Multi-Factor Interest Rate Models: Beyond One Dimension

One-factor interest rate models have a fundamental problem. They assume that a single number, the spot interest rate, drives the entire yield curve. That means all rates of all maturities move together in lockstep. If the spot rate goes up by 1%, every other rate adjusts accordingly. The yield curve can shift up and down, but it cannot twist or tilt independently at different maturities.

In the real world, the yield curve does twist. Short rates can go up while long rates go down. The spread between the two-year and ten-year rate changes all the time. For any product whose payoff depends on this spread, a one-factor model is useless.

Chapter 35 of Wilmott’s book introduces multi-factor models that fix this problem by adding a second (or third, or more) source of randomness.

The Two-Factor Framework

The idea is straightforward. Instead of just modeling the spot rate r, we add a second variable l. This could be a long-term interest rate, the volatility of the spot rate, or some other quantity. Both variables follow their own random walks:

• r has drift u, volatility w, and is driven by random shock dX_1
• l has drift p, volatility q, and is driven by random shock dX_2

The two shocks can be correlated, with correlation coefficient rho. All the coefficients can depend on r, l, and time t.

Since we now have two sources of randomness, we need two hedging instruments (in addition to the contract being priced) to eliminate all risk. So we build a portfolio with one bond, short some amount of a second bond, and short some amount of a third bond. By choosing the right amounts, we kill both random terms.

The resulting pricing equation is a natural extension of what we had before. It has partial derivatives with respect to both r and l, a cross-derivative term (reflecting the correlation), and two market prices of risk, one for each factor. The model is defined by choosing the volatilities w and q, the correlation rho, and the risk-adjusted drifts.

Why Use the Long Rate as the Second Factor?

Wilmott singles out one particularly natural choice for the second factor: the yield on a consol bond, a fixed-coupon bond with infinite maturity. The consol yield serves as a measure of the “long rate.”

The advantage of using a consol yield is that the consol bond is a traded instrument. Its value must satisfy the pricing equation. This means we can derive an explicit expression for the market price of risk associated with the long rate, instead of having to specify it as an arbitrary function.

Recall that in single-factor models, we always had one unknown function, the market price of risk, that we had to estimate or calibrate. With two untraded factors, we would have two unknowns. But if the second factor is a traded quantity like the consol yield, one of those unknowns disappears. That is a big simplification.

Another option is to model the spread between the long and short rate. In the US, there is evidence that the spot rate and the spread are uncorrelated. If that is true, you can eliminate both the market price of risk for the spread and the correlation coefficient, leaving only three things to model: volatilities of the spot rate and the spread, plus the risk-adjusted spot rate drift.

But there is a catch. The long rate and the short rate are not truly independent. In a one-factor affine model, the long rate is just a function of the short rate. Adding a second factor helps, but there are still internal consistency requirements. You cannot model both rates with completely arbitrary dynamics. The volatility of the long rate is constrained by the relationship between the two.

Popular Two-Factor Models

Wilmott reviews several popular models that fall within this framework.

Brennan and Schwartz (1982) model a short and a long rate with lognormal volatility. No closed-form solutions, and worse, rates can blow up to infinity for certain parameters.

Fong and Vasicek (1991) model the spot rate and its volatility. The second factor captures randomness in volatility itself. Hard to calibrate since volatility is unobservable, but also hard to disprove.

Longstaff and Schwartz (1992) use two abstract factors whose linear combination gives the spot rate. Simple products have explicit solutions.

Hull and White extend their one-factor model. The spot rate mean-reverts to a stochastic level, fitted to the initial yield curve. Closed-form solutions exist.

The general affine model covers many of these as special cases. With linear drifts and square-root volatilities, zero-coupon bond prices take a clean exponential-affine form.

Market Price of Risk as a Factor

Here is a creative idea. What if the second factor is the market price of risk itself? Instead of modeling r and some arbitrary l, model r and the market price of risk lambda.

The advantage: if lambda is your second factor, you already know the market price of risk for r (it is lambda!). The only remaining unknown is the market price of “market price of risk risk.” Yes, that is “market price of risk” squared. Not a typo. You have fewer arbitrary functions in your model, which is a good thing if you are suspicious of fudge factors.

There is empirical evidence (which we will see in Chapter 36) that the market price of spot rate risk is highly random, so this modeling choice makes sense from a data perspective. And since the yield curve slope at the short end is related to lambda, modeling r and lambda is equivalent to modeling the yield curve’s behavior near the short end.

Phase Planes: Checking Your Model

Wilmott introduces a beautiful technique from dynamical systems for sanity-checking two-factor models. If you strip away the randomness and look at the deterministic dynamics of your two factors, you get a system of ordinary differential equations. You can plot these in a phase plane, a picture showing how r and l evolve together over time.

The key features to look for are singularities, points where both drift rates are zero. The type of singularity tells you about the model’s long-term behavior.

A stable node draws all nearby trajectories toward it. This means the interest rates have a natural resting point, which is realistic. This is what you get with the Fong and Vasicek model.

An unstable node or saddle point pushes trajectories away. This could mean rates heading off to infinity, which is definitely not realistic.

The phase plane for Brennan and Schwartz reveals the problem mentioned earlier: for certain parameters, rates escape to infinity. The Hull and White model, by contrast, has a “nice” deterministic phase plane with well-behaved trajectories.

When you add randomness back, the trajectories become jagged and unpredictable, but they still tend to follow the general pattern of the deterministic flow. The phase plane tells you where rates want to go on average.

The Yield Curve Swap

The yield curve swap is the poster child for why you need two-factor models. In this contract, one party pays a floating rate based on one part of the yield curve (say three-month LIBOR), and the other pays a floating rate based on a different part (say the two-year swap rate).

The net cashflow depends on the spread between these two rates. In a one-factor model, these rates are perfectly correlated, so the spread has zero volatility. The contract would be trivially priced. But in reality, the spread fluctuates a lot, and that is the whole point of the contract.

You can either model the spread directly as a Normal variable (simpler, more robust) or use a full two-factor model (more consistent but harder and more model-dependent).

General Multi-Factor Theory

The framework extends to N factors. With N factors you need N hedging instruments plus the contract. Tractable solutions exist in the affine family: multi-factor Vasicek and multi-factor CIR both give zero-coupon bond prices that are exponentials of linear functions of the factors. Yields are also linear in the factors. But as Wilmott notes, “It’s hard enough accurately modeling one factor, so you can imagine what a task it is when you have two or more.”

Key Takeaways

One-factor models are too restrictive for products that depend on the shape of the yield curve. Two-factor models allow the yield curve to twist and tilt, capturing the spread between short and long rates. The choice of second factor matters. Using a traded quantity like the consol yield eliminates one market price of risk. Popular models like Brennan-Schwartz, Fong-Vasicek, Longstaff-Schwartz, and Hull-White each make different trade-offs between realism and tractability. Phase-plane analysis provides a visual sanity check on whether your model can blow up. And for practical purposes, some products like yield curve swaps simply cannot be priced with one factor at all.

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