p_i+f_i+\varphi (L_i)+k{\theta }_i,\text{s.t.}\sum _i{q}_i=Q,q_i\le S_i.}$$

The objective is linear and separable in the $q_i$, and the constraints are a simple capacity-plus-equality system. The solution is the greedy rule:

Sort venues by ${\hat{p}}_i$ ascending; set $q_i=S_i$ for the cheapest, then the next, until the residual $Q-\sum q_i$ is absorbed.

Optimality. The objective is a linear functional over a transportation polytope; a basic feasible solution assigns each unit of quantity to the currently-cheapest available capacity, which is exactly the greedy rule. No LP solver is needed - sorting suffices.

Two useful corollaries.

• Fee-aware reordering. The greedy order on ${\hat{p}}_i=p_i+f_i$ differs from the order on $p_i$ whenever fees are large relative to price gaps. The router that sorts on $p_i$ alone is fee-blind and leaves money on the table.
• Phase transition on the fee spread. Venues $i,j$ swap rank when $p_i+f_i=p_j+f_j$, i.e. $f_i-f_j=p_j-p_i$: the fee gap must exceed the price gap to reorder. This is the exact threshold at which a fee-blind router starts misrouting.

Size creates a second-order term via convexity. When $S_i$ is small, the router is forced to venues with worse ${\hat{p}}_i$, so the realized average slides up the merged book. The average all-in cost is the $\hat{p}$-weighted mean of the swept levels; adding the next level raises it (piecewise-linear convexity from page 01).

Cum-fee price is the right ranking variable. Because a taker pays the take fee regardless of venue, the correct ranking statistic is the cum-fee price $p_i+f_i^{\text{take}}$, not the displayed spread (page 04 develops why make/take splits change the displayed spread without changing this). The router must always rank on the number the taker actually pays.

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3. Computational Implementation - a fee-aware SOR engine

Route a 1,000-share buy across fragmented venues by (a) raw quoted price and (b) all-in effective price, and measure the gap. Standard library only; re-executed and reproduced.

Read the numbers. Venue C displays the cheapest quote (99.995) but charges a 2-cent take fee, giving an all-in price of 100.0150 - last among the four on the number that matters. The fee-blind router loads 400 shares onto C first and 600 onto D; the fee-aware SOR loads 800 onto D (all-in 100.0025) and only 200 onto B. Same shares, same prices on the screen - but 0.45 bps apart all-in ($4.50 on 1,000 shares). Scale that to a desk routing millions of shares a day and fee-blind routing is a standing tax.

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4. Failure Modes & First-Principles Breakdowns

1. Ranking on the wrong variable. Sorting on displayed price instead of ${\hat{p}}_i=p_i+f_i$ is the canonical SOR bug. At 0.45 bps per order it looks trivial in one trade and compounds into a measurable P&L drag across a book (see 04 · Fees).
2. Ignoring capacity. A greedy router that forgets $q_i\le S_i$ will “route” more than the venue shows, then be forced into the residual at whatever price is left. Size must bound each leg.
3. Static thresholds. Fees, latencies, and venue availability change intraday; a router with hard-coded venue rankings (rather than re-sorting on live ${\hat{p}}_i$) misroutes exactly when conditions move. The reorder condition $f_i-f_j=p_j-p_i$ is the trigger to re-evaluate.
4. Missing legs. If latency or toxicity is not in ${\hat{p}}_i$, the router prefers a venue that looks cheap but whose fills are systematically worse - the subject of pages 05 and 06.

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5. References

• Johnson, Barry - Algorithmic Trading & DMA (2010). The most concrete practitioner treatment of smart order types, order lifecycle, and routing mechanics.
• Foucault, Pagano & Röell - Market Liquidity (2013)
• Colliard, Jean-Edouard & Foucault, Thierry - “Trading fees and efficiency in limit order markets,” RFS 25(11), 2012.
• Hasbrouck, Joel - Empirical Market Microstructure (2007)

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6. Connected Graph Bridges

• Back: 02 · Fragmentation & the NBBO · Index Hub
• Continue: 04 · Fees & Venue Selection
• Rebates & maker economics: Market-Maker Economics & Rebates
• Sibling execution: Execution Algorithms: VWAP, TWAP, POV