As you might have heard, this offseason is going to be a historic one for top-of-the-market free agents. Manny Machado and Bryce Harper are hitting the free agency market at preposterously young ages, and both of them are likely to secure contracts larger than any previously offered in baseball. It won’t just be the money, though. Machado and Harper are both going to get opt-outs in their contracts. Opt-outs, also known as player options, are all the rage in elite free agent contracts these days. As teams have shied away from exceeding the luxury tax thresholds, they have turned increasingly to player options as a way to give free agents something valuable without increasing their tax bill. The Cardinals will very likely include options on the contracts they offer to Harper and Machado this season, assuming they make a bid for both superstars.

It’s clear that player options have worth for players. If you had an option for a 5 year, 60 million dollar contract or the same contract plus the ability to opt out after three years, the second contract would be the clear choice. How much is it worth, though? Would you rather have 5/70 or 5/60 with an option? What about 6/80 or 5/60 with an option? Knowing how to value these contracts will be a key part of this offseason’s contract negotiations. In this two part series, I’m going to advance a framework for valuing player options. Part 1 will focus on a theoretical model for the extra money a player would need to receive in lieu of an option. Part 2 will focus on practical complications to the theoretical model described below.

**How to Think About Options**

To understand the value of an option, it’s necessary to think about volatility, about the uncertainty of future outcomes. It’s hard to think about volatility precisely because of its inherent unknowability. How many games are the Cardinals going to win next year? We can’t know, obviously. We can have an expectation for a median, and accept that there’s some uncertainty around it, but it’s still not natural to think of next season as a distribution of different outcomes. This distribution, though, is key to understanding the value of an option. Consider Steady Eddie, who knows with certainty that he will be offered exactly a 2 year, 40 million dollar contract in two years. He knows this for sure. For Eddie, there would be no difference between a 4 year, 80 million dollar contract with no player option and the same contract with a player option after two years. There’s no reason for him to exercise the option, no reason for him to need it. If a team gave him a choice of 4/80 with an option or 4/81 with no options, he’d just take the extra million dollars. Hey, a million bucks is a million bucks. Nothing wrong with that.

For an option to shine, we need a player more like Risky Robbie. In two years, Risky Robbie knows with complete certainty that he is going to get one of two contract offers: either a 2 year, 79 million dollar contract or a 2 year, 1 million dollar contract, each with 50% probability. He doesn’t know which of the two contracts he’ll be offered, though, because his skill level is either going to increase or decline precipitously. For Riskie Robbie, a guaranteed contract with an option to opt out after two years is incredibly valuable. With no option, he has an expected value of making 40 million dollars over those two years. By changing his two choices to either opting in and making $40 million over two years (in the scenario where he is offered the tiny contract), or opting out and making $79 million, he raises the amount he makes in expectation over those two years to nearly $60 million, which makes his player option worth something like $20 million. If a team offered Robbie a 4/90 contract with no options or 4/80 with an option after two years, he might take the one with the option. That’s the power of volatility. Even with the same expected value of contracts offered (both players figure to make $40 million on average), volatility changes the value of the option by about $20 million.

One interesting aspect of the player option is that it’s not exactly directional. Getting a player option isn’t betting on your upside, exactly. It’s also hedging your downside. For me, the most useful way to think of it is as a guarantee of at least a minimum contract. When a player signs a contract with a player option, what they’re really doing is making sure they’ll make no less than a certain amount in every year past the option. If you think of a player’s potential future earnings as a distribution, a contract with a player option just cuts out the lower part of that distribution. The team giving out the contract has agreed to accept paying an above-market rate if the player is worse than his option value, while seeing him leave if he’s better than his option value.

**Empirical Valuation**

To simplify things in this section, I’m going to make a series of assumptions. First, we’re going to assume that teams pay a fixed amount of dollars per win above replacement at every point in the future. For the sake of this exercise, we’ll call it 8 million dollars per win. Second, I’m ignoring aging curves for now. Older players decline over time, but exactly how much requires some complicated modeling. Instead, we’re going to focus on players who are expected to be as good next year as they are this year.

With those assumptions out of the way, thinking about a player’s value is pretty straightforward. Consider a player who was worth 3 WAR this year. He’s expected to be worth 3 WAR next year, the year after that, and so on. If you were a team willing to pay him market value for a four year contract, you’d offer him 4 years at $24 million a year (8mm/WAR), a 96 million dollar contract. Given that teams are always going to pay 8 million dollars for a win and that players never age in our example, knowing a player’s age and WAR this year is enough to offer him a contract.

Next, we’ll add a volatility component. To get an idea of how much players’ values change year to year, I took the WAR of every hitter who qualified for the batting title in 2016 and 2017. Next, I compared those numbers to their WAR in the subsequent year. As an example, Matt Carpenter recorded 3.1 WAR in 2017 and 5 in 2018, so his change in WAR was 1.9. This gave me the change in each player’s WAR year-over-year. I then took the standard deviation of these changes in wins, which gives us the annualized volatility of the average baseball player. The standard deviation was 1.9 wins. In other words, about 68% of players finished their second season between 1.9 wins worse and 1.9 wins better than their first season. A quick test of normality shows that the distribution of season-over-season change is pretty normal. If you’re wondering, this year Mookie Betts (+5.1) improved the most while Corey Seager (-5.4) declined the most.

Next, we’re going to take a quick detour into the world of finance. One of the great financial discoveries of the 20th century was that if you know the price of something as well as its volatility (plus some other less important nonsense), you can price the value of an option to buy or sell that asset. It’s known as the Black-Scholes Formula, and its three creators won a Nobel Prize for their efforts. Want a quick look at the formula? Here it is:

Yeesh. Let’s, uh, let’s not use that. The good news is, we can do an absolute ton of simplifying. The only real variables here are ‘underlying price,’ or the player’s current value in WAR, ‘strike price,’ or the dollar value of the option they’re being offered, ‘sigma,’ or the volatility of the player’s WAR, and ‘T-t,’ or time until exercise. We’ll also need to add a term for how many years the option covers- after all, the ability to lock in three years is more valuable than the ability to lock in one. Everything else is operations that can be done in Excel, or in this case Google Sheets.

I’ve built a publicly accessible spreadsheet for calculating the value of a player option here. I’ll provide a quick walkthrough below. Everyone is absolutely welcome to use the sheet, but to avoid too many people trying to change inputs at once, please save down a copy if you want to mess around with it. The calculator’s inputs look like this:

Let’s go through the components quickly:

**Contract $: **This is the amount of money the player option is for, per year. If a contract has a player option for 3 years at 32 million dollars a year, the cell will say 32, as above. The WAR equivalent is shown to the right.

**Fair Value Pay: **This is the player’s current skill level, at $8 million per win above replacement. In the above example, the player is a 5 win player at current, so his fair value pay is $40mm/year.

**Years Until Option: **How many years until the option will be exercised or declined. In the above example, the player has an opt out after three years.

**Years In Option: **The number of years the option covers. In the above example, there’s a three year contract, and then an opt out, and then another three years at $32mm a year.

When you plug numbers into this formula, you’ll get an output like so:

That’s what the player would be willing to pay for the option, but not obligation, to sign a 3-year, 32 million dollar a year contract in three years’ time. Let’s make this example concrete. This offseason, the Cardinals make a player an offer for 6 years at 32 million dollars a year, with an opt out after three years. What is that contract worth? Well, the first three years are locked in at 32 million dollars a year. Assuming the player is a 5-win player, that option on the last three years is worth 17.4 million dollars to him. Accordingly, you can think of the Cardinals’ contract offer as the equivalent of a 3 year, $96 million contract plus a payment of $17.4 million, or a 3-year, $113.4 million contract. They won’t end up paying him that, but in a risk-neutral world, the player would be indifferent between the six-year contract with an opt-out and the sweetened 3-year contract.

One quick note on the model: It’s much easier to think of player options as opt-ins, rather than opt-outs. If you look at a contract as all the guaranteed years, plus giving the player a choice of signing a second contract with value equal to whatever is left on the first one after the opt-out, it’s much easier. For example, the above contract could be called a 6 year, 192 million dollar contract with an opt-out after three years. I find it more convenient to think of that as a 3 year, $92 million contract plus the choice of either free agency or another 3 year, $92 million contract. Considering all the years after the opt-out as a choice against free agency, rather than an existing contract the player must choose to cancel, makes a little more sense in my head, and they function the same way mathematically.

**Problems and Conclusions**

I’ve presented above a simple model for valuing player options. It’s nothing near complete, but it does give a quick and dirty estimate of the value of player options in a way that isn’t at all clear when contract terms are discussed. I’m going to quickly discuss a few limitations, which I’ll elaborate on at further length in Part 2.

First, the amount a team will pay for a win above replacement isn’t constant and also isn’t known. I’ve used a flat $8 million per win for simplicity’s sake, but the option should be worth more due to the fact that dollars per win can change. If we have a huge recession and baseball revenue is cut in half, the ability to lock in a contract becomes way more valuable. Conversely, if hyper-inflation of salaries doubles contracts, opting out would make much more sense.

Second, I’ve assumed a 0 interest rate. This is largely to make the time value of money a bit clearer, as well as to simplify the option math. When a team offers a player a three-year contract for $90 million, it’s a bit less than $90 million of present value. That’s the trick behind contracts with deferred payments. Ignoring interest rates makes option math a lot easier, but could be folded back in. This would create a more complicated but more realistic model. In practice, I don’t think it adds enough to be worth junking up a nice clean spreadsheet, but it would be more precise.

Third, player aging is 100% ignored. I’ve done this for two reasons. One, it’s much easier to work out the option value of a fixed-skill player on a multi-year option than a player whose skill level will vary over those years. Second, I don’t believe that aging curves are sufficient to answer this problem. Player volatility is most likely not constant over time. By this, I mean that older players probably don’t have the same level of year-to-year changes in production and skill level that younger players do. Further empirical study would be required to work out a reasonable model for aging and volatility.

Last, I find it pretty unlikely that multi-year WAR changes are truly independent. A player who sees his level decline markedly in year 1 is probably somewhat more likely to see his skill increase in the next year than the average player, because so many of the components that underlie WAR are mean-reverting. I have absolutely not covered that in this model, and it would result in lower option prices than my model assumes. As a quick test, I looked at the two-year variation in WAR levels for players from 2016 to 2018. It was lower than you’d expect if it was an independent process, though still higher than the one-year variation.

I’ll attempt to address some, though not all, of these limitations in Part 2. I’ll also discuss fancier structures like multiple opt-outs as well as a more philosophical question- whether teams have a theoretical limit to how much they’re willing to pay any player no matter how good, and how much that should affect the value of an option offered to that player.

I know that today’s article has been, shall we say, math-heavy. Thanks for reading along. If you have any questions or comments about my opt-out modeling, please let me know. Until then, happy offseason, and may the Cardinals’ contract offers be both fair and interesting.

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