Option Pricing Model for Prospect Valuation

So, I've talked to Adam a few times about developing an options pricing model for prospect valuation.  As I see the  business side of prospects, their primary value is in years of team control, i.e., the time a team has benefit of the player's services at below open market prices.  In this way, a prospect is similar to an equity option (for instance, purchasing the option to buy a stock at potentially below market rate, at a premium).  Below the jump is my attempt to create a pricing model for prospects.

An option is valued, in its simplest sense, as the time value of the option plus the inherent value.  Google closed 9/23/09 at $498.46.  The Google $500 Oct call is out of the money (i.e., it has no inherent value) and on 9/23/09 was trading at an ask of $16.50, which represents purely time value.  The Google $490 Oct call is in the money and on 9/23/09 was trading at an ask of $21.70; the option has an inherent value of $8.46 and time value of $13.24.  Various option pricing models are used, but time value can be greater or lesser depending on the volatility of the underlying asset (among other things).  That is, all other things being held equal, Google will have greater time value than General Mills because its volatility is greater.  This situation is analogous to, say, live arms or toolsy outfielders, with the staid and boring Marcus Lemon's of the world taking the place of General Mills and the Lasting Milledge's of the world taking the place of Google.

I experimented with adapting Black-Scholes or the Monte Carlo option pricing models to the prospect world, but decided instead on a Drake equation style open-ended solution.  If an option value is equal to time value plus inherent value, that seems to give us a decent framework for valuing a prospect (or other young player)... time value plus inherent value.

Inherent value should be easy enough to value: for players that are currently playing in the Major League, +WPA gives us a value.  So let's say a player is a 2nd year player, producing at 2 +WPA, we could assign an inherent value of $9 million based on 2008's $4.5 million per win estimate at fangraphs.

What of time value?  The most straightforward calculation would be P[S-(tc)], where P is equal to the probability of success, S is equal the open market salary of a comparable free agent, t would be equal to the years of control, and c would be equal to an annualized salary over those years.  For our purposes here, we can call "success" purely subjectively; for instance, if we're using a Type A pitcher as a prospect's ceiling, it's going to affect the other variables.  You can see how high ceiling, toolsy types that are projectable but have a high probability of failure would nonetheless still have a lot of time value.

Prospect/Player value = P[S-(tc)] + I

So, to use an example here, Elvis Andrus had a +WPA of 6.58 for 2009. At $4.5 million per win, that gives him an inherent value of $29.6 million.  That seems really high.  Let's say we think Elvis' success was predicated on repeatable phenomena and we think there's a low likelihood of regression, that we think his annualized average salary will be around $2.6 million over his remaining 5 years, and that a type A free agent shortstop would earn $15 million / year.  That gives us time value of .75[75mm - (5*2.6mm)]= $46.5 million.

So this formula gives us a value of Elvis as $46.5 + $29.6 = $80.2 million.  Strictly from a financial point of view, any trade involving Elvis would have to bring $80.2 million in value back to the Rangers for it to make sense, either through time value in prospects, inherent value in +WPA, and/or some other variable, like, say, increased fan recognition due to a title.

Let's say you want the Cubs to eat 2/3's of Milton Bradley's remaining contract, which would be $14 million (I did a search for "bradley eat salary" and that was a proposal, facetious or not I don't know, from trza, so let's use it).  What kind of prospect would you have to kick in?  It seems like most of what I have seen is proposals of lower ceiling arms.  So let's say we want to kick in a player of the Jon Garland/Nate Robertson/Jarrod Washburn ceiling.  Those players had an annualized average salary of around $8 million for the past 4 seasons.  The names that I've seen kicked around are Kiker and Beavan.  So lower level arms, lots of time still to flame out, let's call it a 25% probability of reaching that ceiling (that's probably being pretty generous).  That gives us P = .25($48 mm - (6*1.04 mm)) + 0, or $10.4 million.  So by that analysis, the Cubs would be paying us about $4 million to free up a roster spot.  Maybe that's fair, I'm not sure.

So what do you think?  Adequate way to quantify a trade involving players under team control?