What is the Ho-Lee model and how does it differ from other term structure models?
My CFA Level II study materials introduce the Ho-Lee model as an arbitrage-free term structure model. I understand the basic idea of modeling rate movements, but how does Ho-Lee specifically work and what makes it 'arbitrage-free' compared to equilibrium models like Vasicek?
The Ho-Lee model is one of the simplest arbitrage-free term structure models and a key concept for CFA Level II. Let's compare it to equilibrium models and walk through its mechanics.
Two Families of Term Structure Models
| Feature | Equilibrium Models (e.g., Vasicek, CIR) | Arbitrage-Free Models (e.g., Ho-Lee, BDT) |
|---|---|---|
| Starting point | Assumptions about rate dynamics | Current market term structure |
| Calibration | Fits parameters to historical data | Fits exactly to today's yield curve |
| Output | May not match today's prices | Always matches today's prices |
| Use case | Understanding rate behavior | Pricing bonds and derivatives |
The Ho-Lee Model
The Ho-Lee model specifies that the short rate evolves as:
dr = theta(t) dt + sigma dW
Where:
- theta(t) = time-varying drift, calibrated to match the current term structure
- sigma = constant volatility of the short rate
- dW = random Wiener process (standard normal shock)
Why Is It Arbitrage-Free?
The magic is in theta(t). Unlike the Vasicek model where drift is a constant mean-reversion term, the Ho-Lee model's drift changes at each time step to ensure the model reproduces today's observed yield curve exactly.
Think of it this way: if today's 2-year spot rate is 4.2% and the 3-year is 4.5%, the Ho-Lee model adjusts its drift parameters so that the tree it generates is perfectly consistent with those market rates. No arbitrage opportunities exist between the model prices and market prices.
Building a Ho-Lee Tree — Whitmore Capital Example
Suppose the current term structure is:
- 1-year spot: 3.80%
- 2-year spot: 4.10%
- 3-year spot: 4.35%
With sigma = 1.50%, the Ho-Lee tree is calibrated so that:
The drift at each step is chosen so that when you price benchmark bonds using backward induction through this tree, you recover exactly the observed spot rates.
Limitations of Ho-Lee:
- No mean reversion — Rates can drift arbitrarily high or low. The Vasicek model has mean reversion, which is more realistic.
- Negative rates possible — Since the model uses a normal distribution for rate changes, rates can go negative. (The CIR and BDT models avoid this.)
- Constant volatility — Real rate volatility varies across maturities and over time.
Comparison Summary:
| Model | Arbitrage-Free? | Mean Reversion? | Negative Rates? |
|---|---|---|---|
| Vasicek | No | Yes | Yes |
| CIR | No | Yes | No |
| Ho-Lee | Yes | No | Yes |
| BDT | Yes | Yes | No |
Exam Tip: CFA Level II typically asks you to identify which models are arbitrage-free and which are equilibrium models. Remember: Ho-Lee and BDT match today's curve; Vasicek and CIR do not necessarily.
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