Why So Many Rate Models Have Formulas: Duffie and Kan's Affine Class

By the mid-1990s the term structure literature had accumulated a strange coincidence. Vasicek's model gives you closed-form bond prices. So does CIR. So do various multi-factor cousins. These models make quite different assumptions, yet they all end up handing you an explicit formula for the price of a bond, which almost never happens by chance in stochastic modelling.

Darrell Duffie and Rui Kan asked the obvious question nobody had properly answered: why? And what exactly is the set of models with this property?

The answer they produced, the affine class, is one of the organising ideas of modern fixed income.

The problem: tractability was a happy accident

Suppose you write down some plausible way for interest rates to move, several factors, some mean reversion, some stochastic volatility, and then ask for the price of a ten-year bond.

In general this is hopeless. You need the expected value of a discounted payoff over every possible path the factors might take, and that expectation is an integral in many dimensions with no closed form. You simulate, you grind, you wait.

Except sometimes you do not. Sometimes an exact formula appears. And it kept appearing in these particular interest rate models, in a very particular shape: the log of the bond price came out as a simple linear expression in the factors.

Nobody knew whether this was luck, or whether there was a rule. If there was a rule, you could design models to satisfy it deliberately, and get tractability by construction rather than by hoping. If you did not know the rule, you were reduced to guessing, writing down a model, and finding out afterwards whether it was solvable.

The key idea via analogy: which shapes tile the floor

Duffie and Kan found the rule, and it is a two-part condition on the factors that drive rates.

Here is the analogy. Ask which shapes of tile can cover a floor with no gaps. Not many can: squares, triangles, hexagons, and a few exotic families. It is not obvious in advance which shapes will work, and most shapes will not. But once someone proves the tiling condition, you stop guessing and start designing.

Duffie and Kan's tiling condition has two parts, and both are about the word affine, which is a slightly grand way of saying "linear, plus a constant." A straight line, essentially.

For a model to be in the tractable class:

1. The drift of each factor must be affine in the factors. Where the factors are heading on average must be a straight-line function of where they currently are. Vasicek's mean reversion qualifies: the pull is proportional to how far the rate is from its target, which is a straight line.
2. The variance of each factor must also be affine in the factors. How much they jiggle, squared, must be a straight-line function of where they are. Vasicek satisfies this trivially (its variance is constant, and a constant is a straight line with zero slope). CIR satisfies it too: its variance is proportional to the level of the rate itself, which is a straight line through the origin. That is exactly the square-root process.

Plus one more link: the short rate itself must be an affine function of the factors.

Get all three, and Duffie and Kan prove the magic happens. The log of every bond price is an affine function of the factors, with coefficients that depend only on maturity and that you get by solving a pair of ordinary differential equations. Ordinary differential equations are, computationally speaking, nothing. Instead of a many-dimensional integral you have a small set of curves you can compute once and reuse.

The class is exactly big enough to be interesting. Vasicek is in it (constant variance). CIR is in it (variance proportional to the level). Multi-factor blends of the two are in it. Models with genuine stochastic volatility, where the jitter of rates is itself a random factor, are in it, which was the paper's headline: you can have stochastic volatility and closed-form bond prices, as long as you keep everything affine.

The paper's other neat contribution is in the title. Duffie and Kan show you can take the abstract, unobservable factors and replace them with actual yields. Pick a handful of maturities, the 3-month, the 2-year, the 10-year, and use those observed yields as the factors themselves. Every other yield on the curve is then a fixed linear combination of them. The model becomes something you can look at: no hidden variables, just yields explaining yields.

Why it mattered

• It turned a coincidence into a design principle. After Duffie and Kan, you do not stumble into tractability. You write down an affine model and you know, before you start, that you will get closed forms. That is the difference between craft and engineering.
• It unified everything that came before. Vasicek, CIR, Hull-White, multi-factor Gaussian models: all revealed as points inside a single family. The paper is the map, and it made the previous twenty years of models legible as a whole.
• It set the agenda for the next decade. Dai and Singleton would classify the affine family and test which corners of it fit the data. Duffee would show where it fails and extend it. Ang and Piazzesi would bolt macroeconomic variables into it. Every one of those papers is working inside the space Duffie and Kan defined.
• The affine trick escaped fixed income entirely. The same structure powers Heston's stochastic volatility model for equities and the reduced-form models used to price credit default swaps. "Affine" is now a general recipe for tractability wherever you need to discount an uncertain payoff.
• Yields as factors made models estimable. Using observable yields rather than latent variables removed a layer of statistical difficulty and made the models something you could actually fit to data without a filtering nightmare.

The honest limitations

• Affine is a straitjacket, and reality is not linear. The whole point is that everything must be a straight line: the drift, the variance, the short rate. Real interest rates almost certainly are not. Rates behave differently near zero than at 8 percent; central banks act nonlinearly; volatility clusters in ways a linear variance function struggles with. Affine models buy tractability by assuming away curvature, and there is a growing body of evidence (quadratic models, regime-switching models, shadow-rate models) that the curvature matters.
• A hidden trade-off between volatility and correlation. This one is subtle and was made precise by Dai and Singleton four years later. If you want a factor to drive stochastic volatility, that factor generally has to stay positive (a variance cannot be negative), and keeping it positive imposes restrictions that limit how the factors are allowed to correlate with each other. So within the affine class you cannot freely have both rich stochastic volatility and rich correlation structure. You trade one against the other, and the data seems to want both.
• The class still forecasts badly. Duffee's 2002 paper showed that standard affine models predict future yields worse than simply assuming yields do not change. The affine structure that gives you clean prices also constrains how risk premia can behave, and that constraint turns out to be empirically wrong.
• Latent factors are hard to interpret. In the abstract version, the factors are unobservable mathematical objects with no economic meaning. Duffie and Kan's yields-as-factors representation helps enormously, but you are still describing rates with rates, not explaining them.
• Estimation is a swamp. Affine models are notorious for likelihood surfaces full of local maxima. Two researchers with the same data and the same model routinely land on different parameter estimates. This got so bad that Joslin, Singleton and Zhu wrote a whole paper (in 2011) whose main contribution was making these models estimable in a reliable, fast way.

The one-line takeaway

Duffie and Kan proved that closed-form bond prices appear precisely when the factors driving rates have straight-line drifts and straight-line variances, defining the affine class that unified every tractable interest rate model before it and framed every one that came after.