NFL Scoring Margin Distributions

This is a follow up from my previous post on building an in-game cover probability model for the NFL.  As a first step in that process, I calculated an expected scoring distribution for any given point spread.  I derived this by building an ordered logistic model off of actual NFL scores, with the Vegas point spread as my independent variable.  In that original post, I shared the expected distribution for a 7 point favorite.  The tool below will provide the distribution for any point spread (from 0 points to 21 points).

The distributions below are self-consistent with the spread, meaning that the median of each distribution is the point spread.  Use the dropdown to see the distribution for any point spread (from 0 to 21 points).

Buying Points

Although the main purpose of building these distributions was for the cover probability model, they could also be used to come up with "fair" prices for what is known as "buying points".  Some sports books will allow you to buy points when betting against the spread.  What this means is that you can tilt the spread further in your favor, for a price.

For example, let's say you are betting on a team that is a three point favorite.  The standard option would be a bet of $110 that would pay you $100 (in addition to your original $110 stake) if the favored team won by more than three points.  If the team won by less than three points (or lost outright), the sports book keeps your $110.  If the team won by exactly three points, the sports book returns your $110 to you (a "push").

If you buy a half point in this situation, you're changing the point spread to 2.5 points. What would be a fair price for this option?  For this calculation, I am assuming that a fair price would have the same (negative) ROI as the original bet on the three point spread.

The ROI for the original bet is -4.1% (45.4% chance of not covering, 9.3% chance of a push, 45.4% chance of covering).  Buying a half point changes things to a 45.4% chance of not covering and a 54.6% chance of covering (these numbers are pulled from the distributions above).  For the same negative ROI of -4.1%, the sports book should charge you $132.50 for a chance to make a $100 profit if the team covers a 2.5 point spread.

The value of $132.50 seems consistent with two existing calculations that are available online.  SBRForums's calculator puts the fair price at $135.30 and the Wizard of Odds chart puts the price at $130 (if I'm interpreting the chart correctly).

I plan on sharing the full table of fair prices in a separate post.
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