I’m going to offer you a once in a lifetime opportunity. I’ve got this great product called a whizzbang and five years from now everybody is going to want it. It’s going to cost you a pretty penny to get your hands on one then—but what if I offered you the chance to buy it at a guaranteed cost of $22 five years from now? There’s no obligation, of course. If, five years from now, you can find a whizbang for cheaper than $22, then you can buy it at that price. But if the whizzbang costs $25 or even $50 you will have the ability to buy it for only $22.

Would you pay $5 today for such an option?

The question, of course, is central to general managers and agents negotiating team, player, or mutual options at the end of contracts. Options work essentially as I have outlined them above. Jason Giambi signed a long-term contract with the New York Yankees at the end of the 2001 season. The contract has an option for 2009 at the price of $22 million.

The Yankees believed at the time that they signed Giambi that there was a chance that his production would be worth more than $22 million. The option gives them the opportunity to lock in Giambi’s production at that price even if his 2008 “fair market value” (and I use the term “fair” loosely) is $25 or $50 million dollars. The Yankees will pay $5 million for this opportunity — in other words, if they decline the option, they have to pay Giambi the $5 million buyout price.

THT’s own David Gassko remarked recently,

A player or team will rarely exercise an option unless the player’s market value is much higher (if it’s a team option) than the option or much lower (if it’s a player option). So really, when either side agrees to an option that the other side can exercise, they’re saying, “Yeah, you can screw me, it’s alright.”

In practice, I think David’s pretty much on the mark. But is there any logic to be applied to pricing options?

Option pricing in finance can be pretty sophisticated, and many of the models cannot be applied to baseball contracts because the assumptions break down (for example, an individual player cannot be divided into many different securities). Even if some of the models can be applied to baseball contracts, I am certainly not clever enough or knowledgeable enough to do so. Still, I’m a process-oriented type, so let’s take a stab at this.

First, we need to settle on a the concept of value. Dave Studeman has developed a neat toy called Net Win Shares Value that will help us out here. In a nutshell, Net Win Shares Value tells us how much money a player is worth above or below his actual salary. For example, Jason Giambi was paid $18 million last year. In this free agent market, that much money can be expected to purchase 12 WSAB. He actually produced 14 WSAB, making his Net Win Shares value $1.5 million—the price it would have cost to purchase those additional two Win Shares *on the free agent market*. That’s an important distinction; in baseball, salary is heavily tied to service time. In this article I’m only going to talk about free agents, and assume that the free agent market is the only way to acquire players.

Options, like piranhas, are a tricky species. They are appended to the end of contracts, often long-term five- or six-year contracts, and so involve—or *should* involve—a fair amount of projection. Because there are several different projection systems (Marcel, ZiPS, PECOTA, not to mention the proprietary projection systems employed by many teams or agents), let’s work in the abstract and consider a probability curve p(NWSV). It might look something like this:

In this situation, there is a probability that the player has a positive or negative Net Win Shares Value, and if all the probabilities are added (that is, if p(NWSV) is integrated over all space) it should equal 100% (one). While I’ve shown it to be symmetrical around Net Win Shares Value = 0 and to have a bell-curve shape, there’s no reason why it should look like this. The picture above is just one of many possible shapes.

Now lets consider the possible outcomes when it comes time to make a decision on exercising the option:

1. The player is projected to have a positive Net Win Shares Value, so exercising the option is a no-brainer.

2. The player is projected to have negative Net Win Shares Value, but this value is greater than the buyout price. This is a tricky situation: The player is not really worth the money, but it would cost more to exercise the buyout and send the player packing. In this case, the buyout is generally referred to as a sunk cost and the option is exercised anyway.

3. The player is projected to have a negative Net Win Shares Value, and this value is less than the buyout price. The option is declined and the player becomes a free agent.

In other words, the option is exercised when the player’s Net Win Shares Value is greater than the option price minus buyout price, and the team “gains” money in the form of Net Win Shares Value. The option is declined when the Net Win Shares Value is less than the option price minus buyout price and the team loses money in the form of the buyout price.

We can put this into a picture and call the curve M(NWSV).

Note that the buyout price is maximum amount of money that can be lost (represented here by gaining a negative amount).

Each Net Win Shares Value has an associated monetary value M(NWSV) and probability p(NWSV); the sum of the monetary values, weighted by its probability (i.e., the integral of p(NWSV) x M(NWSV)), gives the expected monetary value. That might sound a little confusing, but the bottom line is that the expected monetary gain or loss can be determined if one a) can generate a probability curve describing the likelihood of any given Net Win Shares Value and b) knows the parameters of the option and buyout.

If the buyout is priced fairly, there ought not be a situation where the team or player is *sure* to make money. That is to say, the expected monetary gain or loss that I described above can be made to come out to zero by pricing the buyout appropriately. (Of course, if a party is clever, they’ll have a good projection system, determine the fair buyout price, and try to negotiate a more favorable value.)

I think that’s a pretty decent sketch of how an option ought to be priced. The problem is that it will be difficult to compare this to reality because of the difficulty in projecting performance several years down the line. Next time, I’ll consider some (very) simple models and try to hammer out a few general conclusions.

**References & Resources**

I would be remiss if I didn’t point out that my wife helped me a great deal in thinking about this situation. Also thanks to David Gassko for a very nice discussion on this topic.

A good elementary text on options is *An Introduction to Mathematical Finance* by Sheldon Ross. In fact, contained therein is my total knowledge (and then some) about finance.

One correction: I’m told that my description of Net Win Shares Value was slightly off. The next-to-last sentence in the descriptive paragraph should read… “He actually produced 14 WSAB, making his Net Win Shares value $1.5 million—the price it would have cost to purchase those additional two Win Shares *from the pool of all players (free agent or not)*.”

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