Short-Rate Models

Vasicek (1977) · Cox-Ingersoll-Ross (1985) · One-factor SDEs

Family 04Interest ratesSDE$\mathbb{Q}$-measure

Bonds, swaps, and rate derivatives require a model for the short interest rate $r_t$ under the risk-neutral measure. Vasicek (1977) proposed a Gaussian mean-reverting SDE; Cox, Ingersoll & Ross (1985) a square-root diffusion that keeps rates non-negative. Both are one-factor affine models with closed-form zero-coupon bond prices and tractable yield curves. Hull-White (1990) extended Vasicek with time-varying parameters to fit observed curves; HJM (1992) generalised to arbitrary forward-rate dynamics.

Educational Purpose

Educational demonstration. Not investment, hedging, or regulatory advice. One-factor models are stylised; production desks use multi-factor models (G2++, LIBOR market model) calibrated to the full swap/cap/swaption matrix.

The models

OR family: Interest-rate modelling Solver class: SDE simulation Measure: Risk-neutral $\mathbb{Q}$ Realism: ★★★ Exact under model

Vasicek (1977)

Vasicek SDE $$ dr_t \;=\; a\,(b - r_t)\,dt \;+\; \sigma\, dW^{\mathbb{Q}}_t $$

Ornstein-Uhlenbeck process. $a$ = mean-reversion speed, $b$ = long-run level, $\sigma$ = volatility. Short rate is Gaussian and can go negative.

Cox-Ingersoll-Ross (1985)

CIR SDE $$ dr_t \;=\; a\,(b - r_t)\,dt \;+\; \sigma\, \sqrt{r_t}\, dW^{\mathbb{Q}}_t $$

Square-root diffusion. Under the Feller condition $2ab \ge \sigma^2$, rates stay strictly positive. Non-central chi-square transition density.

Zero-coupon bond price (affine form)

Both Vasicek and CIR have closed-form bond prices $P(t, T) = A(t,T) \exp[-B(t,T) r_t]$. For Vasicek:

Vasicek bond price $$ B(t,T) \;=\; \frac{1 - e^{-a(T-t)}}{a} $$ $$ A(t,T) \;=\; \exp\!\left[\,\left(b - \frac{\sigma^2}{2a^2}\right)\bigl(B(t,T) - (T-t)\bigr) \;-\; \frac{\sigma^2}{4a} B(t,T)^2\right] $$

For CIR, $B$ and $A$ have different closed forms involving hyperbolic functions of $\sqrt{a^2 + 2\sigma^2}$. Yield $y(t,T) = -\ln P / (T-t)$.

Long-run distribution (Vasicek)

$$ r_\infty \sim \mathcal{N}\!\left(b, \; \frac{\sigma^2}{2a}\right) $$

Stationary Gaussian. Half-life of mean-reversion = $\ln 2 / a$.

Interactive curve simulator

Yield curve & simulated rate paths

Model

Initial rate $r_0$

4.0%

Mean-reversion $a$

0.50

Long-run $b$

5.0%

Vol $\sigma$

1.5%

Paths $M$

Horizon $T$ (years)

Seed

1y yield

—

5y yield

—

10y yield

—

30y yield

—

Half-life

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Simulated short-rate paths $r_t$. Gold dashed = long-run level $b$; coloured lines = sample trajectories under Euler-Maruyama discretisation.

Closed-form zero-coupon yield curve $y(0, T)$ at the current parameters. Long-end yield converges to the stationary expectation; shape depends on $(r_0, b, a, \sigma)$.

Extensions & limitations

Model extensions

Hull-White (1990): $b \to b(t)$ time-varying to fit initial yield curve exactly.
Black-Karasinski: $\ln r_t$ mean-reverts (keeps $r_t > 0$; no closed-form bond price).
G2++ / two-factor Hull-White: adds a stochastic level factor; better fit to swaption surfaces.
Heath-Jarrow-Morton (1992): specifies forward-rate $f(t,T)$ dynamics directly; most general no-arbitrage framework.
LIBOR Market Model (BGM 1997, Miltersen-Sandmann-Sondermann 1997): discrete forward LIBORs as the primary modelling objects; simulated directly for cap/swaption pricing.
SOFR-based models: post-LIBOR world, backward-looking in-arrears rates require modifications to all classical frameworks.

Limitations

Vasicek rates go negative. Acceptable since 2015 eurozone era, but requires care in pre-2015 markets.
One factor captures only parallel shifts. Multi-factor needed for curve steepening/flattening and butterfly risk.
Calibration is ill-posed. Multiple parameter sets can fit observed instruments; principled regularisation matters.
Volatility surface fitting requires local-vol / stochastic-vol extensions beyond one-factor models.

Key references

Vasicek, O. (1977).

An Equilibrium Characterization of the Term Structure.

Journal of Financial Economics, 5(2), 177–188. doi:10.1016/0304-405X(77)90016-2

Cox, J. C., Ingersoll, J. E. & Ross, S. A. (1985).

A Theory of the Term Structure of Interest Rates.

Econometrica, 53(2), 385–407. doi:10.2307/1911242

Hull, J. & White, A. (1990).

Pricing Interest-Rate-Derivative Securities.

The Review of Financial Studies, 3(4), 573–592. doi:10.1093/rfs/3.4.573

Heath, D., Jarrow, R. & Morton, A. (1992).

Bond Pricing and the Term Structure of Interest Rates.

Econometrica, 60(1), 77–105. doi:10.2307/2951677

Brace, A., Gátarek, D. & Musiela, M. (1997).

The Market Model of Interest Rate Dynamics.

Mathematical Finance, 7(2), 127–155. doi:10.1111/1467-9965.00028

Brigo, D. & Mercurio, F. (2006).

Interest Rate Models — Theory and Practice, 2nd ed.

Springer. ISBN 978-3-540-22149-4.

Filipovic, D. (2009).

Term-Structure Models.

Springer. ISBN 978-3-540-09726-6.