Introducing z-points

“Without question, I’m going to trust my eyes more than the numbers. I never look at the stat sheet. I think stats have value, without question. But it makes me wary if they [the number-crunchers] put value in certain stats that I don’t think have value. A player understands this. I could average 17 points a game on a bad team, because in the last eight minutes of the game, the coach leaves me in the game to get my six or seven points against second-string or third-string guys. And when you’re on a bad team, guys know how to do that. They know how to get their numbers.
“I want to know what happens in winning time. What happens when guys are making plays to win the game, or when a team is making a 10-0 run? Who’s the guy who makes the play to make things happen? I know the look. I know the look that separates aggression from passiveness, dominance from lack of dominance over a person. You can’t get that on a stat sheet. Statistics should only validate what you think subjectively. It shouldn’t create your subjective view.”

Kenny Smith, former NBA player and NBA analyst
Houston Press. November 1, 2007

Well that’s about to change, Kenny!

z-points is a new way of measuring basketball points that takes into account the context in which they are scored. To do so, it considers two dimensions: 1) difference in the scoreboard and 2) time remaining in the game.

Intuitively, we know how these two factors come into play: points should be worth more when they are scored in close situations, and they should also be worth more at the end of games, when there is less time to change the outcome.

z-points accomplish that by using in basketball a parameter from modern financial theory: the delta of a binary option.

Binary options

An option gives you the right (but not the obligation) to buy or sell an asset at a certain price in the future. Let’s say you have an option that gives you the right to buy a stock at $15 in two months, and at maturity the price of the stock is $17. Then you will exercise your right and make a profit of $2.

Now, a binary option is a specific type of option. It is different because the payoff at maturity is either zero or a certain fixed amount (let’s say $1). For example, if 2 months from now the price of the stock in the example above is $15.01, the option will pay us $1 even though the stock price was only .01 above the strike price. If the stock price had been $20 we would still have received $1. But if the price of the stock in two months was anything below $15 (even if $14.99) then our payoff would be zero. So a binary option is a win-or-lose situation determined by a certain threshold.

This is very similar to what happens in a basketball game. Whether a team wins the game by 1 point or by 20 points, the payoff is exactly the same (a win). Likewise, losing by 1 or 20 will give you exactly the same payoff (a loss). In fact, a sports bet is just a binary option.

The delta of a binary option

The delta of a binary option is the change in the value of the binary option for a $1 change in the price of the underlying asset. Basically, what it does is to measure how the option changes in value as time expires and the stock price gets closer or further away from the strike price (remember, the strike price is the reference price that determines whether we will make money).

It turns out that the delta of a binary option reflects very well what we would expect to be the value of points in a basketball game, so it is a great way of valuing points in terms of competitive nature.

Points valued using this model (z-points) are more valuable:

1) the closer the score, and

2) the closer we are to the end of the game;

and less valuable:

1) the higher the difference in the score, and

2) the farther away to the end of the game.

It is the simultaneous combination of these two factors, scoreboard and time remaining, that makes the delta of the binary such a good model for basketball games.

As you can see, the curve is flatter and wider at the beginning of the game: points at this stage don’t have that much value. As the game unfolds, the curve gets steeper and narrower, meaning that, if the game is close, we give more value to points, but if it’s not, then points are worth less. The curve keeps getting steeper and steeper, narrower and narrower, and in the extreme situation when there are only a few seconds remaining, it will award max value to points scored to tie or win the game, but very little value as we separate from a competitive situation.

In short, z-points are just the result of assigning the value of the delta of the binary option to points scored in a game. The model considers that:

- The strike price of the binary option is reached whenever the game is TIED. So if the score is 95-97, then the leading team is +2 above the strike price and the trailing team is -2 below the strike price.
- Time to maturity for the option starts at 1 when the game begins and winds down to zero as we approach the end of the game.

Applying delta to other basketball stats

The binary delta can be applied to any other basketball stats: rebounds, assists, turnovers, steals… however, I will initially use it only for points and assists, that is, for anything that has a direct effect on the score of the game.

Make sure to check out our examples to get a better idea of how z-points work.

z-point stats are automatically calculated from play-by-play data and do not involve any subjective views or opinions.