A single mean-reverting square-root SDE underlies a surprisingly large family of finance models. The state variable changes; the equation does not.
the CIR template
dX = κ(θ − X) dt + σ√X dW
non-negative · mean-reverts to θ at speed κ · vol scales with √X · affine, closed-form pricing
along the yield curve — same SDE, different point
X = r(t)
short rate
dr = κ(θ−r)dt + σ√r dW
CIR 1985 · τ→0 rate · entire curve from one SDE
X = ℓ(t)
long / consol rate
dℓ = κ(θ−ℓ)dt + σ√ℓ dW
Brennan–Schwartz · τ→∞ end of the curve
X = θ_t
central tendency
dθ = α(μ−θ)dt + η√θ dW
BDF, Chen · stochastic target r reverts toward
X = v(t)
forward-rate vol
dv = κ(θ−v)dt + ξ√v dW
SV-LMM, SV-HJM · vol driver for forward / LIBOR rates
X = s(t)
spread / basis
ds = κ(θ−s)dt + σ√s dW
credit spread, swap spread · non-negative differences
X = X₁, X₂, …
latent curve factors
dXᵢ = κᵢ(θᵢ−Xᵢ)dt + σᵢ√Xᵢ dWᵢ
Duffie–Kan affine · level / slope / curvature drivers
outside rates — the Heston branch
X = v(t) — instantaneous variance of an asset
Heston 1993
dv = κ(θ − v) dt + σ√v dW₂
structurally identical to the short-rate SDE — only the label on the state variable changes
paired with the asset SDE to form Heston's 2-SDE model:
dS = μS dt + √v S dW₁, corr(dW₁, dW₂) = ρ