Normally Distributed Rates and the Ho-Lee Model: Time-Dependent Drift

A callable bond's price depends on what rates will do over the next five to thirty years, not just where they are today. But if you price it using a model that implies rates will always average, say, 6%, you will systematically mis-price the option embedded in the bond — because any real yield curve tells you that market participants expect something different. The Ho-Lee model solves this problem: by choosing its drift $\lambda(t)$ date by date, it absorbs whatever combination of expectations, risk premium, and convexity is embedded in today's yield curve and prices every derivative consistent with that curve.

This lesson builds the Ho-Lee model from first principles, starting with its two predecessors — Model 1 (zero drift, normally distributed rates) and Model 2 (constant drift) — and showing why each is insufficient before the time-dependent drift resolves the problem.

• state the SDEs for Model 1, Model 2, and Ho-Lee and write down their binomial-tree representations;
• compute the terminal distribution of the short rate under each model and describe its mean and variance;
• explain why Model 1's term structure is always downward-sloping and Model 2's is still too rigid;
• describe how the Ho-Lee drift $\lambda(t)$ is calibrated to exactly fit the initial forward curve;
• state the closed-form Ho-Lee bond price formula $P(t,T) = e^{A(t,T) - B(t,T)\,r(t)}$ with $B(\tau) = \tau$ and compute $A(t,T)$ from the initial forward curve;
• identify the cost of fitting to the term structure (loss of parsimony, economic interpretability).

1. 1. Model 1: zero drift, normally distributed rates
2. 1. Model 2: constant drift as risk premium
3. 1. The Ho-Lee model: time-dependent drift
4. 1. The Ho-Lee bond price formula
5. 1. When to use Ho-Lee versus equilibrium models