CIR / Square-Root Term Structure Model

Definition

A no-arbitrage continuous-time term-structure model in which the instantaneous short rate $r_{t}$ follows a square-root (Feller) diffusion:

$d r_{t} = κ (θ - r_{t}) d t + σ r_{t} d W_{t}$

with mean-reversion κ, long-run mean θ, and volatility scale σ. Bond prices admit an exponential-affine closed form: $P (t, T) = e x p (A (T - t) - B (T - t) r_{t})$ , where A and B solve a Riccati ODE. The process is non-negative (Feller condition: $2 κ θ \geq σ^{2}$ ) and conditional volatility scales with $r_{t}$ .

Intuition

The square-root diffusion is the simplest one-factor short-rate process that is non-negative, mean-reverting, and heteroskedastic (volatility rises with the level of rates). These three properties are the textbook minimum for a term-structure model that respects the zero lower bound and matches the empirical fact that interest-rate volatility is high when rates are high. The exponential-affine bond pricing formula gives tractable yields, durations, and risk premia in closed form.

Formal notation

Short rate SDE: $d r_{t} = κ (θ - r_{t}) d t + σ r_{t} d W_{t}$
Risk-neutral drift: $κ_{Q} (θ_{Q} - r_{t})$ , market price of risk linear in $r_{t}$
Bond price: $P (t, T) = e x p (A (τ) - B (τ) r_{t}), τ = T - t$
$A (τ), B (τ)$ solve a system of ODEs (Riccati for B, integral for A)
Yield: $y (t, τ) = - l o g P (t, T) / τ = (B (τ) r_{t} - A (τ)) / τ$ — affine in $r_{t}$

Variants

Multi-factor CIR: stack independent square-root processes (e.g. 2-factor or 3-factor models). Yields remain affine in the factor vector.
Affine class (Duffie-Kan, Dai-Singleton): generalization to N-factor processes with affine drift and affine instantaneous variance — CIR is the canonical pure-square-root member.
Regime-switching CIR (Bansal-Zhou 2002): parameters $κ, θ, σ$ , and the market price of risk become functions of a latent Markov regime; bond prices are exponential-affine within each regime, with regime-shift premia entering the cross-regime no-arbitrage restrictions. Bond pricing typically uses a log-linear approximation across regime trajectories.
Log-linear approximation variant: when closed-form bond prices are not available (e.g. multi-regime, multi-factor with non-affine extensions), bond prices are approximated as exponential-affine in the factor vector via a first-order log expansion. Accurate for short maturities; approximation error grows with maturity and with parameter spread across regimes.

Comparison

vs. Vasicek: CIR gains non-negativity and level-dependent volatility, at the cost of more complex bond pricing ODEs. Vasicek allows negative rates and constant volatility.

vs. Affine multi-factor (Dai-Singleton): CIR is a special case; multi-factor affine generalizes to richer dynamics at the cost of identifiability.

vs. HJM / forward-rate models: CIR is a short-rate model; HJM models the entire forward curve directly. CIR is more parsimonious; HJM is more flexible but has degenerate finite-dimensional structure under restrictive assumptions.

When to use

Pricing default-free bonds when non-negativity of rates is essential.
As a building block for credit, mortgage, or asset-pricing models that need a tractable, non-negative short-rate process.
When the empirical motivation requires level-dependent volatility (e.g. high-rate periods are also high-volatility periods).
As the within-regime engine of a regime-switching no-arbitrage model (Bansal-Zhou 2002).

Known limitations

Single-factor versions cannot match the full yield curve simultaneously (level, slope, curvature).
Constant transition matrix and constant $κ, θ, σ$ cannot reproduce time-varying conditional volatility patterns observed in yields (this is exactly the failure that Bansal-Zhou 2002 fixes via regime switching).
Single-factor CIR is rejected by Campbell-Shiller expectations-hypothesis regressions.
Feller condition can be violated by estimated parameters in finite samples, requiring boundary handling.

Open problems

General closed-form pricing for regime-switching CIR remains intractable (only log-linear approximations are available); convergence/error bounds at long maturities are not fully characterized.
Extending the affine/CIR family to non-Gaussian shocks (jumps, fat tails) while preserving tractable pricing.

Key papers

bansal-zhou-2002-term-structure-regime-shifts — embeds CIR inside a 2-state Markov regime-switching framework, captures conditional volatility and EH violations, identifies regimes with NBER business cycles.

My understanding

CIR is the workhorse one-factor non-negative term-structure model and the natural starting point for any no-arbitrage extension. For our CRE project, the relevance is indirect but load-bearing: the Riccati ODE structure that gives CIR its closed form is exactly the structure we exploit in compute_quadratic_pricing_factors_msvar for asset prices conditional on regime paths. The Bansal-Zhou regime-switching extension is the canonical precedent for combining CIR with a latent Markov chain, and it documents both the gains (volatility, correlation, EH) and the cost (log-linear approximation error growing with maturity), both of which we rediscover independently in research/pricing_approximation/.

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