Yield Curve Fitting: Making Models Match Reality
In the last chapter, we saw one-factor models for interest rates. You pick a model, choose some parameters, and out comes a theoretical yield curve. But here is the problem: that theoretical yield curve almost certainly does not match the actual yield curve you see in the market. And if your model gives wrong prices for plain vanilla bonds, how can you trust it to price anything more complex?
Chapter 31 tackles this head-on. The solution is yield curve fitting (also called calibration): tweaking parameters in your model until its output matches market prices. The chapter explains how to do it, why you might want to, and then makes a surprisingly strong argument for why it is conceptually problematic. This is one of those places where Wilmott shows both the practitioner’s necessity and the theorist’s discomfort.
Why Fitting Matters
The building blocks of bond pricing are delta hedging and no arbitrage. To hedge an interest rate derivative, you buy and sell bonds. You must trade at market prices because those are the only real prices. Your model generates theoretical bond prices from the spot rate model. If those theoretical prices are far from market prices, your hedging strategy collapses.
You cannot buy a bond at a theoretical price. You buy at the market price. So if your model says a 10-year bond is worth 95 and the market says 92, your hedge ratios are wrong. Your P&L will drift. The model becomes useless for practical purposes.
Yield curve fitting fixes this by forcing the model to produce the correct prices for liquid instruments like zero-coupon bonds and swaps. Once the simple instruments are priced correctly, you can have some confidence that the model’s prices for more complex instruments are at least in the right neighborhood.
Ho and Lee: The Simplest Fitting
The Ho and Lee model has the risk-neutral spot rate following:
dr = eta(t) dt + c dX
The volatility c is constant. The drift eta is a function of time, and this is the knob you turn to fit the yield curve.
The zero-coupon bond price has a known analytical form involving eta. To match market prices, you set the theoretical price equal to the market price for each maturity and solve for eta(t). The answer involves differentiating the log of market discount factors twice with respect to maturity.
In practical terms, you look at today’s yield curve, compute some derivatives, and out comes the function eta*(t) that makes everything match. It is clean, explicit, and easy to implement.
The function eta*(t) at the short end is directly related to the slope of the yield curve. The slope of eta*(t) at the short end is related to the curvature of the yield curve. These relationships are exact and tell you a lot about what the fitted parameters look like.
Hull and White: Extended Vasicek
The Ho and Lee model has no mean reversion, which is a significant weakness. The Hull and White model (also called the extended Vasicek model) fixes this:
dr = (eta(t) - gamma*r) dt + c dX
Now there is mean reversion with speed gamma. The parameters gamma and c can be estimated from historical data. The time-dependent parameter eta(t) is then chosen to fit the yield curve, just like in Ho and Lee but with mean reversion in the mix.
The fitting procedure is slightly more involved. You end up with an integral equation for eta*(t), but differentiating twice with respect to maturity gives an explicit formula. The result is similar in spirit to Ho and Lee, just with exponential decay terms from the mean reversion.
The Case Against Fitting
This is where the chapter gets really interesting. Wilmott lays out the argument against yield curve fitting, and it is compelling.
Here is the core problem. Suppose you fit your model today at time t*. You get the function eta*(t) that makes everything match perfectly. Now come back a week later. Refit the model to the new market yield curve. What do you find?
If the model were correct, the function eta*(t) should be essentially unchanged. The old part should have just dropped off the short end, and the rest should remain the same. This is because the model’s own dynamics should predict how the yield curve evolves.
This never happens. When you refit, the new eta*(t) looks completely different. It is essentially just a translation of the old function in time. The shape repeats because the yield curve has not fundamentally changed, but the model predicted it would change. This proves that the model is wrong.
Wilmott walks through why this happens with a careful analysis. The slope of a typical yield curve is large and positive at the short end (short rates are usually lower than long rates). Then the curve flattens out, which means it has large negative curvature.
These properties of the yield curve translate directly into properties of the fitted function. A large slope means eta*(t) starts very high. A large negative curvature means eta*(t) has a steep negative slope at the short end. The typical fitted function looks like it starts high, drops quickly, and then oscillates around some lower level. The oscillations are often just numerical noise.
If the model were correct, a few months later you would see the yield curve flatten dramatically (because eta was supposed to drop). But yield curves do not flatten that fast. They tend to maintain their shape. So when you refit, you get essentially the same high-start, steep-drop pattern, just shifted forward in time.
This is a fundamental inconsistency. The model tells you one thing about the future (rates should change to match eta), but reality does something completely different (rates stay in the same general pattern).
The Philosophical Tension
So yield curve fitting is inconsistent. But can you avoid it? Not really, if you want to hedge. You need your model to give you the right prices for the instruments you are hedging with. If theoretical and market prices diverge, your hedges are wrong.
Wilmott summarizes the tension beautifully. From a practical perspective, you must calibrate because you cannot hedge with mispriced instruments. From a modeling perspective, calibration is dangerous because it hides the model’s deficiencies. You are essentially papering over the cracks.
The deeper issue is that no one-factor model can capture the high slope and curvature that is typical of real yield curves. The models “may” give reasonable results when the yield curve is fairly flat, but that is not the usual case.
Other Models
The chapter briefly mentions two other popular models:
Black, Derman and Toy (BDT): The risk-neutral spot rate follows a lognormal process with two time-dependent functions. This allows fitting both zero-coupon bond prices and their volatilities. There are no explicit analytical solutions, but numerical fitting is straightforward.
Black and Karasinski: A more general version of BDT with three time-dependent parameters, allowing even more flexibility in fitting. These models are popular because the fitting can be done with simple numerical schemes like binomial trees.
The same criticism applies to all of them. Fitting is inconsistent regardless of which one-factor model you use. The refitted parameters change too much over time, proving the model is wrong.
A Thought on Speculation
Wilmott ends with an intriguing aside. What if you are not hedging but speculating? In that case, you do not care about matching market prices because you are not using bonds as hedging instruments. You want to know the real expected present value of cashflows, not the risk-neutral value.
For speculation, you would use the real drift u (not the risk-neutral drift u - lambda*w). You would not fit the yield curve at all. Instead, you would model the real behavior of interest rates using historical data. This gives a “value” that represents the expected return under the actual probability measure.
This is a refreshingly honest perspective. Most textbooks pretend that risk-neutral pricing is the only game in town. Wilmott acknowledges that it is the right approach for hedging, but if your goal is different (speculation, risk management, scenario analysis), you might want a different approach entirely.
Key Takeaways
Yield curve fitting is a necessary evil. You cannot hedge fixed-income derivatives without it, but it introduces fundamental inconsistencies into your model. The fitted parameters change dramatically when you recalibrate, proving that the underlying one-factor model is wrong.
The practical advice: fit the model, use it for pricing and hedging, but never believe that the fitted parameters have any real meaning beyond “making today’s prices come out right.” They are a snapshot, not a prediction.
The deeper lesson: if you are uncomfortable with the inconsistency (and you should be), the fix is not better fitting. The fix is a better model. Multi-factor models, HJM models, and other approaches try to address this. But even those come with their own problems, as we will see in later chapters.
For now, the one-factor fitted model is a workhorse. It works well enough in practice, people make money using it, and that is the ultimate test. But keep your eyes open about its limitations.
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