Interest rate models
Vasicek and CIR side by side: a fan of short-rate paths pulled towards the long-run level, with the mean reversion speed under your thumb — and the zero line showing exactly where the two models part company.
12 of 40 paths went negative
dr = a(b − r)dt + σ dW. The pull towards b has half-life ln2/a ≈ 1.4 years — crank a up and the fan snaps to the gold line; ease it off and paths wander for years. Start r₀ far from b to watch the reversion do its work. Vasicek’s constant volatility is tractable but lets rates go negative; CIR’s √r diffusion shrinks the noise as rates approach zero, and under the Feller condition never touches it. Hull-White is Vasicek with b made time-dependent to fit today’s yield curve — same picture, calibrated anchor.
What a term-structure model has to do
The CM2 checklist for a good model — arbitrage-free, realistic rates, mean reversion, tractable bond prices, fits the observed curve — is really a set of trade-offs, and the two classic models sit at different corners. Vasicek buys full tractability (normal rates, closed-form bonds) at the price of negative rates with positive probability: watch the counter as σ rises. CIR spends a little tractability on the √r diffusion, which strangles volatility near zero so rates stay non-negative — and under the Feller condition 2ab ≥ σ² the origin is never even touched. Neither fits today’s yield curve exactly as given; Hull-White fixes that by letting the long-run level drift with time, which is why it gets called “Vasicek calibrated”. The mean reversion itself is the economics: rates are pulled towards a long-run norm at speed a, with half-life ln2/a — the slider that makes that abstraction feel physical.
The same machinery elsewhere. The driving noise is the Brownian motion these paths share with share prices, and what rates do to fixed interest is the bond calculator’s territory. Memori is a flashcard app built by actuarial students, with a ready-made CM2 set in the shop. Join the beta.
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