Executive Stock Options and Concavity of the Option Price
Bibliographic record
Abstract
Accounting for grants of executive stock options (ESOs) now requires that they be treated as an expense and valued at their fair values at the time of issue. But unlike traded options, maturity dates for ESOs are uncertain. They can not be exercised until a vesting period has passed, but after that, exercise may take place over a wide range of dates. Because the Black-Scholes model is nonlinear in time to expiration, simply putting the expected value of the date of exercise into the formula as the option maturity will produce a bias. It is commonly believed that this bias is positive, i.e., an option priced at the expected exercise date will be worth more than the mean value of a set of options exercised at dates uniformly distributed over the exercise period. Boyle and Scott discuss this problem and show, among other things, that there will be a bias, but it can go in either direction as a function of the other model parameters. The way to eliminate the bias is to value the option within a framework, such as a lattice model, in which the exercise decision is modeled specifically. The true expected life for accounting purposes should then be the implied time to maturity, that is, the maturity input that makes the Black-Scholes equation produce the same value as the lattice model. TOPICS:Options, simulations
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.023 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.003 | 0.006 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.009 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".