Skip to content
Derivatives Pricing and Computational Finance / Binomial Trees, Barrier and American Options

Derivation: The CRR Parameters from Moment Matching

The Binomial Model states that the three calibration equations have the solution u=β+β2−1u = \beta + \sqrt{\beta^2 - 1} with β=12(e−rΔt+e(r+σ2)Δt)\beta = \tfrac12(e^{-r\Delta t} + e^{(r+\sigma^2)\Delta t}), approximately eσΔte^{\sigma\sqrt{\Delta t}}. Here we derive both claims from scratch, then repeat the calculation with real-world drift μ\mu to show the tree geometry is preference-free while the probability carries the measure. This derivation is a model exam answer for "derive the CRR parameters".
Learning objectives. By the end of this note you should be able to:
  • eliminate pp from the mean and variance equations to obtain a quadratic in uu;
  • verify β≥1\beta \ge 1 so the square root is real;
  • derive the small-Δt\Delta t expansion u≈eσΔtu \approx e^{\sigma\sqrt{\Delta t}};
  • state the general-γ\gamma (tilted tree) solution.

In this lesson

  1. Step 0 — The system
  2. Step 1 — Eliminate pp
  3. Step 2 — Use (E3) and solve the quadratic
  4. Step 3 — The small-Δt\Delta t approximation
  5. Step 4 — The Shreve/Hull route (drift μ\mu, expansion first)
  6. Step 5 — General γ\gamma (the tilted tree)

Before you start

The full lesson, with worked examples and exercise solutions, is for members.

Read the full lesson

New to SelfMadeQuant.com? See membership options or browse the Derivatives Pricing and Computational Finance syllabus.