Early Exercise & Stopping

Timing geometry.

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The Formal Theorem

V = max(Exercise, Wait)

Analytical Intuition.

American Options are the Geometry of Choice. You can exercise any time. This creates a Free Boundary problem?finding the optimal price at which to stop waiting and take the profit.
CAUTION

Institutional Warning.

Always worth at least as much as European options. The extra value is the 'Early Exercise Premium.'

Academic Inquiries.

01

When to exercise early?

When the benefit of cash today (dividends/interest) outweighs the value of future volatility.

Standardized References.

  • Definitive Institutional SourceHull, J.C. (2021). Options, Futures, and Other Derivatives.
  • Hull, J.C. Options, Futures, and Other Derivatives.
  • Shreve, S.E. Stochastic Calculus for Finance I & II.

Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Early Exercise & Stopping: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/financial-mathematics/early-exercise-stopping-theory

Dominate the Logic.

"Abstract theory is just a movement we haven't seen yet."