Interest-rate models — Vasicek, CIR, HJM

Interest-rate modelling is one of the deepest applications of stochastic calculus in finance. Unlike equities where one asset has one volatility, rates form a curve — different maturities, all moving together. Short-rate models, the HJM framework, and forward-rate models are three lenses on the same problem: how to model the no-arbitrage evolution of the entire yield curve.

Why rates are harder than stocks

• A stock is one number; the yield curve is a function from maturity to rate.
• Bonds of all maturities must price consistently — no arbitrage across the curve.
• Rates exhibit strong mean-reversion (unlike stock prices).
• Negative rates are possible (some models force positivity, others don't).
• The historical / risk-neutral measure distinction is more subtle for rates than for equities.

Vasicek (1977)

dr = θ(μ - r) dt + σ dW

Ornstein-Uhlenbeck short rate. Closed-form bond price formula:

P(t, T) = A(t, T) exp(-B(t, T) r(t))
B(t, T) = (1 - exp(-θ(T-t))) / θ
A(t, T) = exp((B(t,T) - (T-t))(θ²μ - σ²/2) / θ² - σ² B(t,T)² / (4θ))

Strengths: tractable, closed-form everything. Weakness: rates can go negative (no constraint to non-negative).

Cox-Ingersoll-Ross (1985)

dr = θ(μ - r) dt + σ √r dW

Square-root volatility ensures r ≥ 0 (provided Feller condition 2θμ ≥ σ²). Still closed-form for bond prices (non-central χ² distributions appear). Used in fixed-income pricing libraries; basis for the Heston volatility model.

Hull-White (1990)

dr = (θ(t) - a r) dt + σ dW

Vasicek with time-varying mean θ(t). The θ(t) function is chosen so that the model exactly reproduces the observed term structure today — calibration to the current curve becomes possible. Standard model on bank fixed-income desks.

HJM (Heath-Jarrow-Morton 1992)

Model the entire forward curve f(t, T) as a stochastic process for each maturity T:

df(t, T) = α(t, T) dt + σ(t, T) dW

Critically, the drift α is not free — no-arbitrage forces α(t, T) = σ(t, T) ∫_t^T σ(t, u) du. This is the HJM drift condition. The framework subsumes all short-rate models and lets you specify volatility structures directly.

LIBOR market models (BGM, Brace-Gatarek-Musiela 1997)

Model discretely-compounded forward rates (LIBORs) directly. Each forward rate is lognormal under its own forward measure. The dominant interest-rate model class on derivatives trading floors. Calibrates well to cap and swaption volatility surfaces. Now adapting to RFRs (SOFR, SONIA) post-LIBOR.

Affine term-structure models

A general family with bond yields affine in a small state vector x(t):

P(t, T) = exp(A(T-t) + B(T-t)ᵀ x(t))

Vasicek, CIR, Hull-White, multi-factor extensions all fit. State-of-the-art academic and central-bank yield-curve work. Empirical leading examples: Ang-Piazzesi (2003) macro-finance, Cochrane-Piazzesi (2005) bond risk premia.

Risk-neutral vs real-world rates

Risk-neutral rate dynamics are calibrated to bond prices (no arbitrage). Real-world dynamics — used for VaR, stress, scenario design — are estimated from historical data. The two differ by the market price of duration risk, which has been historically positive (the term premium). Decomposing observed yields into expected-rate paths and term premia is one of the central tasks of empirical fixed-income macro.

Application to African / EM curves

EM yield curves typically show:

• Higher and more volatile term premia than DM.
• Greater regime-dependence (currency crises, IMF programs, political shocks).
• Thinner liquidity at the long end, requiring careful calibration.
• Stronger interaction with inflation dynamics and FX.

Modelling Kenyan or Nigerian yield curves typically uses a multi-factor affine model with macro state variables (inflation, FX, growth) plus latent term-structure factors. The mathematics is the same as DM; the parameter estimates are very different.