What's the difference between equilibrium and arbitrage-free term structure models?

Term structure models generate the evolution of interest rates over time, which is essential for valuing bonds with embedded options. CFA Level II groups them into two families with fundamentally different philosophies.

Equilibrium Models

These models describe how interest rates move based on economic theory. They start from assumptions about the economy and derive the term structure as an output. The model may or may NOT match today's observed yield curve.

Vasicek Model:

dr = a(b - r)dt + sigma x dz

• Mean reversion: Rates are pulled toward long-run mean b with speed a
• Constant volatility: sigma doesn't depend on the rate level
• Flaw: Allows negative interest rates (which was considered a flaw before 2010s negative-rate environments)

Cox-Ingersoll-Ross (CIR) Model:

dr = a(b - r)dt + sigma x sqrt(r) x dz

• Also mean-reverting with the same speed parameter
• Volatility scales with sqrt(r): When rates are near zero, volatility shrinks, preventing negative rates
• More realistic but harder to calibrate

Arbitrage-Free Models

These models are calibrated to EXACTLY match today's observed yield curve. They don't derive the curve from theory — they take it as a given and model how it evolves over time.

Ho-Lee Model:

dr = theta(t)dt + sigma x dz

• Time-dependent drift theta(t): Chosen to match the current term structure exactly
• Constant volatility
• Allows negative rates
• Simplest arbitrage-free model

Black-Derman-Toy (BDT) Model:

Used in binomial interest rate trees

• Calibrated to match both the current yield curve AND current volatility structure
• Log-normal: rates are always positive
• Most commonly used for valuing callable/putable bonds in practice and on the exam

Key Comparison:

Exam Tip: If a question asks which model type is better for pricing a callable bond, the answer is arbitrage-free (BDT). If it asks which model implies mean reversion as an economic property, the answer is equilibrium (Vasicek or CIR).

Explore term structure modeling in our CFA Level II Fixed Income course.