# CN-Arbitrage Spread — Methodology

> Index code: `CN-ARB-SPREAD`  
> Type: Spread / friction index  
> Frequency: Weekly  
> Status: Published v0.2

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## 1. Purpose

Measure quote dispersion for the **same model (compare group) across providers** — routing value, market friction, and information failure — not “absolute expensive or cheap.”

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## 2. Sample

Each week, for each `compare_group_id`:

- chat × token × `healthy` only
- Effective providers in group $k \ge 2$
- Blended price $p = 0.7\cdot\text{input}+0.3\cdot\text{output}$

Yields $\{p_{g,1},\ldots,p_{g,k}\}$.

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## 3. Within-group spreads

For group $g$:

$$
\begin{aligned}
s^{\text{range}}_g &= \max p_g - \min p_g \\
s^{\text{rel}}_g &= \frac{\max p_g - \min p_g}{\mathrm{median}(p_g)} \\
s^{\text{IQR}}_g &= Q_{0.75}(p_g) - Q_{0.25}(p_g)
\end{aligned}
$$

**Headline metric**: relative range $s^{\text{rel}}_g$ (dimensionless; comparable across models).

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## 4. Market level (primary unit: %)

Over eligible groups $G_t$ (prefer $|G_t|\ge 20$):

$$
S_t = \mathrm{median}_{g \in G_t}\big(s^{\text{rel}}_g\big)
$$

**Headline unit: percent** (publish $100\times S_t$, e.g. `34` = 34% median relative range). Absolute range $s^{\text{range}}$ may be an auxiliary field in USD / 1M tokens.

Optional parallels: mean relative range; open-only / closed-only $S_t$.

### Optional: index points

$$
I_t = 100 \times \frac{S_t}{S_{t_0}}, \quad t_0 = 2026\text{-}06\text{-}20
$$

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## 5. Interpretation

| Path of $S_t$ | Possible reading |
|-----------------|------------------|
| Rising | Wider same-model cross-cloud quotes → more routing / comparison value |
| Falling | Quotes converge → stronger competition or better information |
| Falls with Token Median | Deflation plus intensifying competition |
| Median falls while Spread rises | Some vendors lag cuts → wider arb window |

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## 6. Rebalance and revisions

No constituent rebalance; universe tracks the panel. Recompute if historical CSVs are revised.

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## 7. Limitations

- Region, SLA, rate limits, and residency are not stripped — some “spread” is product differentiation.
- Mixing OpenRouter with first-party can inflate dispersion; a “ex-aggregator” robust cut is optional.
- Not a settlement price for a tradable spread contract.

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## 8. Relation to other indices

Use with CN-Token Spot: Spot is the **level**; Arbitrage is the **routable gap**.