Vasicek, CIR and Hull-White models explained

CM2 · Financial Engineering & Loss Reserving · 12 min read

What a short-rate model is for

The term-structure arithmetic tells you what the yield curve is today. It says nothing about where it will be, and anything whose payoff depends on future interest rates — a bond option, a swaption, a guaranteed annuity rate, the reserve for any long liability — needs that.

A one-factor short-rate model supplies it, by describing the instantaneous risk-free rate r(t) as a stochastic process. One number drives everything: given the process for r, the whole yield curve at every future date follows, because a bond price is an expectation of discounting at r along the way. That is the appeal and also the central limitation, because a single factor means the whole curve moves together, and real curves twist.

Mean reversion, which all three share

An interest rate does not behave like a share price. A share can wander anywhere; a short rate that reached 40% would be pushed back down by policy, and one at −5% would be pushed back up. The rate is pulled towards a level rather than drifting freely, and the models say so:

dr_t = α(μ − r_t) dt + (diffusion) dW_t

In words: the expected change in the rate is proportional to how far it currently sits below its long-term level, so the further away it is the harder it is pulled back.

Read the drift term carefully, because it is the whole idea. If r < μ the bracket is positive and the rate is expected to rise. If r > μ it is expected to fall. And at r = μ the drift is zero, so μ is the level the process has no tendency to leave — the long-term mean.

α sets how fast. It has units of one over time, and the interpretable version of it is the half-life of a deviation:

half-life = ln 2 / α

At α = 0.25 that is 2.77 years: a rate two percentage points below its long-term level is expected to have closed half the gap in under three years. Quoting the half-life rather than α is the quickest way to say whether a fitted speed is plausible.

The three models

Vasicek

dr_t = α(μ − r_t) dt + σ dW_t

Constant volatility. This is the simplest mean-reverting model there is, and it is almost entirely tractable: the short rate is normally distributed at every future date, bond prices have a closed form, and so do options on them.

That normality is also its defect. A normal distribution puts probability everywhere, so Vasicek allows the short rate to go negative, and not as a remote curiosity — the worked example below puts the chance at a few tenths of a per cent on realistic parameters. For many years that was treated as a fatal objection. Since several central banks have in fact set negative rates it is a milder criticism than it was, and the honest modern answer to an exam question is that it is a defect in a model of a rate that cannot go negative, such as one bounded by physical cash, and a feature in a model of a policy rate that can.

Cox-Ingersoll-Ross

dr_t = α(μ − r_t) dt + σ√r_t dW_t

One change: the volatility is multiplied by the square root of the current rate. Everything else is Vasicek.

That one change does two things at once, and seeing both is the point of the comparison. As the rate approaches zero, √r approaches zero, so the randomness switches itself off exactly where it would otherwise push the rate below zero — while the drift term, α(μ − 0) = αμ, is still pulling firmly upwards. The rate therefore cannot go negative, and provided 2αμ ≥ σ² it cannot even reach zero.

It is also more realistic in a second way that is easy to miss: it makes volatility higher when rates are higher, which is what is observed. A one percentage point move is a much bigger event at 2% than at 12%, and Vasicek treats the two as equally likely.

The cost is tractability. The rate is no longer normal — it follows a non-central chi-squared distribution — so the algebra is heavier, although bond prices still have a closed form.

Hull-White

dr_t = α(μ(t) − r_t) dt + σ dW_t

Vasicek with a long-term mean that is a function of time. That looks like a small generalisation and is in fact the one that makes the model usable in practice.

Here is the problem it solves. Vasicek and Cox-Ingersoll-Ross have three constants between them, so they can produce only a limited family of yield curve shapes — and today's actual curve, with its humps and kinks, is almost never one of them. A model that misprices the bonds you can see cannot be trusted on the options you cannot. Letting μ(t) vary adds a whole function's worth of freedom, and it can be chosen so that the model reproduces today's observed curve exactly.

That property has a name: the model is arbitrage-free with respect to the initial curve, or simply "calibrated to the market". It is why Hull-White and its relatives are what practitioners actually use, and why the first two are best understood as the ideas Hull-White is built from.

What Vasicek predicts, exactly

Because the Vasicek rate is normal, its distribution at any future date can be written down. Starting from a known rate r₀:

E[r_t] = μ + (r₀ − μ)e^−αt
Var[r_t] = (σ²/2α)(1 − e^−2αt)

In words: the expected rate starts where you are and decays exponentially towards the long-term mean, and the variance grows from zero towards a ceiling.

Both formulae are worth reading as statements rather than as algebra. The mean closes the gap by a constant proportion each year, which is what mean reversion means. The variance does not grow without limit, unlike a share price's — it levels off at σ²/2α, the stationary variance, because the pull back towards μ eventually balances the randomness pushing away from it. A mean-reverting process has a long-run distribution; a random walk does not.

That ceiling is also the easiest sanity check on a fitted model. If the stationary standard deviation implies rates routinely 6 percentage points from their mean, the parameters are wrong whatever the fit statistics say.

Where marks are lost

Getting the sign of the drift backwards. It is α(μ − r), which pushes up when the rate is low. Writing α(r − μ) gives a process that runs away from its mean rather than towards it, and the resulting model is explosive rather than stationary.

Saying Cox-Ingersoll-Ross "has no negative rates because of the drift". Both models have the same drift. The square root in the diffusion is what does it, by removing the randomness at zero. A question asking why usually wants exactly that distinction.

Claiming Vasicek cannot fit the observed curve "because it has too few parameters" and stopping there. Say which freedom is missing: a constant long-term mean cannot bend the curve the way a time-dependent one can, which is precisely what Hull-White adds.

Using the stationary variance at a short horizon. σ²/2α is the limit. At one year on the parameters below the standard deviation is 1.33%, not the stationary 2.12% — using the limit would overstate the one-year uncertainty by more than half.

Forgetting that σ here is an absolute volatility. In Vasicek, σ is in units of rate per square root of time, so 1.5% means 1.5 percentage points, not 1.5% of the current rate. In Cox-Ingersoll-Ross the √r changes the meaning again, so the two models' σ values are not comparable.

Questions students ask

Why only one factor?

Because it is the simplest thing that can work, and a great deal of the variation in a yield curve really is one factor — the whole curve moving up or down together typically accounts for most of the observed movement. What one factor cannot do is let the short end and the long end move differently, so it cannot price anything whose value depends on the shape changing, and that is the reason two-factor and multi-factor models exist.

Is a negative rate a model failure or not?

It depends what the rate represents. If it is a rate at which an investor could instead hold physical cash at no cost, a materially negative rate is a genuine arbitrage and the model is wrong. If it is a central bank deposit rate, where holding cash has storage and insurance costs, negative rates are real and have been observed. The examinable answer names the objection, notes that it has been overtaken by events, and says which of the two situations is being modelled.

How are these parameters estimated?

Two different ways, and they disagree. Fitting the historical path of the short rate gives the real-world parameters. Fitting today's bond and option prices gives the risk-neutral ones, which is what pricing needs. The gap between them is the market price of interest rate risk, and using historical parameters to price a derivative is a standard and expensive mistake.

Does mean reversion mean rates are predictable?

Only in the weak sense that a rate far from its long-term mean is more likely to move back than further away, and the half-life says how slowly. Over a horizon short relative to that half-life the model is close to a random walk, and the expected move is small next to the uncertainty. On the figures below, after one year the rate is expected to have risen 44 basis points with a standard deviation of 133, so the drift is a fifth of the noise.

A worked example

Take a Vasicek model with a reversion speed of α = 0.25, a long-term mean of μ = 6%, a volatility of σ = 1.5%, and a short rate today of r₀ = 4%.

All four parameters are invented for this example. They are not calibrated to any market.

Step 1: how fast does it revert?

half-life = ln 2 / 0.25 = 0.6931472 / 0.25 = 2.7726 years

So the 2 percentage point gap between 4% and 6% is expected to be half closed in under three years. That is a moderate speed: slow enough that the current rate matters for a decade, fast enough that it does not matter for thirty years.

Step 2: the ceiling on the uncertainty

stationary variance = σ²/2α = 0.015² / 0.5 = 0.000225 / 0.5 = 0.00045
stationary standard deviation = √0.00045 = 2.12132%

However far ahead you look, this model never puts the short rate's standard deviation above 2.12 percentage points. Read that as a plausibility check on the parameters: it says the rate will spend most of its time between roughly 2% and 10%, which is a defensible range, and if it had come out at 0.3% or 15% the parameters would need revisiting.

Step 3: the distribution at three horizons

t	E[r_t]	sd[r_t]	P(r_t < 0)
1 year	4.44240%	1.33064%	0.00042
5 years	5.42699%	2.03239%	0.00379
20 years	5.98652%	2.12127%	0.00239
limit	6.00000%	2.12132%	0.00234

The one-year row, worked out in full:

E[r₁] = 0.06 + (0.04 − 0.06)e^−0.25 = 0.06 − 0.02 × 0.7788008 = 0.06 − 0.0155760 = 0.0444240
Var[r₁] = 0.00045 × (1 − e^−0.5) = 0.00045 × 0.3934693 = 0.0001770612
sd[r₁] = 0.0133064

The check: solve for the mean a second way

The closed form for the mean can be verified without using it. Taking expectations of the stochastic differential equation kills the dW term, leaving an ordinary differential equation for m(t) = E[r_t]:

dm/dt = −α(m − μ), with m(0) = 0.04

Stepping that forward numerically in two million steps gives 0.0444240 at t = 1 and 0.0542699 at t = 5. The closed form gives 0.0444240 and 0.0542699. Seven decimal places, by a route that uses no exponential at all.

Reading the negative-rate column

The one-year figure is Φ(−0.0444240 / 0.0133064) = Φ(−3.3385) = 0.00042 — about one chance in 2,400.

Two things in that column are worth noticing. First, the probability is small but not negligible, and on a model used to value a thirty-year guarantee a one-in-250 chance of a negative short rate at the five-year point will show up in the answer. Second, and more interesting, the probability peaks in the middle: 0.04% at one year, 0.38% at five, 0.24% at twenty. Early on the variance has not had time to grow; much later the mean has risen to 6% and pulled the distribution clear of zero. The worst point is where the variance has grown but the mean has not yet arrived, and that is a genuine feature of the model rather than an arithmetic slip.

What Cox-Ingersoll-Ross would do with the same parameters

Replace σ dW with σ√r dW and the whole column above becomes zero: the rate cannot go negative. Check the stronger condition too:

2αμ = 2 × 0.25 × 0.06 = 0.03
σ² = 0.015² = 0.000225
0.03 ≥ 0.000225, comfortably

So with these parameters the Cox-Ingersoll-Ross rate never even touches zero, let alone crosses it. The condition is 2αμ ≥ σ² and it fails only when the volatility is large relative to the upward pull at the origin, which these parameters are nowhere near.

Note what has not changed: the drift is identical, so the mean path above is unchanged to a first approximation and the half-life is still 2.77 years. The square root altered the tails, not the central tendency, and that is exactly the right way to describe the difference in an exam.

And what Hull-White would add

Neither model above can be made to reproduce an observed yield curve, because between them they have three constants. Suppose today's curve is the one from the term-structure arithmetic: one-year 5.0%, two-year 5.6%, three-year 6.0%. A Vasicek model starting at a 4% short rate with a 6% long-term mean implies its own curve, and that curve will not be those three numbers. Changing α, μ and σ moves it around but cannot bend it into an arbitrary shape.

Hull-White replaces the constant μ with a function μ(t) and chooses that function so the implied curve matches the observed one at every maturity. The cost is that μ(t) is no longer a quantity with an economic meaning you could argue about — it is whatever it has to be — and the model has stopped explaining the curve and started accepting it. For pricing a derivative off that curve, accepting it is exactly what you want.

Where this fits in CM2

This page explains one idea from CM2 Financial Engineering & Loss Reserving. The full CM2 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.

Short-rate models: Vasicek, Cox-Ingersoll-Ross and Hull-White compared