How Interest Rates Actually Behave: Empirical Evidence
We have spent several chapters building interest rate models. Vasicek, CIR, Hull and White, Ho and Lee, Black-Derman-Toy. Each one chosen for its nice mathematical properties, clean closed-form solutions, and easy calibration. But here is the uncomfortable question Wilmott asks in Chapter 36: do any of these models actually match what interest rates do in the real world?
The answer is sobering. Most popular models were designed for tractability, not realism. In this chapter, Wilmott takes US spot rate data from 1977 to 1994 and builds a model from the ground up, letting the data speak instead of forcing it into a pretty formula.
The Problem With Popular Models
Look at the standard one-factor spot rate models. They all assume the spot rate follows a stochastic differential equation of the form:
dr = u(r,t) dt + w(r,t) dX
The differences between models come down to the choices for the drift u and volatility w. Vasicek uses constant volatility. CIR uses volatility proportional to the square root of r. Ho and Lee makes things time-dependent. Hull and White adds mean reversion with time-dependent parameters.
All of these choices were made because they lead to zero-coupon bond prices of the nice exponential-affine form: Z = exp(A(t,T) + B(t,T)*r). That form makes calculations easy. But nobody ever checked whether interest rates actually behave this way. Or rather, when people did check, the results were not encouraging.
Step 1: What Does Volatility Really Look Like?
The first question Wilmott tackles is the volatility structure. Many models assume volatility of the form w(r) = c * r^beta, where beta determines the type of model. Beta = 0 gives constant volatility (Vasicek, Ho and Lee). Beta = 1/2 gives square root volatility (CIR). Beta = 1 gives lognormal volatility (BDT).
The famous study by Chan, Karolyi, Longstaff and Sanders (1992) estimated beta = 1.36 for US data. Wilmott does his own analysis: bucket the data by rate level, compute average squared changes per bucket, and plot on a log-log scale. The slope gives 2*beta.
The result: beta is approximately 1. Spot rate volatility is close to lognormal, meaning the relative change dr/r is roughly independent of the rate level. This puts BDT ahead of Vasicek and CIR. Vasicek’s constant volatility is clearly wrong. CIR is better but still underestimates how much volatility increases with rate level.
Step 2: What About the Drift?
Estimating drift from data is statistically much harder. The drift is a smaller effect, buried in noise. Wilmott takes a clever alternative: instead of measuring drift directly, he looks at the steady-state probability density function of the spot rate. If rates revert to some average level, then over time, the distribution of where rates have been should settle into a stable shape.
Why is this useful? Because there is a mathematical relationship (through the Fokker-Planck equation) between the steady-state distribution, the volatility, and the drift. If you know two of these three, you can figure out the third. We already know the volatility. If we can estimate the steady-state distribution from data, we get the drift for free.
From US LIBOR data, Wilmott estimates the steady-state distribution by simply counting how often rates fall into various buckets. The resulting histogram looks like a lognormal curve. He fits it with a lognormal distribution with two parameters: a = 0.4 and a mean rate of about 8%.
The drift that comes out of this analysis is mean-reverting, pulling rates back toward 8%. By using the steady-state distribution to find the drift, you guarantee that your model cannot do anything pathological in the long run, like letting rates grow without bound or go permanently negative.
Step 3: The Market Price of Risk
We now have the real-world volatility w(r) and the real-world drift u(r). But to price bonds, we need the risk-neutral process, which means we need the market price of risk lambda(r).
There is no information about lambda in the spot rate data alone. Lambda connects the real world to the risk-neutral world, and it shows up in the yield curve. Specifically, the slope of the yield curve at the short end is directly related to the risk-adjusted drift (u - lambda*w)/2.
Wilmott computes this slope from one-month and three-month US LIBOR data and plots it against the spot rate. The result is striking and troubling.
For one-factor models, the risk-adjusted drift u - lambda*w should be a single-valued function of r. But the plot shows nothing of the sort. The same spot rate can correspond to wildly different yield curve slopes at different times. This is strong evidence that one-factor models are fundamentally inadequate.
Nevertheless, pressing forward with the one-factor assumption and fitting an average through the cloud of data points, Wilmott finds that lambda(r) = -40 * r^(beta-1) is a reasonable fit for US data.
What the Yield Curve Slope Tells Us
The estimated market price of risk leads to some important conclusions.
First, the risk-adjusted drift is dominated by the lambda*w term, not by u. In other words, the risk-neutral process is very different from the real process. The risk-neutral spot rate mean-reverts to a much higher level than the real 8%. This means the theoretical yield curve generated by the model has a steep slope.
Second, any one-factor model faces an impossible trade-off. If you fit the short end of the yield curve (matching the slope correctly), the long rate comes out way too high. If you fit the long rate, the short end slope is too shallow. You cannot get both right with a single factor.
Wilmott compares three curves: the actual market forward rate curve, the expected future path of the real spot rate (with lambda = 0), and the theoretical forward rate curve when you fit the yield curve slope. The gap between the market curve and the real expected path shows just how much of the yield curve is driven by risk premia rather than expectations about future rates.
BDT Comes Out Ahead
After all this empirical analysis, Wilmott concludes that among popular models, Black-Derman-Toy is the closest to reality for US data. Its lognormal volatility structure matches the data well, and its risk-adjusted drift is a reasonable approximation.
But Wilmott emphasizes that BDT was originally chosen for computational convenience (nice tree structure) and data fitting ability, not because anyone looked at the data and said “this is the right model.” It happens to work well empirically, almost by accident. And the philosophy behind BDT (fitting time-dependent parameters to market data) has its own dangers. Those parameters constantly need refitting, which means the model is always slightly out of date.
Forward Rate Curves on Average
Wilmott goes further with a global test. For a time-homogeneous model, the real-world average of the forward rate curve must satisfy a specific integral equation involving both real and risk-neutral probability distributions. The question is whether a positive risk-neutral steady-state distribution exists that satisfies this equation. For US data, Wilmott could not find one. Another nail in the coffin for one-factor models.
Toward Two-Factor Implied Models
Wilmott closes with a roadmap for two-factor empirical modeling: fit volatilities for both factors from data, exploit the fact that the spot rate and the spread are uncorrelated (at least in the US), extract the market price of risk from the yield curve slope, determine real drifts from the joint steady-state distribution via the two-factor Fokker-Planck equation, and calibrate deterministic dynamics using phase-plane analysis. Building from data helps avoid the pathologies (like rates going to infinity) that plague some popular two-factor models.
Key Takeaways
Real interest rates have near-lognormal volatility (beta around 1), which means popular models like Vasicek (beta = 0) and CIR (beta = 0.5) are off the mark. The drift can be estimated indirectly through the steady-state distribution, giving a mean-reverting process around 8% for US data. The market price of risk, extracted from yield curve slopes, is large and variable, and the data strongly suggests that one-factor models cannot simultaneously match the short and long ends of the yield curve. Among standard models, BDT comes closest to matching empirical evidence, but the honest conclusion is that one-factor models should be used with care. They do not describe reality in any quantitative way. A proper model probably needs at least two factors, and it should be built from data rather than chosen for mathematical convenience.
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