Market model and risk-neutral valuation established that an option price is a discounted risk-neutral expectation. This topic computes that expectation by discretising the randomness itself: two successor states per time step. The Cox–Ross–Rubinstein (CRR) tree is the simplest complete-market discretisation of geometric Brownian motion, and its great virtue is that the two contract features that break the Black–Scholes closed form — early exercise and barriers — cost almost nothing extra on a tree. Along the way we meet stopping times, the free-boundary problem, the linear complementarity formulation of American options, and the Brownian-bridge correction for barrier Monte Carlo.
By the end of this note you should be able to:
- derive the binomial parameters from moment matching against risk-neutral GBM, including the CRR solution and the tilted tree;
- write out and execute the backward-induction algorithm for European and American options, quote its cost, and read tree Greeks off the first two layers;
- formulate the American option as an optimal stopping problem, a free-boundary PDE, and a linear complementarity problem — and explain how the three are related;
- price barrier options on trees and grids (boundary conditions), use in–out parity, and correct Monte Carlo for continuous monitoring via the bridge survival probability;
- carry out the Longstaff–Schwartz regression example by hand.
In this lesson
- Contents
- The big picture
The full lesson is for members.
Read the full lessonNew to SelfMadeQuant.com? See membership options or browse the Derivatives Pricing and Computational Finance syllabus.