MR 14 – The Art of Term Structure Models – Drift

A short-rate model describes how the instantaneous, continuously compounded interest rate moves through time. Once you specify that movement under the risk-neutral measure, the prices of zero coupon bonds, coupon bonds, swaps and interest rate options all follow from it, so the whole term structure and every fixed income derivative valued off it inherit the assumptions you made about the short rate. Building a useful model is therefore less about one grand formula and more about choosing and combining a handful of components, each of which buys a particular feature at a particular cost. Deciding which components a given application needs is a matter of judgment, and that judgment is what practitioners mean when they describe term structure modeling as an art.

Every model in this lesson writes the change in the short rate over a small interval of time as the sum of two parts. The first part is the drift, the expected change per unit of time. The second is a random shock scaled by a volatility parameter. This lesson holds the random part fixed as a normally distributed shock with a constant basis-point volatility and varies only the drift. The sequence runs as follows:

• Model 1: no drift at all, so the rate is equally likely to rise or fall.
• Model 2: a constant drift, read as a risk premium plus any expected trend.
• The Ho-Lee model: a drift that is allowed to differ from one period to the next, chosen so the model reproduces market prices.
• The Vasicek model: a drift that depends on the level of the rate itself, pulling it back toward a long-run value, which is called mean reversion.

One distinction runs through all of it. The drift in a risk-neutral process is not the same as the drift investors truly expect. Under the true (real-world) measure the drift reflects genuine expectations of where rates are heading. Under the risk-neutral measure, which is the one used for pricing, the drift also absorbs a risk premium, because investors demand compensation for bearing interest rate risk. Market prices depend only on the total risk-neutral drift, not on how it splits between expectation and premium, but splitting it is still a useful sanity check on whether a model’s parameters make economic sense.

The simplest possible model lets the short rate wander with no tendency to move up or down. Writing r for the short rate, Model 1 is

Here dr is the change in the rate over a short interval dt measured in years, and σ is the basis-point volatility, quoted as an annual standard deviation of rate changes in rate units rather than as a percentage of the rate. The term dw is a normally distributed shock with mean zero and standard deviation equal to the square root of dt. Calling the shock dw, as though it were itself a change, is simply the notation the field has settled on, and nothing in the calculations below depends on understanding why.

Two consequences follow straight away. A shock with mean zero gives a rate change with mean zero, so Model 1 has no drift. And scaling a shock whose spread is √dt by σ gives a rate change with standard deviation σ√dt, a quantity usually shortened to the standard deviation of the rate.

For pricing, the continuous process is approximated by a binomial tree. In each period the rate either rises by one standard deviation or falls by one standard deviation, each with probability one half. Over two periods the generic Model 1 tree has three terminal nodes:

Rising then falling lands on the same value as falling then rising, so the middle node at date 2 is shared and the tree recombines. Because the step is a constant 0.326%, the numerical example produces the tree in Figure 1.

Checking that the tree matches the process

The tree is a valid approximation if, over one step, its mean change and variance equal those of Equation 14.1. Moving from date 0 to date 1, the change is +σ√dt in the up state and −σ√dt in the down state, so

The first line is just the definition of variance. Taking the square root of σ²dt gives σ√dt, so both the drift and the volatility of the tree coincide with the process.

Matching the first two moments is not the same as matching the distribution. In the continuous process a shock of 0.15 moved the rate to 6.35%, a value the tree cannot reach, since after one month the tree allows only 6.506% or 5.854%. The approximation improves with the number of steps: as more and more two-way branches are stacked, the spread of terminal nodes and their binomial probabilities approach the normal distribution of the process.

The distribution of the short rate at some future horizon, viewed from today, is called its terminal distribution. In Model 1 the rate after T years is normal with mean r₀ and standard deviation σ√T. Models whose terminal distribution is normal are known as normal or Gaussian models. With r₀ = 6.18% and σ = 1.13%, the one-year terminal distribution has a standard deviation of 1.13%, so a range of three standard deviations either side of the mean runs from 2.790% up to 9.570%, with 3.920%, 5.050%, 7.310% and 8.440% marking the intermediate standard deviations. Over a single year, then, a negative rate is practically impossible.

The problem is that the normal distribution has no lower bound. Economically, a substantially negative short rate makes little sense, because a lender who is offered a negative return can simply hold cash and earn zero. (The rate can dip slightly below zero when a deposit or security is safer or more convenient than holding physical cash, but not by much.) Since the standard deviation grows with the square root of the horizon, the chance of a negative rate grows the further ahead you look.

When negative rates matter

Whether that tail ruins the model depends on what it is used for. A coupon bond’s value depends mainly on the average path of rates, and a small amount of probability below zero barely moves that average, so a normal model can still price coupon bonds sensibly. The trouble comes with instruments whose payoff is lopsided toward low rates. Consider a 10-year call option on a long-term coupon bond struck at a yield of 0%. It pays off only if rates fall below zero, so a model that puts too much weight on negative rates will hugely overvalue it. The more a security’s value is concentrated in the low-rate states, the more dangerous a Gaussian model becomes, and the issue is sharpest when the current level of rates is already low.

Change the distribution

The first approach replaces the normal distribution with one that cannot go below zero, or that makes negative values unlikely. A lognormal short rate is the classic example: since the logarithm of the rate is normal, the rate itself is always positive. The catch is that changing the distribution changes the behaviour of the model everywhere, not only near zero. In a lognormal model, for instance, volatility in basis points scales with the level of the rate, which may give volatilities that do not suit the application. So the cure for a problem at low rates reshapes the dynamics at every rate level.

Floor a shadow rate at zero

The second approach, associated with Fischer Black (“Interest Rates as Options”, 1995), keeps whatever distribution is preferred, normal included, and then replaces every negative node in the tree by zero. The rates in the original tree are called shadow rates, and the floored rates are called the observed rates. When the shadow rate falls below zero the observed rate sits at zero, and it can stay there for some time until the shadow rate climbs back into positive territory.

The economic story behind this is attractive. The observed rate is held at or above zero only because investors can always switch into cash. The shadow rate, by contrast, reflects aggregate decisions about saving, investment and consumption, and there is nothing to stop those forces implying a negative equilibrium rate. Of course, flooring changes the distribution of the observed rate relative to the shadow rate, but the change is confined to the region around zero and below it. That is the key advantage over switching distributions: the dynamics at ordinary rate levels are left untouched.

A tree like Figure 1, extended over many periods, can be used to price fixed coupon bonds and so to derive the model’s par, spot and forward rate curves. In the numerical example the starting rate of 6.18% was chosen so that the model’s 10-year semiannually compounded par rate equals the market’s, at 6.086%, using U.S. dollar swap data from February 16, 2001. That one point matches by construction. Almost every other model rate is well away from the market, which is strong evidence that a single parameter on the rate level cannot reproduce even a simple market curve.

Convexity is the only force on the curve

Model 1’s term structure slopes downward: roughly flat at short terms, then falling at longer ones. With zero drift, nothing pushes expected rates up or down, so the decline comes entirely from convexity. Bond prices are convex in rates, so uncertainty about future rates raises expected bond prices and lowers the yields that match them, and the effect grows with maturity. To measure it for a given term, compare the model par yield at the chosen volatility with the par yield the identical model gives when volatility is zero.

Convexity values from Table 14.1 of Tuckman and Serrat; the ratio row is derived.

The 113 basis point input is realistic: options on the data date implied exactly that volatility for the 10-year swap rate. The convexity magnitudes themselves should be treated with caution, because the size of the convexity effect depends on the model, and Model 1 is not a credible description of how rates actually behave.

A flat term structure of volatility

Because every rate in Model 1 is driven by the same shock with the same 113 basis point volatility, the standard deviation of par rate changes is 113 basis points per year at every maturity. Market implied volatilities on the data date were not flat: they were humped, rising over the first few years of term and then declining. A flat model volatility curve misses that shape entirely.

A one-factor model of parallel shifts

The model has a single factor, the short rate. If it rises by 10 semiannually compounded basis points, every rate on the curve rises by 10 basis points, as the closed-form spot rate solution for Model 1 confirms. Every hedge built on Model 1 therefore assumes parallel shifts, with the whole curve moving together by equal amounts.

Market term structures usually slope upward, and one natural explanation is a risk premium: investors want extra return for holding longer bonds, which shows up as an upward drift in the risk-neutral short rate. Model 2 adds a constant drift λ to Model 1:

The expected change per period is now λdt, and the standard deviation is still σ√dt. The section’s parameters are r₀ = 5.138%, λ = 0.229% and σ = 1.10%.

Raising the mean of the terminal distribution pushes it away from zero, so a positive constant drift makes negative risk-neutral rates less likely than they were in Model 1.

Splitting the drift into expectation and premium

A risk-neutral drift of 1.9 basis points a month is consistent with many stories. The market might expect the short rate to climb by 1.9 basis points a month with no premium at all. It might expect a rise of 1 basis point and require a premium of 0.9 basis points. Or it might expect the rate to fall by 0.1 basis points while demanding a premium of 2 basis points. All three produce identical prices.

The Model 2 tree

The tree simply shifts every node of the Model 1 tree up by the accumulated drift, with probabilities of one half on each branch. It still recombines, and its one-step mean and variance match Equation 14.5.

With the example parameters, the date 1 nodes are 5.4746% and 4.8395%, and the date 2 nodes are 5.8113%, 5.1762% and 4.5411%. Each is the Model 1 style spread of 0.3175% per step around a centre that rises by 0.0191% a month.

How well does Model 2 fit?

Here r₀ and λ are calibrated to the 2-year and 10-year par swap rates, and σ is set at the average implied volatility of those two par rates. The model curve tracks the market much more closely than Model 1 did. The weak point is the size of λ. Read as a pure risk premium, a drift of 0.229% against a volatility of 110 basis points implies a Sharpe ratio of about 0.229 / 1.10 = 0.21, which is high for interest rate risk. Read partly as an expected rise in rates, it asks us to believe that rates keep rising steadily even at the long end of the curve, which is hard to justify. Both difficulties point to the same diagnosis: the model is still too rigid to explain the curve’s shape in an economically sensible way. Forcing it through the 2-year and 10-year points is probably why the model’s 30-year par rate overshoots the market by about 25 basis points.

A constant drift does not alter the features that come from the random part. Model 2 still has a flat term structure of volatility, convexity effects of a similar size, and parallel shifts. Models 1 and 2 are both equilibrium models, because no attempt is made to make them reproduce the initial term structure exactly.

The Ho-Lee model generalises Model 2 by letting the drift take a different value in each period. Writing λ_t for the drift at time t,

The annualised drift might be -20 basis points in the first month, 20 basis points in the second month, and so on. A drift that changes from date to date in this way is a time-dependent drift. Exactly as with a constant drift, each period’s value is some mix of risk premium and expected rate change.

In the tree, λ₁ moves the date 1 nodes and λ₁ + λ₂ accumulates into the date 2 nodes:

Calibrating the drifts to market prices

Each λ is a free parameter, so the drifts can be chosen, one maturity at a time, to reproduce market prices of bonds and other fixed cash flow instruments. With monthly steps the procedure runs as follows:

1. Set r₀ equal to the market’s one-month rate.
2. Search for λ₁ so that the two-month spot rate produced by the tree equals the market two-month spot rate.
3. Holding λ₁ fixed, search for λ₂ so that the tree’s three-month spot rate equals the market’s.
4. Repeat for each later date until the tree is complete.

Every input rate is matched by construction, which is what makes Ho-Lee an arbitrage-free model. What the fitting does not change is just as important. As in Model 2, the drift does not touch the random part, so the Ho-Lee model keeps the flat term structure of volatility and the parallel-shift factor structure of Model 1.

Whether a model should reproduce today’s market prices exactly is the central question when choosing between an arbitrage-free model, such as Ho-Lee, and an equilibrium model, such as Model 2 or Vasicek with a constant long-run level. The honest answer is that it depends on the job the model is built to do.

Use 1: quoting illiquid securities

Suppose a client asks a swap desk for a rate on a swap maturing in three years and four months, while the only liquid quotes are for three-year and four-year swaps, or perhaps only two-year and five-year swaps. The desk can price the odd maturity with an arbitrage-free model fitted to the liquid quotes, using the model essentially as an interpolation tool. This can beat curve-fitting methods such as linear or quadratic interpolation. In a fitted model the intermediate rate reflects the expectations and risk premium embedded in the neighbouring swaps plus the convexity that follows from the volatility assumption, which is a financial argument. In a purely mathematical fit the answer is largely dictated by the functional form chosen, which has no economic justification. The advantage holds only if the model’s assumptions are sound, however. A poor volatility assumption, for example, gives a poor convexity estimate, and the arbitrage-free model may then do worse than a simpler technique.

Use 2: pricing and hedging derivatives

Market makers and proprietary traders often take underlying prices as given. A dealer in options on a 10-year bond typically assumes the bond itself is fairly priced, leaving any view on the bond itself to a separate analysis. Since an arbitrage-free model matches many traded prices by construction, it is the natural tool for valuing derivatives relative to their underlyings.

Two warnings about fitting

The case for fitting rests on the idea that market prices contain a great deal of information about future rates. That is reasonable, but it comes with two caveats.

• Calibration cannot rescue a poor model. If parallel shifts are a bad description of reality for the application, bolting a time-dependent drift onto a parallel-shift model so it matches today’s prices leaves it just as unsuitable as before.
• Market prices may not be fair with respect to rates. Supply and demand imbalances, taxes and liquidity can distort particular securities, classes of securities or maturity ranges. A fitted model wrongly attributes these distortions to the rate process.

Picture a large bank dumping its holdings of bonds or swaps with about seven years to maturity. Prices around that maturity fall and yields rise. It would be a mistake to conclude that expected rates seven years ahead have gone up. The seven-year securities are fair in a narrow sense, because the liquidity pressure of the moment requires lower prices, but they are not fair relative to the expected path of rates and the market risk premium. That is exactly why investors and traders might buy them cheaply and hold them until the liquidity event has passed, hoping to sell at a profit.

The same point can be put in terms of the drift. A risk-neutral drift has only two sources: expectations and the risk premium. A fitted model that uses one drift between years 15 and 16 and a quite different drift between years 16 and 17 implicitly claims either that today’s expectation of the one-year rate 15 years out differs from that 16 years out, or that the risk premium changes in a specific way between those two years. Neither claim is very believable, so a fitted drift that jumps sharply from year to year is most likely absorbing effects that have nothing to do with interest rates.

When equilibrium models are required

If the purpose is to judge bonds or swaps against one another, taking a large set of their prices as given defeats the purpose, because an arbitrage-free model concludes by construction that all those prices are fair relative to each other. Investors choosing among securities, market makers seeking value in their choice of hedges, and traders hunting temporary mispricings therefore need equilibrium models.

In practice the boundary is blurred. A model might use a deterministic drift for the first few years to reflect near-term rate forecasts and a constant drift after that. Another might accept the market’s 2-, 5-, 10- and 30-year rates as inputs, treating the most liquid points as fair while leaving the model free to value everything else. Blending the two approaches well is a large part of the modeller’s craft.

If the economy gravitates toward an equilibrium set by fundamentals such as the productivity of capital and long-run monetary policy, short-term rates should show mean reversion. When the rate is above its long-run level the drift is negative and pulls it down; when it is below, the drift is positive and pushes it up. Beyond being a reasonable description of short rates, mean reversion lets a model capture several features of term structure behaviour in an intuitive way. The Vasicek model (O. Vasicek, 1977, the paper that launched the literature on short-rate models) writes the risk-neutral process as

Here θ is the long-run value, or central tendency, of the short rate under the risk-neutral process, and the positive constant k is the speed of mean reversion. The drift is proportional to the gap between the current rate and θ, so the further the rate strays, the harder it is pulled back.

A word of caution: mean reversion is a strong assumption. Long histories of rates from stable markets may look mean reverting precisely because no catastrophe occurred, which is why the long history exists at all. Hyperinflation destroys a currency and its interest rates, and the data series simply stops. As the sharpest critics put it, rates mean revert until they do not.

Separating expectations from the risk premium

As with every risk-neutral process, the drift in Equation 14.8 mixes expectations and premium, and prices do not care about the mix. To judge whether parameters are sensible, though, assume the true process reverts to a long-run level r_∞, and that the risk premium enters as a constant drift λ, as in Model 2. Then

This matches Equation 14.8 exactly when

Many different pairs of r_∞ and λ give the same θ, and hence the same risk-neutral process and the same market prices. That is the formal version of the point that prices cannot tell expectations and premium apart.

A θ of 15.339% looks startling, but it is a risk-neutral level: a premium of 0.229% divided by a slow reversion speed of 0.025 adds 9.160% to the true long-run rate of 6.179%.

The obvious way to build a Vasicek tree is to apply the expected change plus or minus one standard deviation at every node. The first step from 5.121% works out as

Repeating the same rule from each date 1 node gives four date 2 values. From 5.5060% the drift is 0.025 × (15.339% – 5.5060%) / 12 = 0.0205%, producing 5.8902% and 5.1628%. From 4.7786% the drift is larger, 0.025 × (15.339% – 4.7786%) / 12 = 0.0220%, producing 5.1643% and 4.4369%.

Why the naive tree does not recombine

The volatility components of the two mixed paths offset each other exactly, as in Model 1, but the drift components do not. Since 4.7786% lies further below θ than 5.5060% does, the drift at the lower node is bigger, so the up move from 4.7786% lands at 5.1643%, slightly above the 5.1628% reached by the down move from 5.5060%. The tree branches into four, then eight nodes, and so on, which quickly becomes unmanageable.

Forcing recombination

Many schemes produce a recombining Vasicek tree; one simple scheme works as follows. Keep the first step as it is. Then fix the centre node at date 2 to be the expected value of the process after two steps, using the fact that volatility adds nothing to the expectation:

The centre node is set at 5.1635%. The unknowns are now the top node r_uu with its up probability p, and the bottom node r_dd with probability q of moving up to the centre. From 5.5060% the process requires an expected rate and standard deviation of

The two branches out of 5.5060% must therefore satisfy a mean condition and a standard deviation condition:

In the Vasicek model the gap between the current rate and its goal θ shrinks exponentially at rate k. That one fact delivers closed forms for the mean of the terminal distribution and for the half-life of the factor. The expected short rate after T years is

The expectation is a weighted average of today’s rate and the long-run value, and the weight on today’s rate decays exponentially at a speed set by k.

The half-life is a far more intuitive gauge than k itself. A rate with a half-life of 27.73 years closes half of its distance to θ in that time, so its expected value 30 years ahead sits roughly midway between 5.121% and 15.339%. Larger values of k mean shorter half-lives and faster reversion.

Why volatility grows more slowly than the square root of time

The volatility parameter of 126 basis points is quoted per year, but with mean reversion that does not mean the terminal standard deviation scales with the square root of the horizon. That scaling holds only without mean reversion. With it, every shock is partly undone by the pull toward θ, so the terminal standard deviation rises more slowly, reaching only 353 basis points at 10 years against 398 basis points without reversion.

For the curve fit, r₀ and θ were calibrated to the 2-year and 10-year market par rates, so the Vasicek rate curves look qualitatively like those of Model 2. The reversion speed could have been used to improve the fit further, but it was reserved for shaping volatility instead. The conclusion is that, as calibrated, the model is probably too inflexible to produce the variety of term structures seen in practice.

A downward-sloping term structure of volatility

Mean reversion changes the volatility picture dramatically. With k and σ chosen to match the implied volatilities at 10 and 30 years, the model’s par rate volatilities fall steadily with term. It matches the market at those two maturities but overstates volatility at shorter ones. That is an improvement on the flat structure of Model 1, yet the model slopes downward everywhere and so cannot capture the hump in market volatility.

Smaller convexity at long maturities

Lower volatility for long par rates means less convexity. Compare the convexity effects of the two models side by side:

Convexity values from Tables 14.1 and 14.2 of Tuckman and Serrat; the difference column is derived.

Up to 10 years the Vasicek convexity is slightly larger, because its volatility of 126 basis points exceeds 113 and the reversion is slow enough to matter little over those horizons. By 30 years the damping of volatility has taken over, and the Vasicek effect of about 75 basis points is far below the roughly 135 basis points of the model without reversion.

Not a parallel-shift model

Raising the short rate by 10 basis points lifts very short spot rates by about 10 basis points, but longer rates by less; the 30-year spot rate rises by only about 7 basis points. So, unlike Models 1 and 2 and Ho-Lee, the Vasicek model does not produce parallel shifts, and its single factor has a downward-sloping shape.

Mean reversion as the lifespan of economic news

A helpful reading treats each move in the short rate as the arrival of some economic news. Because the rate always ends up back at the same long-run goal whatever happens, the effect of any shock must fade over time. Long-lived news alters the market’s outlook for the economy many years ahead, as with a technological innovation that lifts productivity. Short-lived news alters the outlook only for the near term, as with weak holiday retail sales blamed on unusually cold weather. On this reading, the reversion parameter measures how long news lasts: a small k (a long half-life) means news lingers for many years, while a large k (a short half-life) means it fades quickly. Real news comes in both kinds, which is what multi-factor models are designed to capture.

The same lens explains the shape of volatility and of the factor. Every rate on the curve blends current conditions, measured by the short rate, with long-run conditions, measured by θ. Without mean reversion only current conditions matter, so a shock to the short rate moves all rates equally, yielding parallel shifts and flat volatility. With mean reversion, short maturities are dominated by current conditions and long maturities by long-run conditions. A shock to the short rate therefore moves short rates more than long rates, which is why both the volatility curve and the factor slope downward.