# Using a Wall Street Technique on Baseball Salaries

The Black-Scholes model for calculating the premium of an option was introduced in 1973 in a paper entitled, “The Pricing of Options and Corporate Liabilities” published in the *Journal of Political Economy*. The formula was developed by three economists – Fischer Black, Myron Scholes and Robert Merton. Scholes and Merton were awarded the 1997 Nobel Prize in Economics for their work in finding a new method to determine the value of derivatives. Fischer Black was not awarded the prize as he had died in 1995 and the Nobel Prize is not given posthumously.

The Black-Scholes model is still used today on Wall Street to calculate the theoretical price of European put and call options, ignoring any dividends paid during the option’s lifetime. This article will seek to apply this current tool of finance to baseball contracts.

The Black-Scholes model makes certain assumptions, many of which will be significant advantages for our analysis, including:

- The options are European and not American.
- In European style options, the option can only be exercised at expiration, while American options can be exercised at any time during the exercise period. As baseball contracts do not allow a player to opt out during the season and only at the conclusion of the season, baseball contracts have a European style option.

- No dividends are paid out during the life of the option.
- Since baseball players do not pay cash dividends, this actually makes it easier to use the Black-Scholes model rather than other option models that include dividend payouts.

- Efficient markets
- Since all players and teams have the same information, this is met by major league baseball.

- No commissions
- This is not perfectly true for major league baseball as draft choices are sometimes given up when a free agent contract is signed by teams who do not have a protected draft choice. Still, I relax this requirement for this analysis.

- The risk-free rate is known and constant.
- Used in finance as the risk-free opportunity cost of money. Can be used in our analysis as well.

- The volatility of the underlying asset (i.e. the player in question) is both known and constant.
- Given aging patterns, volatility is likely to accelerate as a player ages. In addition, future injuries are unknown at the time a contract is signed.

- Follows a lognormal distribution;
- Returns on the underlying are normally distributed. Baseball players probably don’t meet a strict definition of having lognormal distribution of their returns, as a random variable which is log-normally distributed takes only positive real values, and each years a few players do wind up with negative WAR. However, the vast majority of seasonal WAR values are positive and we can assume that we are pretty close to meeting this requirement.

The formula below, which can be skipped, takes the following variables into consideration:

- Current underlying price (the salary without the option)
- The option’s strike price (the annual salary if the player does not exercise his opt-out)
- Time until expiration, expressed in percentage form as the number of season’s until the opt-out can be exercised
- Implied volatility (older, more injury-prone players may have more volatility)
- Risk-free interest rate

D1 = 1/(σ*sqrt (T-t)) * [(ln(S/K)+(r+ ((σ^{2}/2))*(T-t)]

D2 = D1- σ *sqrt (T-t)

And C = N(D1)S-N(D2)Ke^{-r(T-t)}

C is the Call premium, S = current price, T-t = time until option matures, K = option strike price, r = risk free interest rate, N=cumulative standard normal distribution, σ = standard deviation or underlying volatility of the returns (as measured in WAR) of the ballplayer in question, and ln = natural log.

Calculating the call premium is essentially divided into two parts: the first part, *N(D1)S*, multiplies the price by the change in the call premium in relation to a change in the underlying price. This part of the formula shows the expected benefit of purchasing the underlying outright. A player “buying” the opt-out option (buying a call in Wall Street terms on himself) in his contract may do well if either his future expected value increases (he has a good year), or the dollar per WAR calculation, done each offseason at FanGraphs, increases appreciably more than expected.

The second part, *N(d2)Ke^-r(T-t)*, provides the current value of paying the exercise price upon expiration (remember, the Black-Scholes model applies to European options that are exercisable only on expiration day, which for us means the off season). The value of the call option is calculated by taking the difference between the two parts, as shown in the equation. So, if a player can make $X as called for in his contract and not exercising an opt-out but can expect to make $Y by opting out and $Y > $X, he has a financial incentive to opt out.

For those intimidated by the math, many internet sites offer free Black-Scholes model calculations where one merely has to enter the variables and the calculation is done for you. For this article, I used thiswebsite calculator:

A player option means that the player is “long” his own performance, and the team is “short,” the player paying for the option with a lower contract amount and the team getting compensated for the risk by paying less money. If the player does well, his value will rise and the player hopes to sign a larger contract the following season.

Now let’s start to analyze the value of a real opt out for a real player. I chose Masahiro Tanaka, for many reasons, not the least of which was his contract had a single option, and the writer sees him pitch every five days for the Yankees. The contract Tanaka signed before the 2014 season was for seven years with a $22 million annual salary for the first six years and a $23 million contract in year seven. The option was a player option whereby Tanaka could leave after year four, leaving three years and $67 in guaranteed money on the table.

Using the web site shown above, I attempt to estimate the value of that call option. In other words, how much did Tanaka “pay” by accepting a lower value than he would have gotten otherwise?

- The “stock price” is $65,774,416.26
- The present value of what the contract would have been without the player option (call option) built in.
- Of course, with present value, the first item of controversy will be what discount rate to apply to the future cash flows. As stated on page 123 of the current Collective Bargaining Agreement,
- If the terms of a Contract are confirmed by the Association and the Office of the Commissioner before the Imputed Loan Interest Rate for the first Contract Year covered by the contract is available, the Imputed Loan Interest Rate shall be the annual “Federal mid-term rate” as defined in section 1274(d) of the Internal Revenue Code for the month preceding the month in which terms are confirmed.

- Turning to Section 1274(d) of the IRS Code, we find that,
- (ii) in the case of a debt instrument with a term over three years but not over nine years, the Federal mid-term rate; and (iii) in the case of a debt instrument with a term over nine years, the Federal long-term rate. Section 1274(d)(1)(B) requires the Secretary to determine during each calendar month the foregoing rates that shall apply during the following calendar month. The IRS publishes the AFRs for each month in the Internal Revenue Bulletin, as described in section 1.1274-4(b) of the Income Tax Regulations.

- Based on this, the discount rate I used was 1.75 percent.
^{5}Note that increasing the discount rate would lower the cash price, just as higher interest rates lower the price of bonds. - For the numerator of the present value of the contract, I started with the actual contract, which included the player option, and added $2 million to each of the years covered by the option:

Date | Year of Contract | Cash Paid | PV Factor | PV of Cash Flow |
---|---|---|---|---|

7/1/2014 | 1 | 22,000,000 | 1.0175 | 21,621,621.62 |

7/1/2015 | 2 | 22,000,000 | 1.0353 | 21,249,750.98 |

7/1/2016 | 3 | 22,000,000 | 1.0534 | 20,884,276.15 |

7/1/2017 | 4 | 22,000,000 | 1.0719 | 20,525,087.12 |

7/1/2018 | 5 | 24,000,000 | 1.0906 | 22,005,900.87 |

7/1/2019 | 6 | 24,000,000 | 1.1097 | 21,627,421.00 |

7/1/2020 | 7 | 25,000,000 | 1.1291 | 22,141,094.39 |

TOTAL | 161,000,000 | 150,055,152.13 |

The cash price for the years in question would therefore be $65,774,416.26. As shown above, should the opportunity cost of the player option be paid by lowering the cash value in other years, the present value of the impacted cash flow can be calculated by adjusting the relevant cell(s).

- The exercise price is $60,367,018.55

The “exercise price” is the present value of the pre-determined guaranteed salary that the player is foregoing by exercising the option. Using the same 1.75 percent interest rate used above, Tanaka’s exercise price was $60,367,018.55, shown by summing the present value figures in red as presented below in Table 2:

Date | Year of Contract | Salary | PV Factor | PV of Cash Flow |
---|---|---|---|---|

7/1/2014 | 1 | 22,000,000 | 1.0175 | 21,621,621.62 |

7/1/2015 | 2 | 22,000,000 | 1.0353 | 21,249,750.98 |

7/1/2016 | 3 | 22,000,000 | 1.0534 | 20,884,276.15 |

7/1/2017 | 4 | 22,000,000 | 1.0719 | 20,525,087.12 |

7/1/2018 | 5 | 22,000,000 | 1.0906 | 20,172,075.80 |

7/1/2019 | 6 | 22,000,000 | 1.1097 | 19,825,135.92 |

7/1/2020 | 7 | 23,000,000 | 1.1291 | 20,369,806.84 |

TOTAL | 155,000,000 | 144,647,754.42 |

Note that increasing the discount rate from 1.75 percent would lower the exercise price of the option.

- The time to maturity is four years,
- Tanaka had to wait four years until he was able to exercise his option. Note that the model actually increases the value of the call the longer to exercise date. While this makes sense in the stock market where profits and dividends are expected to increase over longer time horizons, given an aging body, this may make less sense in the baseball world. Offsetting the aging body is the inflationary value of WAR, which has been documented to be increasing at an annual rate well in excess of the risk-free interest rate.

- The annual risk-free interest rate is .11 percent.
- Referenced is the yield of the one-year Treasury security on Jan. 17, 2014, the last business day before Tanaka signed his contract.
- Note, the higher the annual risk-free interest rate, the greater the price of the call option. Being able to reference different risk-free rates over time will provide significant value going forward for users of the model.

- The annualized volatility I used was 15 percent.
- Please note that the greater the volatility of the athlete, the greater the value of the option. I quickly guesstimated from current aging tables to be 15 percent. Spending additional time on determining the volatility of performance by position type, body type, pitcher-type (flamethrower vs junk baller) could yield significant improvements in the practical use of this model going forward.

Rounding the dollar figures off from 1 and 2 above to the nearest 10-thousands place (entering 65.77 for example in the stock price cell) and using the other figures shown above shows that the value of the player-option Tanaka received was theoretically worth $10.67 million at the time Tanaka signed his contract.

How much did Tanaka “pay” for his option? That would be the difference between the present value of the contract’s cash flows in the cash option shown in Table 1 and the present value of the cash flows in the contract with the player option, as shown in Table 2. That difference is $5,407,397.71, rounded to $5.4 million to those scoring at home. Since the value provided by the option was over $10 million, and only $5.4 million was paid for the option, Tanaka was smart to take the offer with the option.

How much would the Yankees have had to offer Tanaka to avoid the player option? That would be a contract worth the $144.6 million in present value shown in Table 2 plus another $10.67 million of present value.

Of course, these are based on estimates of the cash value, the discount rate and volatility. Should the parties in the room actually be able to compare the offer without the option and an offer with the option, a player and his agent could then determine the theoretical value of the option and if it is greater than the differences in the two offers, choose that one. Conversely, the negotiating parties on the team’s side can do the same and seek to lower the value of the disproportionately valuable offer and adjust so that there is no theoretical difference between the two offers made to the player.

### Conclusion

This paper has attempted to apply a widely used, somewhat standard measure of valuation of options and apply the model, as created by noble prize-winning economists, to the valuation of player options in a baseball contract.

Future analysis may be enhanced with determining the expected annual volatility of player performance by position, and by age, to help further refine a standard finance model to the measurement of option valuation in baseball.

In addition, the world of finance has numerous models for pricing various financial instruments. Future analysts are encouraged to attempt to use models and calculations developed and utilized in the financial world to the National Pastime.

### References:

- Wikipedia
- 2017-2021 Basic Agreement
- IRS, “Adjusted Applicable Federal Rates And Adjusted Federal Long-term Rates”
- IRS, “Determination of Issue Price in the Case of Certain Debt Instruments Issued for Property”
- Matt Swartz, FanGraphs, The Recent History of Free-Agent Pricing”
- U.S. Dept. of Treasury, “Resource Center: Daily Treasury Yield Curve Rates”
- Mitchel Lichtman, The Hardball Times, “How do baseball players age? (Part 1)”
- Mitchel Lichtman, The Hardball Times, “How do baseball players age? (Part 2)”

As an actual Wall Street quant, this piece is incorrect. Black-Scholes is designed to calculate the cost of a hedging strategy, given the assumptions you listed above. But to be able to hedge, you have to be able to buy and sell the underlying. In the case of a stock option, the underlying (i.e. the stock) can obviously be bought and sold easily. But since no liquid market exists for Tanaka’s future, Black Scholes does not apply.

The player is actually either “selling” his future (signing without an option) in the free player market or buying his future back with a player option.

the whole ‘liquid market’ thing throws a wrench into economics the same way that ‘friction’ does to physics

As opposed to being a quant I am a Wall Street banker. Even though this situation does not perfectly fit with the assumptions requried for Black Scholes, this piece is not entirely incorrect. This case is similar to small-cap public equities being valued using the Capital Asset Pricing Model. The market for those equities is not nearly as liquid as the market for large-cap equities and therefore does not satisfy all of the requirements to use CAPM. However, we generally apply an appropriate size premium to adjust for this. Similarly, one could use a simple liquidity premium here to compensate… Read more »

Small cap public equities can still be bought and sold, just with nonzero transaction costs, so Black-Scholes mostly holds. Masahiro’s value cannot, so Black-Scholes is irrelevant.

Another assumptions for CAPM is that markets are prefectly liquid, which small cap equity markets are not. So by applying your logic, it should be irrelevant, which it is not. Your logic is flawed. Also, Tanaka’s value can be bought and sold through a trade, exactly how the underlying would be bought and sold just using a different currency (i.e. other players rather than stricly cash), which I believe someone in this section mentioned already. Although there is no way to isolate the specific years of Tanaka’s value (years 5-7) to which the option applies, it is still possible to… Read more »

The website above requires an annual risk free interest rate. Also, note the higher the risk free rate, the value of the call option increases.

Not a quant, so I don’t entirely understand your point. Honestly curious question: Are you saying that Black Scholes doesn’t apply bc Tanaka can’t buy/sell his in-contract years himself? Because there is cartainly a liquid market for Tanaka’s contract – if NYY include $20m cash in a trade, literally every team is interested in trading for him.

There would need to be i.e. a liquid futures market in Masahiro Tanaka’s value. So you’d need to be able to bet on his value going up and down at will without major transaction costs. Black-Scholes isn’t relevant for assets that can’t be easily and limitlessly bought and sold.

Thanks for educating me on the CBA discount rates. I suspect baseball clubs use a much higher internal cost of capital used for budgeting and planning (7% – 15%?) than the 2% long-term AFR. Could this be driving the decline in players’ share of revenue? If a company invests for 10% annual return with players’ salaries as a meaningful expense, while players’ only expect a 2% annual return, this spread implies companies’ expect salaries to be a progressively lower part of operating expenses over time. Am I thinking about this coherently?

Agree that the internal cost of capital used by financial executives at ball ball clubs is much higher than the long-term AFR. Suspect that, as has been proven in many industries, wage increases lags industry profits. Since regional TV networks, newer ball parks, and Internet revenues have sky-rocketed during the last decade or so, that that is what has been driving the player’s shrinking piece of the baseball pie.

Thanks for this, interesting read. One question what’s the basis for the addition of the two million when deriving the spot price?

“For the numerator of the present value of the contract, I started with the actual contract, which included the player option, and added $2 million to each of the years covered by the option”

The original difference of $2 million was more for discussion and to use a reasonable example. I chose $2 million as it was the difference at the time of Tanaka’s contract and Cliff Lee, then the highest paid pitcher in the US. With the tables presented, and the link to the Black-Scholes calculator provided in the article, readers are invited to easily change the assumptions to re-run both this and other contracts. Obviously, only the parties at the negotiating table would know the exact difference of what the final negotiated difference was between what the contract would have paid without… Read more »