Interest-Rate Models: The Curve You Fit and the Risks You Keep

Interest-rate models serve two related but distinct purposes: describing plausible rate dynamics and valuing contracts consistently with current market prices. A model can fit today’s curve exactly while giving restrictive future co-movements, or capture an economic mean-reversion idea while missing today’s curve.

This reference consolidates short-rate models, calibration trees, multifactor extensions and forward-rate frameworks. Workbook numbers are teaching parameters, and simulated paths are not forecasts of future policy rates.

Start with the object being modelled

A short-rate model describes the instantaneous rate $r_t$. A forward-rate model describes rates for future intervals or the entire instantaneous forward curve. The chosen state variables determine which movements the model can represent directly.

Under a pricing measure, a zero-coupon bond paying one unit at $T$ has value

$$P(t,T)=\mathbb E_{\mathbb Q}\left[ \exp\left(-\int_t^T r_sds\right)\mid\mathcal F_t\right].$$

In words: The bond price is the expected future discount factor under pricing probabilities, given information available now.

The discount factor is random because the future rate path is random. Replacing it with the exponential of the expected integrated rate generally misses convexity.

Historical estimation targets physical dynamics. Pricing calibration targets the distribution required to reproduce relevant market instruments under valuation assumptions. Mean-reversion and volatility parameters need not be identical under the two measures.

Vasicek provides a tractable Gaussian benchmark

The Vasicek model is

$$dr_t=a(b-r_t)dt+\sigma dW_t,$$

where $a>0$ is mean-reversion speed, $b$ the long-run rate and $\sigma$ additive rate volatility. Its conditional mean is $b+(r_t-b)e^{-ah}$, and its conditional variance over horizon $h$ is $\sigma^2(1-e^{-2ah})/(2a)$.

The Gaussian distribution permits negative rates. Whether that is a limitation depends on the intended variable and market setting. The model’s analytic structure is useful for testing pricing and simulation systems.

Zero-coupon prices take exponential-affine form $P(t,T)=A(t,T)e^{-B(t,T)r_t}$, with

$$B(t,T)=\frac{1-e^{-a(T-t)}}a.$$

The coefficient $B$ controls sensitivity to the current short rate. The function $A$ incorporates the remaining parameters and horizon. With constant parameters, the resulting initial term structure has a restricted shape and generally cannot fit an arbitrary observed curve exactly.

A workbook example separates assumptions from outcomes

One retained workbook specifies initial short rate 6%, long-run mean 6.5%, mean-reversion speed 0.15 per year and additive volatility 1.2 percentage points per square root year, with a five-year horizon. Under those illustrative parameters,

$$\mathbb E[r_5]=0.065+(0.06-0.065)e^{-0.15(5)} \approx6.264\%.$$

In words: The expected rate moves from its starting level toward the assumed long-run level over the stated horizon.

The corresponding conditional standard deviation is approximately 1.93 percentage points. The mean is a model expectation given the stated measure and parameters, not an actual five-year rate forecast.

A separate repeated workbook uses initial rate 8%, equilibrium reference 6%, strike 6%, one-million-unit notional and 300 simulation iterations. These are different teaching inputs. Merging the workbooks should preserve that distinction rather than average their parameters or treat the displayed paths as independent empirical studies.

CIR changes behaviour near zero

The Cox–Ingersoll–Ross model replaces additive volatility with square-root volatility:

$$dr_t=a(b-r_t)dt+\sigma\sqrt{r_t}\,dW_t.$$

The diffusion shrinks as the rate approaches zero. With appropriate nonnegative initial conditions, the process remains nonnegative; the usual Feller condition $2ab\geq\sigma^2$ ensures the zero boundary is unattainable from a positive start in the standard setting.

Failure of that condition does not automatically imply negative rates. It changes boundary accessibility. A naive Euler discretisation can nevertheless generate negative numerical states, so simulation needs a scheme consistent with the square-root process.

The retained workbook uses $a=0.15$, $b=0.065$ and CIR diffusion coefficient 0.06. Then $2ab=0.0195$, exceeding $\sigma^2=0.0036$. Its instantaneous volatility at an initial rate of 6% is $0.06\sqrt{0.06}$, about 1.47 percentage points per square root year. That is not the same quantity as a 6% additive rate volatility.

Time-dependent shifts fit today’s curve

A one-factor Hull–White representation is

$$dr_t=[\theta(t)-ar_t]dt+\sigma dW_t.$$

The deterministic function $\theta(t)$ is chosen to fit the initial term structure under the model and convention. Mean reversion and volatility then govern future dynamics. Exact initial curve fit is therefore achieved through a time-dependent calibration function rather than by claiming one constant equilibrium level explains every maturity.

A one-factor Gaussian model has restrictive instantaneous co-movement across maturities: all rate shocks come from one random driver. It can price many instruments efficiently, but it cannot generate independent level and slope shocks at the same instant.

Mean-reversion speed settings in the retained workbook

In the basic one-factor relation, volatility at maturity $T$ scales like $\sigma e^{-aT}$. Higher $a$ concentrates volatility nearer the short end and changes long-maturity option behaviour.

Two factors improve curve flexibility

A two-factor Gaussian model can represent the short rate as a deterministic shift plus two mean-reverting factors. Different reversion speeds allow one factor to decay quickly and another slowly. Their correlation controls how joint shocks combine.

The retained illustrative parameters include a fast speed 0.5, slow speed 0.03, volatilities 1.0 and 0.7 percentage points, and correlation minus 0.9. These inputs describe a teaching setup, not an estimated universal rate relationship.

Additional factors improve flexibility but increase calibration complexity. Several parameter combinations may fit similar option prices while producing different extrapolated dynamics. Calibration residuals, parameter stability and sensitivity to instrument selection should all be reviewed.

Black–Karasinski instead models a logarithmic short-rate state in its standard positive-rate formulation. Its state transformation changes both boundary behaviour and numerical calibration. A shifted extension or alternative model may be required when the relevant rate can be negative.

Trees convert rate dynamics into backward induction

A calibrated tree specifies nodes, branching probabilities and a time-dependent adjustment that reproduces initial discount factors. At each step, a bond value is the appropriately discounted conditional average of successor values.

The retained tree exercise uses half-year steps and six steps to form a three-year horizon. It checks branching probabilities and uses a shift to fit the curve. Negative probabilities, failure to reproduce input bond prices or dependence on arbitrary truncation are implementation failures requiring resolution.

Exercise rights can be incorporated by comparing continuation and exercise values at each eligible node. Coupon dates, reset dates and exercise dates should align with the tree timeline or receive an explicit interpolation treatment.

HJM imposes no-arbitrage on forward-rate drift

The Heath–Jarrow–Morton framework models the instantaneous forward rate $f(t,T)$. In a one-factor money-market-measure formulation,

$$df(t,T)=\alpha(t,T)dt+\sigma_f(t,T)dW_t,$$

and no arbitrage constrains drift to

$$\alpha(t,T)=\sigma_f(t,T)\int_t^T\sigma_f(t,u)du.$$

The volatility structure therefore determines the permitted drift under that measure. A different numeraire changes the drift representation. HJM is a framework with many possible volatility choices, not one uniquely specified model.

The full curve is naturally a high-dimensional object, although particular volatility structures admit finite-dimensional Markov representations. It would be too strong to say that every HJM implementation must be computationally infinite-dimensional.

Discrete forward-rate models align with quoted instruments

A market-model approach specifies forward rates over discrete accrual intervals. Under each rate’s associated forward measure, a suitable model can have a simple driftless representation; under a common simulation measure, interactions generate nonzero drifts.

Historical teaching models often use legacy interbank forward rates and lognormal option quotations. Their mathematical lessons remain useful, but those benchmark conventions should not be presented as the current default for every contract. Overnight compounding, observation shifts, payment timing and multi-curve relationships can require a different implementation.

Model selection should follow the product, calibration instruments, required curve co-movement, boundary behaviour and computational budget. An exact fit to today’s curve is necessary for many valuation uses, but it is only one part of a credible model.

Matching instantaneous volatility does not match the full law

To compare a square-root model with an additive Gaussian model at initial rate $r_0$, one can set the Gaussian diffusion coefficient equal to $\sigma_{\mathrm{CIR}}\sqrt{r_0}$. This matches the initial instantaneous variance but not its future state dependence.

After the rate moves, CIR volatility changes with the square root of the new state, while Gaussian volatility remains constant. Their distributions, tail probabilities and option prices can therefore differ even when an initial local-risk comparison looks nearly identical.

The workbook contains a note describing a very small bond-yield difference in one particular matched example. That observation is specific to its parameters and horizon. It should not become a general statement that the functional form of the diffusion never matters.

Calibration weights encode priorities

A calibration may minimise squared price errors, relative price errors, implied-volatility errors or errors divided by a bid-ask measure. These objectives weight instruments differently. An equal price error is not equally important for a low-premium option and a high-premium option, while an implied-volatility error can be unstable when vega is small.

Instrument selection also determines which model features are identified. A narrow set of expiries may leave mean reversion weakly constrained. Adding long-expiry or different-tenor options can change the fitted dynamics even when the original instruments still reprice well.

The calibration report should list inputs, weights, residuals and parameter constraints. Repeating the optimisation from several starting points helps identify local minima or weakly identified directions. Smooth day-to-day parameters can be desirable, but a smoothing penalty changes the objective and should be acknowledged.

A rate tree has both geometry and calibration

Tree geometry specifies the state spacing and branching structure. Calibration adjusts state levels or probabilities to reproduce the initial curve and, where intended, option information. Mixing the two stages can make it difficult to diagnose whether a failure comes from the stochastic approximation or the curve fit.

At each node, branching probabilities should sum to one and remain admissible. Discount factors should be applied over the correct step length, and cash flows should occur at their actual dates. A visually recombining tree is not enough: it must preserve the intended moments and pricing relationships.

Comparing trees under the same initial curve and volatility assumptions is more informative than comparing outputs with different inputs. The retained workbook’s tree-comparison structure is useful precisely because it encourages isolating the effect of the model and numerical representation.

Interest-rate models should be tested against their use

Different instruments require different model features

Stress testing should vary mean reversion, volatility shape, factor correlation and curve extrapolation where those inputs matter. The resulting range is often more informative than a single calibration result presented without model-risk context.

Option-quote conventions are part of calibration

The retained workbook includes an at-the-money swaption example with forward and strike both 6.5%, one-year expiry, lognormal volatility 25% and annuity 4.2. These quantities must be interpreted together. The 25% is proportional lognormal volatility, not 25 percentage points of normal rate volatility. The annuity converts the option value on the swap rate into value per unit notional under the selected convention. Comparing that quote with a normal-volatility input requires a price-consistent conversion, not simply reusing the same numerical volatility in a different formula.

A one-factor model may be sufficient for one task and inadequate for another. Adding factors increases expressive power but also increases calibration uncertainty and implementation burden. The preferred model is the least complicated one that represents the material risks and contractual features with acceptable residual error.