MR 16 – The Vasicek and Gauss+ Models

The Vasicek model, published by Oldrich Vasicek in 1977, was the first short-rate term structure model and it is still the cleanest place to learn how such models behave. Its central assumption is mean reversion: the short-term rate is pulled toward a long-run level, so a rate sitting below that level is expected to drift upward and a rate sitting above it is expected to drift downward. Under risk-neutral probabilities, the rate obeys the stochastic differential equation below, where dr is the move in r over a tiny interval.

The change has two pieces. The first is the drift, k(θ − r)dt, which equals the mean reversion parameter k multiplied by the gap between the long-run value θ and today’s rate. A wider gap or a larger k produces a stronger pull. Every quantity in the model is quoted per year, and the dt factor scales the annual drift down to whatever slice of time is actually passing, so a month uses dt = 1/12.

The second piece is the random shock σdw. Here dw is a normally distributed increment with a mean of zero and a standard deviation of √dt, so the shock itself is normal with a mean of zero and a standard deviation of σ√dt.

Bond prices may embed a risk premium, and a constant risk premium looks exactly like an extra drift in the short-term rate. Suppose that under real-world (true) probabilities the short-term rate reverts to r_∞, and that investors demand a known premium of λ per year. Adding that premium as a drift term and regrouping gives the following.

Comparing the regrouped expression with Equation 16.1 shows that the risk-neutral long-run level θ is the real-world long-run level plus the premium divided by k. The consequence is a basic identification problem. Prices pin down only θ, and an endless set of pairs of r_∞ and λ produce the same θ, so the same risk-neutral process and the same prices. Observing prices alone cannot tell an analyst how much of a high θ reflects genuine rate expectations and how much reflects compensation for bearing rate risk. Separating the two always needs an extra assumption, which is a theme that returns in the Gauss+ macro trading discussion.

So a very high risk-neutral long-run rate in a slowly reverting model need not imply that anyone actually expects rates to reach that level.

A large part of the Vasicek model’s appeal is that most of its rates and prices come out in closed form, whereas more complex securities in richer models need numerical tools such as binomial trees. Four results are especially useful. In each, t is measured in years from today and r₀ is the current short-term rate.

The expected rate is a weighted average of where the rate starts and where it is heading, with the weight on r₀ decaying as e^−kt. At t = 0 it equals r₀, and as t grows without limit it approaches θ. The variance starts at zero and climbs toward a ceiling of σ²/(2k). Without mean reversion the standard deviation would instead be σ√t, growing forever, so reversion narrows the range of likely future rates at every horizon.

The forward rate equals the expected short-term rate minus a convexity term, since 1 + e^−2kt − 2e^−kt is simply (1 − e^−kt)². That convexity term grows with t and with σ, pulling long forwards below expectations. The spot rate has the same structure, with averaged versions of the decay factors, because a spot rate is an average of forward rates out to its term. A useful corollary follows: whenever the forward curve lies above the spot curve, spot rates are rising with term.

A mean reversion parameter such as k = 0.0165 is hard to interpret on its own, so practitioners often convert it into a half-life: the time it takes for half of any shock to the short-term rate to disappear from expectations. Equation 16.4 shows that the influence of r₀ on the expected future rate decays by the factor e^−kt. A shock to r₀ therefore fades at that same rate, and the half-life h is the time at which the factor reaches one half.

With k = 0.0165, h = 0.693147 / 0.0165 = 42.01 years, just over 42 years. That number has an intuitive reading: a shock to today’s short-term rate keeps shaping rate expectations for decades, and the expected path needs decades to cover even half the distance from today’s level to θ.

The same conversion is applied to every factor in the Gauss+ model. Its estimated parameters, listed with their half-lives below, show a clear ordering from fast to slow.

Speeds and half-lives in years from Tuckman and Serrat’s estimates on US Treasury zero coupon yields, January 2014 to January 2022; the months column is derived.

Differentiating Equation 16.6 with respect to r₀ shows that every forward rate moves by e^−kt basis points for each basis point move in the short-term rate. Because the short-term rate is the only source of randomness in the model, the volatility of the forward rate of term t is just σe^−kt, and the curve of factor sensitivities across terms has exactly the same shape as the curve of forward rate volatilities. Both decline smoothly with term.

That declining shape matches one empirical regularity: at longer terms, observed volatility does fall as maturity increases. It fails at the short end, however. Actual short-term rate volatility is far lower than the 95 basis points the model assigns to it, because central banks hold short-term rates pegged to a policy target. A one-factor mean-reverting model cannot reproduce both a low volatility at the very short end and a declining pattern further out, and this is the gap the Gauss+ model is designed to close.

The model also lets an analyst split each forward rate into three parts by using Equation 16.3 to replace θ. Writing the expectation under real-world probabilities gives the decomposition below, which follows directly from Equations 16.3 and 16.6.

With r₀ = 2%, λ = 0.125% and r_∞ = 3.424%, the expectations component rises only mildly over the coming years, because the real-world target is not far above today’s rate. Out to about 10 years, forward rates climb much faster than expectations because the 12.5 basis point annual premium keeps accumulating. Beyond that, the convexity component, which enters negatively, grows quickly enough to offset expectations and the risk premium together and makes forward rates turn down with term.

The Vasicek model still has a few practical roles, and its simplicity is the main reason. A single factor explains a large share of the variability of the term structure, particularly at longer maturities. The parameters r₀, k and θ can be calibrated together to approximate the curve’s shape together with the pattern of factor sensitivities across maturities, and σ can be set to match an implied option volatility at one chosen point on the curve.

Where the model can be used

A natural application is long bonds with a call feature that starts after an intermediate lockout period. Here the model can match noncallable bond prices for maturities from 10 to 30 years and also the volatility most relevant to the call option, that of 10-year rates. Sensitivities are found by shifting r₀, repricing, and turning the price changes into DV01s or durations. When a bond of one maturity is hedged with a bond of another, the hedge works only as well as the model’s shape of factor sensitivities across terms.

A second use is trading the long end. Resetting r₀ daily so that the model and market 10-year rates agree, a trader can treat gaps between market and model prices of longer bonds as rich or cheap signals, hedged with model ratios. Success depends on whether the model captures the steady-state curve shape and on how reliable its sensitivity curve is.

Why the model is rarely used on its own

Three weaknesses limit wider use. One factor cannot reproduce the many curve shapes seen in markets, so calibrating to one point leaves every other point to the model. A second factor adds considerable explanatory power, and shorter and intermediate bonds cannot be hedged well without it. Finally, the model forces factor sensitivities and volatilities to decline monotonically with term, whereas empirical volatilities typically rise quickly with term and then flatten or decline. So the model cannot price options spread across the curve consistently, and it cannot reliably hedge a bond sensitive to one segment of the curve with a bond sensitive to another.

The Gauss+ model is popular among practitioners for proprietary relative value and macro trading and for hedging. Its assumptions are intuitive, and it balances tractability against the need to capture realistic curve dynamics. It has three factors: the short-term rate r, a medium-term factor m and a long-term factor l. In its cascade form, each factor reverts toward the next one down the chain.

The α parameters are speeds of mean reversion, σ_m and σ_l are volatilities, ρ is a correlation and μ is a constant. The two random increments dW¹ and dW² are independent. Given this structure, both m and l can be read as rates.

The economic reading of each factor

The short-term rate reverts quickly toward m, which stands for business cycle and monetary policy conditions. The medium-term factor reverts more slowly toward l, which reflects long-run expectations of inflation and the real rate, themselves shaped by slow trends in demographics and production technology. Finally, l reverts slowest of all toward μ, which, as with θ in the Vasicek model, mixes a long-run expectation of the short-term rate with a risk premium. The estimated speeds should line up with this ordering, and they do.

Equation 16.9 mirrors central bank practice. A central bank such as the Federal Reserve keeps its policy rate pegged at a target over short intervals and moves that target as the business cycle and monetary conditions evolve. In the model, r has no shock over an instant dt but is pushed gradually toward m. In reality the policy rate jumps in discrete steps on meeting dates, while expectations of future rates evolve smoothly; the model approximates the discrete path with a continuous one starting from today’s rate.

The missing volatility term in Equation 16.9 is the most important design choice in Gauss+. Mean-reverting factors with their own shocks generate a term structure of volatility that slopes downward from the shortest term, exactly as in Vasicek. Observed and implied volatility curves instead show a hump: volatility is low at the very short end, because central banks anchor policy rates, then rises to a peak at intermediate or longer terms before easing off. By giving r no direct shock, Gauss+ keeps short-term rate volatility low while the shocked medium- and long-term factors generate the rise and the slow decline further out.

The name follows from the same feature. “Gauss” signals that rates are normally (Gaussian) distributed. In most normal models, the number of factors equals the number of sources of risk. Gauss+ is unusual: describing the state of the world requires knowing three state variables, r, m and l, yet only two random increments, dW¹ and dW², drive the system. The plus sign marks the presence of a state variable that is not itself a source of risk.

Equations 16.9 to 16.12 describe risk-neutral dynamics, which are what pricing needs. With extra assumptions, an implicit risk premium can be identified. The application developed later assumes that only the long-term factor earns a risk premium.

Both the medium- and long-term factors trend toward their targets while fluctuating around those trends. For l the fluctuation over dt is σ_ldW¹. Since dW¹ is normal with mean zero and standard deviation √dt, the instantaneous fluctuation of l has mean zero and volatility σ_l√dt.

The random part of the medium-term factor looks more involved, but it is built so that m has volatility σ_m√dt and correlation ρ with l. Since dW¹ and dW² each have standard deviation √dt and are uncorrelated, the variance of the shock to m is σ_m²(ρ² + 1 − ρ²)dt, which gives the first result.

Only the dW¹ part of the shock to m is shared with l, so the covariance picks up just that piece.

The estimated values are σ_m = 109.2 bps, σ_l = 96.4 bps and ρ = 0.212, so the two shocked factors are only mildly positively correlated.

Calculating factor changes is a matter of evaluating Equations 16.9 to 16.11 over a small interval. Each change has a drift equal to minus the speed times the distance from the target, scaled by dt; the medium- and long-term factors also receive their shocks. Because the short-term rate has no shock, its change over the interval is fully determined once r and m are known.

Translating factor changes into forward rate changes

The estimated model also says how much each forward rate responds to a one basis point change in each factor. Moves in r matter only at the shortest terms. The medium-term factor dominates forwards around two to three years ahead, so it behaves like the market’s view of monetary policy over that horizon. The long-term factor peaks in influence at terms of six to eight years and is the only driver past 10 years. As an illustration, a 1 basis point rise in l lifts the seven-year forward by 0.9 basis points, so the 28.87 basis point change in Example 7 would move that forward by about 26 basis points.

Industry practice for estimating term structure models varies widely, and the method shown here is a simpler alternative to the maximum likelihood methods that dominate the academic literature. Its inputs are daily observations, January 2014 to January 2022, of the fed funds target and of Treasury-derived zero coupon prices from the Gurkaynak, Sack and Wright dataset, which the Federal Reserve makes public.

Data choices and daily fitting

Consistent with the model’s interpretation, r is set each day equal to the fed funds target rate. A general collateral repo rate would fit a Treasury curve more naturally in theory, but repo rates show occasional idiosyncratic jumps that complicate estimation with little offsetting benefit. Once the parameters are estimated, m and l are chosen each day so that model and market agree on two forwards, the one-year rate starting in two years and the one-year rate starting in 10 years, alongside the fed funds target. Those two forwards are therefore fair by construction, and the rest of the curve can be judged against them. Note that m and l are not those forward rates themselves; their relationship to every forward depends on the mean reversion parameters. A larger α_r passes changes in m and l into the curve faster. A larger α_m makes m converge to l faster and lessens its effect on longer yields. A smaller α_l slows the convergence of l to μ and makes its effect on longer yields more similar, or parallel, across terms.

The three estimation stages

Stage one targets the mean reversion parameters. Changes in zero yields of each term are regressed on changes in the two-year and 10-year zero yields. For the three-year yield the estimated loadings are 0.91 (two-year) and 0.22 (10-year); for the 15-year yield they are −0.20 (two-year) and 1.10 (10-year). Because the model’s implied regression coefficients depend only on α_r, α_m and α_l, and not on volatilities, these three can be chosen to match the empirical coefficients as closely as possible. The fitted values match closely and lie within statistical confidence intervals.

Stage two chooses σ_m, σ_l and ρ so that the model’s term structure of yield volatilities, in annual basis points, matches the empirical one; again the fit is excellent. Stage three picks μ to minimize the sum of squared errors between market and fitted yields across every day in the sample.

The speeds follow the expected order: the central bank reaction is fastest, then the medium-term factor, then the long-term factor. Mean reversion dampens the factor shocks, so zero yields end up less volatile than the σ_m and σ_l estimates themselves. A μ above 10% may look high, but l moves toward it extremely slowly and μ includes a risk premium.

Running the daily fit through history recovers a time series for each factor. Although estimation used data from January 2014, the parameters can be applied back to January 2007 to extract factors over January 2007 to January 2022. For readability the long-term factor is shown shifted forward 10 years, which makes it roughly comparable to expectations of the short-term rate a decade out.

Compared with the two-year forward rate, the medium-term factor behaves differently in two regimes. When rates are well above zero, m tracks the two-year forward closely, which supports reading it as the market’s forecast of the policy rate about two years ahead. Near the zero lower bound, however, the forward rate cannot fall much further while m can drop deep into negative territory. In that sense m acts as a forward shadow rate: a forward-looking measure of where short-term rates are expected to go, and one that can be traded explicitly within the model. This is a different idea from the shadow rate in models where it represents what the short-term rate would be today if it were not bounded above zero (see, for example, Wu and Xia, Bauer and Rudebusch, and Kim and Singleton).

The medium-term factor also works as a leading indicator. It climbed sharply from early 2014, roughly two years ahead of the first Federal Reserve hikes, which illustrates that m moves with expectations while r follows only when the central bank acts.

Relative value, in the language of a factor model, means a position with little or no net factor exposure. The textbook version buys one undervalued security that should soon return to model value and sells nearby fair or overvalued ones as a hedge, earning the convergence. In practice individual securities often stay cheap or rich for long periods, and neighbouring securities tend to be cheap or rich together. So most opportunities take a different form: receive forwards that sit above model values in one region, pay forwards that sit below model values in another, expect both to converge quickly, and size the legs so factor exposure is small. Speed of convergence is a central consideration throughout.

Building a fitting error signal

One framework tracks, day by day, the gap between the market nine-year forward and its model counterpart; each daily gap is a fitting error. A signal is then the five-day moving average of the fitting error minus its 40-day moving average. If the demeaned fitting error is positive, the nine-year forward is high relative to the model. If the signal is also negative, the errors have started falling back toward the model, and receiving that forward may be attractive.

Removing factor exposure

On its own, a received nine-year forward is heavily exposed to m and l, so its profit would depend on factor moves. One remedy is to combine several attractive trades and size them so that net factor exposure is minimal. Another is to diversify across mean-reverting trades. Errors at the nine-year and five-year points tend to move together, so paying one and receiving the other is attractive, and the difference between the nine-year and five-year signals turns out to be strongly mean reverting, one of the most important properties of a good relative value trade.

A macro trader deliberately holds factor exposure, the opposite of the relative value approach. Basic versions bet that the level or slope of the curve will move differently from what prices imply. A more ambitious one, which has attracted growing interest, trades the long-run level of the short-term rate. Long-term forward rates combine expectations, a risk premium and convexity, and within Gauss+ these can be separated with the help of one strategic assumption.

Convexity is relatively easy to compute in any term structure model. Separating expectations from the risk premium is much harder, and the academic approaches (for example Adrian, Crump and Moench, Ang and Piazzesi, Cieslak, and Kim and Orphanides) all have shortcomings. The approach here assumes that expectations of the short-term rate stop changing beyond some horizon: a forecaster’s view of the policy rate 15 years out is the same as the view for 20 or 30 years out. So any difference among forward rates beyond that horizon must come from the risk premium and convexity, not from expectations, which lets the premium be backed out of observable rates.

What the estimated risk premium shows

Applying this assumption with the estimated parameters gives a daily series for the risk premium in the 10-year forward rate. A reading of 60 basis points means that 60 basis points of that forward rate are attributed to risk premium on that day. The premium often moves opposite to rates: when rates fall, the curve usually steepens, and the model reads the steepening as a higher premium. This fits the low-inflation decades in which government bonds rallied when risky assets fell: as rates near their floor, bonds hedge those losses less well, so they are worth less and the premium rises.

Once the premium is known, the market’s implied long-run rate follows by taking the 10-year forward, deducting the premium for that term and adding back the convexity for that term, as Example 9 did. This series can be compared with an independent estimate that the Federal Reserve Bank of Cleveland builds from its own real rate and inflation projections. The two track each other quite well, and the gap between the market-implied view of the long-run rate and the economist-generated fundamental view can serve as a basis for outright long or short bond positions.

Both models are Gaussian, mean-reverting and tractable, and both separate a long-run target into an expectation and a premium. They differ in how many factors they use, where randomness enters and therefore which curve shapes and volatility patterns they can reproduce.

A single factor explains much of the variation in long rates, but hedging shorter bonds, valuing options spread across maturities and separating policy expectations from long-run views all need the separate medium- and long-term factors and the unshocked short-term rate that Gauss+ supplies.