Pricing 2-Leg SGPs with the r-Adjustment Model

Sometimes you don't have enough paired game history to use conditional probability. Maybe it's a new player pairing, a role change mid-season, or you're analyzing a trade deadline acquisition.

For these situations, you need the r-adjustment model—a way to price SGPs using market probabilities plus your own correlation assumption.

When to Use This Method

The r-Adjustment Formula

When you don't have reliable paired samples, use this correlation adjustment on the binary legs:

Where:

• pA = Probability that Leg A hits (de-vigged)
• pB = Probability that Leg B hits (de-vigged)
• r = Your assumed correlation coefficient

r is the knob that moves you from independence (r = 0) to stronger positive or negative dependence.

Understanding the Formula Components

Let's break down what each part does:

Independence Baseline

pA × pB

This is what you'd get if the legs were completely independent.

Correlation Adjustment

r × √(pA(1-pA) × pB(1-pB))

This adjusts for dependence:

• r > 0: Adds probability (positive correlation)
• r = 0: No adjustment (independence)
• r < 0: Subtracts probability (negative correlation)

The square root term is the maximum possible adjustment given the marginal probabilities.

Step-by-Step Worked Example

Let's price the Williams-Loveland SGP using the r-adjustment method.

Given Information

Step 1: Calculate the Independence Baseline

pA × pB = 0.478 × 0.345 = 0.165 (16.5%)

Fair odds at independence: +506

Step 2: Calculate the Maximum Adjustment Term

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

Step 3: Price with Different r Values

Step 4: Compare to Market

The market offers +305, which corresponds to roughly r ≈ +0.35.

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

• If you believe r > 0.35: The SGP may be +EV
• If you believe r < 0.35: The SGP is -EV
• If you believe r ≈ 0.35: The SGP is fairly priced

Choosing Your r Value

Here's a practical guide for estimating r when you don't have data:

Positive Correlation Scenarios

Negative Correlation Scenarios

Near-Zero Correlation

The Role Change Problem

SGPs are most dangerous when a player's role is changing. Both the marginal hit rate AND the dependence structure can shift.

Conditioning on Usage State

One practical approach is to condition on a usage indicator. Define:

• T = High-usage game (e.g., 7+ targets)
• A = Leg A hits
• B = Leg B hits

Then use the law of total probability:

Example: Loveland's Expanding Role

From the Williams-Loveland data:

• Among A-games (10 games where Williams hit 2+ TDs):
  • High targets (T) occurred in 4 games → P(T|A) = 0.40
  • Within A & T: Loveland scored in 3 of 4 → P(B|A,T) = 0.75
  • Within A & not-T: Loveland scored in 2 of 6 → P(B|A,¬T) = 0.333

With full-sample P(T|A) = 0.40:

P(B|A) = 0.75 × 0.40 + 0.333 × 0.60 = 0.50

But if you believe Loveland's role has grown and P(T|A) is now 0.55:

P(B|A) = 0.75 × 0.55 + 0.333 × 0.45 = 0.562

Joint probability: 0.588 × 0.562 = 0.330 (fair ≈ +203)

SGP Pricing Sensitivity Table

This table shows how different r assumptions affect fair price for the Williams-Loveland SGP:

Market = +305 → Book implied r ≈ +0.35

Practice Exercise

Key Takeaways

1. r-adjustment formula: P(A ∩ B) = pA × pB + r × √(pA(1-pA) × pB(1-pB))
2. r is your lever — adjust based on the relationship between legs
3. Independence (r = 0) is usually wrong for same-game props
4. Compare your r to the book's implied r to find value
5. Role changes require adjustments — condition on usage states when relevant