Is there one statistical measure that can determine with finality the value of each player? Something that can be used to compare Kevin Garnett to Bob Cousy, or Jason Kidd to Kenyon Martin? Many attempts have been, usually a linear combination of various officially reported statistics, sometimes on a per-minute or per-possession basis. I have looked in depth previously at net team points, and its many variants. Today, I present the principles of what may be the ultimate foundation for evaluating player performance: spread options. Itís not just one statistic; itís a whole new field of study.

Spread Option Valuation and Basketball

                                         
Philip Maymin
Basketball News Services  

Is there one statistical measure that can determine with finality the value of each player? Something that can be used to compare Kevin Garnett to Bob Cousy, or Jason Kidd to Kenyon Martin. Many attempts have been, usually a linear combination of various officially reported statistics, sometimes on a per-minute or per-possession basis. I have looked in depth previously at net team points, and various off-shoots. Today, I present the principles of what will be the ultimate foundation for evaluating player performance. Itís not just one statistic; itís a whole new field of study.

Like physicists searching for a grand unified theory of everything that would combine relativity and quantum mechanics, we can first list the properties we would like our ultimate statistic to have. The ultimate statistical theory of basketball:

         Should reflect the difference between actual performance and expected performance of a fill-in player. Note that it should not reflect the difference between the actual performance of a player and the expected performance of that player. Otherwise, consistency is punished. Nor should it reflect the difference between the actual performance of a particular player and the actual performance of another particular player, because then you are missing two expectations, and also encouraging teammates to cheer against their replacements, causing unnecessary team friction.

         Should NOT differentiate between teammates who share the exact same minutes. If two players are on the court at the exact same times, and one of them has a vacuous stat line while the other has lit up the scoreboard, it could be the case that the vacuous one drew so much attention, or set such great picks, that without him, the other guy wouldnít have accomplished anything. Basketball is by definition a team game, and though it is often clear by watching a game who is the greatest player, it is not always, and we are looking for an objective measure of worth.

         Should reward only the net gains of the team, offense minus defense, in evaluating players. A player who leaks like a sieve on defense but scores consistently on offense may be a worse player than someone who doesnít score a lot, but can shut down the entire opposing team, e.g. Bill Russell.

         Should be based on some kind of per-minute or per-possession evaluation to partially alleviate the potential bias created by coachesí decisions on playing time.

         Should weigh more important shots more heavily. A game-winning bucket in the final seconds is of far greater importance than one in the middle of the second quarter, even during the seventh game of the Finals. There is more attention by both teams on the final possession if it can decide the outcome of a game, and the best players are, or should be, on the court.

The concept Iím about to introduce is easier to understand from the background of net team points, which is just the amount by which a playerís team outscored the opposition during the time the player was on the court. This is sometimes called plus-minus, but because plus-minus can also refer to a variant of this in which the net score earned by the team during the playerís time on the bench is subtracted, I prefer to refer to it more clearly as net team points. As an aside, the plus-minus variant where the performance of a playerís replacement is subtracted is terrible for team morale, as it encourages players to hope for their replacements to play poorly. Letís just hope the Dallas Mavericks do not have such a statistic in the contract for their players; it would destroy the team.

Net team points solves virtually all of the bullet points above, but it misses the final two. Net team points per minute or per possession solves all but one, but it is an important one: net team points (or even net team points per minute) rewards garbage time equally with important crunch time performance. The approach Iím going to outline now solves those problems.

The approach I propose as a new foundation for evaluating players is to calculate the option value they provide. Two examples will clarify. Suppose youíre a team that consistently allows your opponent, any opponent, 80 points a game. Your own scoring is more sporadic, sometimes getting only 60, sometimes as many as 100. Then you are essentially long a call option on your offense; that is to say, you win games if you score more than 80, much like the holder of a call option on stock struck at $80 makes money if the stock rises above $80. Standard Black-Scholes option valuation means that in this situation, you would like your volatility to increase. In other words, you want to play your volatile players, your streaky players, the ones that sometimes score 30 and sometimes score nothing. Why? For the same reason that if youíre long a call option on a stock, you want that stock to be bouncy: in most cases, higher volatility increases the probability that it ends up above the strike price.

Alternatively, suppose you are a team that consistently scores 90 points, regardless who guards you or what defense they play. On the defensive end, however, you are sometimes an unstoppable force and sometimes a sieve. Then in this case you are essentially short a call option on your defense; that is to say, you win games if you keep your opponents under 90, much like the seller of a call option on stock struck at $90 makes money if the stock stays below $90. Standard option valuation means that in this situation, you would like the volatility of your defense to decrease. In other words, you want to play your safe and consistent defensive players. Why? For the same reason that if youíre short a call option on a stock, you want that stock to be less bouncy: it most cases, lower volatility decreases the probability that it ends up above the strike price.

Putting these two concepts together is the key. Essentially, the game of basketball is a package of options, one long and the other short, where they in general are looking to have risky offense and conservative defense. There is a particular type of option that describes this specifically, called a spread option.

A spread option pays off the difference between two risky assets. The initial strike price on the spread option is zero, because the score starts out zero-zero. You could express the value of a spread option, if you want to, in terms of the (risk-neutral) probability of your winning the game, i.e., in terms of the probability of your offense scoring more points than your defense allows. Then each point scored can be expressed in terms of the change to this implied probability of winning the game.

This means that a game-winning shot in the final seconds can change the implied probability of winning from 0% to 100%, the largest possible increase. Meanwhile, scoring the first point of the game on a free throw may change the probability of winning from 50% to 51%. In other words, the crunch time points end up counting more, which is one of the desired features of a panacea statistic. Finally, we can look at the change in implied winning probability on a per-minute of playing time basis for each player.

In the very near future, I will be presenting the results for the regular season so far, at which players tend to participate in probability-enhancing runs.