Reverse-Engineering Book Implied Correlation

SGP prices embed dependence. If you know the marginal probabilities and observe the SGP implied probability, you can solve for the book's implied r—the correlation assumption baked into the price.

This is the most powerful diagnostic tool for SGP betting.

The Implied r Formula

Where:

• p_mkt = SGP market implied probability
• pA, pB = De-vigged probabilities for each leg
• Result = The correlation the book is assuming

Why This Matters

Knowing the implied r answers the only question that matters:

How much dependence is already baked into the price?

Once you know this, you can compare it to:

1. Your own r assumption (from intuition or domain knowledge)
2. Historical data (from game logs)
3. Industry benchmarks (typical correlations for similar props)

Worked Example 1: Positive Dependence (Williams + Loveland)

The Setup

Step 1: Calculate Independence Baseline

pA × pB = 0.478 × 0.345 = 0.165

Step 2: Calculate the Denominator

√(pA(1-pA) × pB(1-pB)) = √(0.478 × 0.522 × 0.345 × 0.655)
                        = √(0.0564)
                        = 0.237

Step 3: Solve for Implied r

r_implied = (0.247 - 0.165) / 0.237
          = 0.082 / 0.237
          = +0.346

Interpretation

The book is pricing this SGP with r ≈ +0.35 (moderately strong positive correlation).

Your question: Is the true correlation higher or lower than 0.35?

• If you believe r > 0.35 → SGP may be +EV
• If you believe r < 0.35 → SGP is -EV
• If you believe r ≈ 0.35 → Fairly priced

Worked Example 2: Negative Dependence (Nacua + Stafford)

Sometimes SGPs combine legs that work against each other. Let's analyze an NFL example with negative correlation.

The Setup

Why This Might Be Negative

Think about the game script:

• High pass attempts often signal a negative game script (trailing, playing from behind)
• In those situations, Nacua is typically the primary target and gets fed the ball
• So when Stafford goes Over attempts, Nacua is more likely to go Over receptions too
• This makes Nacua Under + Stafford Over = conflicting outcomes

Calculate Implied r

Independence baseline:

pA × pB = 0.543 × 0.473 = 0.257

Denominator:

√(0.543 × 0.457 × 0.473 × 0.527) = √(0.0619) = 0.249

Implied r:

r_implied = (0.230 - 0.257) / 0.249
          = -0.027 / 0.249
          = -0.108

Validate with Actual Data

From 19 games of 2025 season data:

Empirical joint probability: 4/19 = 0.211

Using conditional probability: P(Nacua Under | Stafford Over) = 4/11 = 0.364

Actual r = (0.211 - 0.257) / 0.249 = -0.185

The Verdict

Worked Example 3: Same-Player SGP (Brandon Miller)

Same-player SGPs feel obvious—and books know it. Let's see how expensive they really are.

The Setup

The Intuition

This feels like strong positive correlation because:

• Made 3-pointers directly contribute to points (3 each!)
• If Miller hits 3+ threes, that's 9+ points already
• He only needs 12 more to hit Over 20.5

Calculate Implied r

Independence baseline:

pA × pB = 0.541 × 0.459 = 0.248

Denominator:

√(0.541 × 0.459 × 0.459 × 0.541) = √(0.0616) = 0.248

Implied r:

r_implied = (0.463 - 0.248) / 0.248
          = 0.215 / 0.248
          = +0.867

Interpretation

The book is pricing this with r = 0.867—near-perfect positive correlation!

When Same-Player SGPs Can Have Value

You need r > 0.867 to find value at +116. This could happen if:

• The player is in a usage-heavy role tonight
• Matchup strongly favors scoring efficiency
• Game pace projection is unusually high

But honestly? It's rare. The book knows what you know.

The Diagnostic Framework

Here's how to use implied r as a diagnostic tool:

Step 1: Calculate the Book's r

Use the formula to extract r_implied from any SGP price.

Step 2: Compare to Benchmarks

Step 3: Validate with Data

If possible, calculate the empirical r from game logs and compare to the book's assumption.

Step 4: Make Your Decision

• Your r > Book's r: Potential +EV
• Your r < Book's r: Likely -EV
• Your r ≈ Book's r: Fairly priced, pass

Practice Exercise

Key Takeaways

1. Implied r = (p_mkt - pA×pB) / √(pA(1-pA)×pB(1-pB))
2. This tells you what correlation the book is assuming
3. Compare to your own r estimate to find value (or avoid traps)
4. Same-player SGPs typically have very high implied r (0.50-0.85)
5. Negative dependence can create traps if book underestimates the negative correlation
6. Always validate with data when possible