Best of Both Worlds: Hull-White and the Extended Vasicek Model

By the late 1980s the interest rate modelling world had a clean split, and both halves were unsatisfying.

On one side sat Vasicek and CIR: models with sensible economics, mean-reverting rates, closed-form bond prices, and a fatal defect, they could not reproduce the yield curve you saw on the screen. On the other sat Ho and Lee: a model that fit today's curve perfectly and had almost no economic content at all, with rates that wandered off without any pull back toward normality.

John Hull and Alan White's 1990 paper is the weld. It takes the good half of each and produces a model that is still, decades later, a genuine industry default.

The problem: fitting the curve versus behaving sensibly

You cannot price an interest rate derivative with a model that misprices the underlying bonds. That is Ho and Lee's point and it is unanswerable. Any model used on a real desk must reproduce today's observed yield curve exactly.

But you also cannot use a model that thinks 30 percent interest rates in twenty years are as likely as 4 percent ones. Mean reversion is not an optional decoration; it is the single most important empirical fact about interest rates. A model without it will systematically misprice anything long-dated, because it wildly overstates how far rates can drift.

So: fit the curve, and mean-revert. Before Hull and White, you picked one.

The key idea via analogy: a moving post for the elastic lead

Recall Vasicek's picture: the short rate is a dog on an elastic lead, tied to a post. The dog wanders randomly, and the further it strays, the harder the elastic pulls it back to the post.

The problem is that the post is nailed to one spot. That is why Vasicek cannot fit the curve: with the post fixed, the model can only produce a small family of yield curve shapes, and the real curve is not usually one of them.

Hull and White's move is almost embarrassingly simple to state: put the post on wheels. Let the post move along a prescribed path through time. The dog still has its elastic lead, still gets pulled back, still mean-reverts, all the good economics survives. But now you get to choose the path of the post.

And here is the payoff. Hull and White show you can solve for the path of the post that makes the model reproduce today's yield curve exactly. You do not guess it, you do not fit it by trial and error. There is a formula. Feed in the observed curve, out comes the required path, and the model now prices every bond on the screen correctly, by construction, while still mean-reverting properly.

This is the Ho-Lee trick (take today's curve as given, let no-arbitrage force the drift) applied to the Vasicek process (which mean-reverts). Hence the model's other name: extended Vasicek. Hull and White do the same for the CIR square-root process, showing the same extension works there too, though it is much less tractable.

They go one step further. Because the mean-reversion speed and volatility can also be allowed to vary through time, you can calibrate the model not just to today's bond prices but also to the prices of interest rate options, matching the volatilities the market is quoting for caps and swaptions as well as the curve itself. That makes it a proper pricing tool: it agrees with the market on everything it can see, and only extrapolates where it has to.

The extended Vasicek version keeps something precious: it stays analytically tractable. Bond prices have closed forms. European bond options have closed forms (via Jamshidian's decomposition). You can build a recombining trinomial tree for the messier products. That combination, correct prices, correct behaviour, and fast computation, is why it won.

Why it mattered

• It became the default one-factor model on real desks. Swaptions, callable bonds, Bermudan structures, mortgage models: for decades, if you asked what model was under the hood, "Hull-White" was the single most common answer. It remains a staple.
• It cleanly separated the two jobs of a model. After Hull-White, everyone understood that a model has a dynamic part (how rates behave: mean reversion, volatility) and a calibration part (a time-dependent function whose only job is to make today's prices come out right). That separation is now second nature, and it is the reason modern models are written the way they are.
• It is fast. Closed forms and recombining trees mean you can revalue a large book overnight, or intraday. Elegance that you cannot compute is useless on a trading floor. Hull-White is the model that made calibrated no-arbitrage modelling practical.
• It made the trade-offs explicit. By comparing option prices across models, the paper showed clearly how much your answer depends on which model you pick, which was a sobering and useful message.

The honest limitations

• Rates can go negative. The extended Vasicek version is Gaussian, so nothing stops the short rate from going below zero. For twenty years this was the standard criticism. Then rates actually went negative in Europe and Japan, and the models that could not handle it (lognormal ones, CIR) were the ones in trouble. Hull-White aged extremely well by accident.
• Still only one factor. All rates on the curve are driven by one random shock, so they are perfectly correlated. The curve can shift, but it cannot independently twist. That makes the model structurally unsuited to anything whose payoff depends on the shape of the curve, spread options, curve steepeners, CMS spread products. Two-factor extensions exist for exactly this reason.
• Perfect fit hides model error. Because the time-varying post path absorbs any discrepancy between model and market, the model will fit today's curve no matter how wrong its dynamics are. Sometimes the fitted path of the post looks bizarre, a sign that the model is being tortured into agreement rather than genuinely describing the market. A perfect fit is not evidence that the model is right.
• Calibration is not unique. Mean reversion speed and volatility trade off against each other. Very different parameter pairs can fit the same set of option prices while implying quite different behaviour for exotic products. This is a real, live problem: two desks with the "same" Hull-White model can disagree substantially on a Bermudan swaption.
• Volatility structure is limited. A single mean-reversion speed and a single volatility function cannot reproduce the full, rich pattern of how implied volatility varies across both option expiry and underlying tenor. You will fit some of the swaption grid well and the rest badly, and which part you fit well is a choice you have to make.

The one-line takeaway

Hull and White let Vasicek's mean-reverting rate be pulled toward a post that moves along a path chosen to reproduce today's yield curve exactly, marrying sound dynamics to a perfect market fit and producing the model that has priced interest rate derivatives on real desks ever since.