Ho-Lee And Black-Derman-Toy Models

Earlier interest-rate models like Vasicek assumed the short-term interest rate follows a particular process and then derived what the whole yield curve should look like from that assumption — but the resulting curve rarely matches the curve actually observed in the market on a given day, which is unacceptable if you're trying to price a bond option consistently with the bonds themselves already trading. Ho-Lee and Black-Derman-Toy flip the approach: instead of deriving the curve, they build it into the model as an input, guaranteeing the model reproduces today's observed curve exactly by construction.

The Ho-Lee model is the simplest version: the short rate moves as a random walk with a time-varying drift term, and that drift is chosen period by period specifically so the model's implied bond prices match the market's actual bond prices today. Its main weakness is that rates can go negative and volatility is constant across all maturities, both unrealistic. The Black-Derman-Toy model improves on this by modeling the log of the short rate (so rates stay positive) and additionally calibrating a time-varying volatility term to match the market's observed volatility term structure (from cap or swaption prices), not just the yield curve level.

Both are "no-arbitrage" models in the sense that they're fit to reproduce today's market prices exactly, as opposed to "equilibrium" models like Vasicek or CIR that impose an economic story about rate dynamics and let the curve emerge as a consequence, potentially not matching the actual market curve at all.