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Interest rate modelling

Compare short-rate, HJM and market models, distinguish forecasting from risk-neutral pricing, and see what calibration does through an exact Vasicek forecast.

Interest rate modelling is the mathematical description of how interest rates and the yield curve evolve through time. Quants use it to forecast funding costs, generate risk scenarios, price rate derivatives and compute hedges. The first choice is the question being answered: whether the task needs real-world probabilities or probabilities used for pricing.

Forecasts and prices use different probabilities

A model under the physical measure P\mathbb{P} describes real-world uncertainty. Its drift parameters can be estimated from historical data, with out-of-sample checks of forecast errors. A model under the risk-neutral measure Q\mathbb{Q} supports no-arbitrage valuation. Its probabilities incorporate risk pricing; they are not automatically forecasts of observed frequencies.

For example, in an idealised default-free, single-currency market with continuously compounded short rate rsr_s, a unit zero-coupon bond has price

P(t,T)=EQ ⁣[exp⁡ ⁣(−∫tTrs ds)∣Ft],P(t,T)=\mathbb{E}^{\mathbb{Q}}\!\left[\exp\!\left(-\int_t^T r_s\,ds\right)\mid\mathcal{F}_t\right],
where Ft\mathcal{F}_t is information available at time tt. This averages discounted paths, rather than discounting at a forecast of the terminal rate. The conditional expectation must use the pricing dynamics. Historical estimates cannot simply be inserted as Q\mathbb{Q} parameters without specifying how risk premia connect the two measures. Privault's bond-pricing notes give the underlying expectation formula.

Three starting points for the same curve

FamilyWhat is modelled directly?What the choice makes easier
Short-rate modelsThe instantaneous rate, possibly with additional factorsCompact state dynamics and bond valuation. Examples include Vasicek, CIR and Hull–White.
Heath–Jarrow–Morton (HJM)Instantaneous forward rates across maturitiesStarting from today's forward curve and specifying its volatility. The no-arbitrage drift is constrained by that volatility.
Market modelsForward rates over specified accrual periods, or swap ratesConnecting dynamics to caplet or swaption quotes. The historical LIBOR market model is a standard example.
These are related representations, not three unrelated theories. A short-rate model induces forward-rate dynamics. In a market model, a forward rate can be driftless under its own payment-date forward measure; that does not make every rate driftless under one common measure. See Privault on HJM and Haugh on market models. The LIBOR name identifies a model's historical construction, not a prescription for today's contract benchmark.

A rate forecast with known parameters

Take a fictional Vasicek process under P\mathbb{P}:
drt=κ(θ−rt) dt+σ dWtP,dr_t=\kappa(\theta-r_t)\,dt+\sigma\,dW_t^{\mathbb{P}},
with r0=4%r_0=4\%, θ=3%\theta=3\%, κ=0.5\kappa=0.5 per year and σ=0.01\sigma=0.01 in decimal-rate units per square-root year. Time is measured in years; rtr_t is an annual continuously compounded short rate, and WtPW_t^{\mathbb{P}} is Brownian motion. With fixed parameters, the exact conditional distribution is Gaussian:
EP[rT∣r0]=θ+(r0−θ)e−κT,VarP(rT∣r0)=σ22κ(1−e−2κT).\begin{aligned} \mathbb{E}^{\mathbb{P}}[r_T\mid r_0] &=\theta+(r_0-\theta)e^{-\kappa T},\\ \text{Var}^{\mathbb{P}}(r_T\mid r_0) &=\frac{\sigma^2}{2\kappa}(1-e^{-2\kappa T}). \end{aligned}
At T=2T=2, the mean is 3.3679%, the variance is 0.00008646650.0000864665 in squared decimal-rate units, and the standard deviation is 0.9299 percentage points. Exact Gaussian quantiles give a central 95% interval of 1.545% to 5.190%. These are original inputs evaluated using the distribution in Privault §17.1.
Exact fictional Vasicek forecast under the physical measure P: rate starts at 4%, the two-year mean is 3.3679%, and the central 95% interval is 1.545% to 5.190%. The dashed long-run mean is 3%. Uncertainty widens as the mean falls.
Exact fictional Vasicek forecast under the physical measure P: rate starts at 4%, the two-year mean is 3.3679%, and the central 95% interval is 1.545% to 5.190%. The dashed long-run mean is 3%. Uncertainty widens as the mean falls.
Original analytical illustration with fixed known parameters. The band contains 95% of the marginal rate distribution at each horizon; it is not a 95% guarantee for an entire rate path and excludes parameter uncertainty and model error.
Mean reversion pulls the conditional mean downward here, while uncertainty widens. It does not require each realised rate to fall. Gaussian dynamics also permit negative rates. Most importantly, this P\mathbb{P} forecast does not determine a bond price: the pricing drift has not been specified.

What calibration proves—and what it leaves open

Fitting today's curve and fitting rate-option prices are separate tasks. A time-homogeneous Vasicek model generally cannot match an arbitrary initial curve. Hull–White adds a deterministic time-dependent drift term to fit that curve; this alone does not identify the volatility dynamics or guarantee a fit to every option quote. Figlewski's model comparison explains this distinction.

A useful assessment therefore asks which quotes were fitted, how stable the parameters are, and whether the model captures the curve movements and option risks the position needs. Matching selected prices does not validate real-world forecasts or every hedge. A single-factor model has only one instantaneous source of randomness; additional curve-shape risks may require more factors.

Contract conventions also matter. Floating-rate projection and collateral discounting need not share one curve. A model built on textbook single-curve identities needs an explicit extension when those curves differ; Cuchiero, Fontana and Gnoatto develop such a multiple-curve framework.
For the forward-curve mechanics, continue with SelfMadeQuant's Forward Rate Curve Dynamics and the HJM Specification. The public page is a preview; the full lesson requires membership.

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  • interest rates
  • fixed income
  • term structure
  • forecasting
  • derivatives pricing
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