An American option adds one word to the contract — exercise anytime — and changes the mathematics completely. The price is a supremum over exercise strategies; the PDE becomes an inequality coupled to a constraint; the boundary between "exercise now" and "wait" is itself unknown. This note develops three equivalent formulations — optimal stopping, free-boundary problem, linear complementarity problem (LCP) — and the numerical methods for each. Shreve treats the binomial American as the discrete Snell envelope; TCF 4.5–4.6 carries the continuous-time LCP machinery.
Learning objectives. By the end of this note you should be able to:
- define a stopping time and state the optimal-stopping characterisation of the American price;
- write the dynamic programming recursion and connect it to the binomial algorithm;
- describe the free-boundary problem for the American put, including smooth pasting;
- derive the Black–Scholes inequality and formulate the LCP;
- explain why early exercise of a put can be optimal but never is for a non-dividend call.
In this lesson
- The optimal stopping formulation
- Dynamic programming and the tree/Bermudan view
- The free-boundary problem
- The Black–Scholes inequality
- Linear complementarity problem
- Computing American options
Before you start
The full lesson, with worked examples and exercise solutions, is for members.
Read the full lessonNew to SelfMadeQuant.com? See membership options or browse the Derivatives Pricing and Computational Finance syllabus.