How Cox–Ingersoll–Ross Short-Rate Algorithms Keep Interest Rates Nonnegative: Square-Root Diffusion, Feller Boundary, Exact Simulation, Bond Prices, Calibration and Model Failure

Reader question: Can an interest-rate model mean-revert like an Ornstein–Uhlenbeck process while making large negative short rates structurally difficult or impossible?

The Cox–Ingersoll–Ross model, usually called CIR, changes one term in the diffusion and changes the geometry of the process. Instead of constant volatility, instantaneous volatility is proportional to the square root of the current short rate. As the rate approaches zero, diffusion shrinks.

This article owns the positive mean-reverting short-rate problem: an initial short rate + mean-reversion parameters + square-root volatility + risk-neutral pricing assumptions → transition distributions, simulated paths, zero-coupon bond prices, yield curves and diagnostics for boundary behaviour and calibration failure.

It does not own arbitrary yield-curve fitting, swaption smile calibration, HJM forward-curve dynamics or Hull–White Gaussian calibration. Those are separate pages. The purpose here is mathematical: why the square-root diffusion is special, how exact simulation works and where the model breaks.

This is public mathematical education. It is not personalised financial advice, a forecast of future rates or a recommendation to trade any fixed-income instrument.

1. The CIR stochastic differential equation

Under a chosen probability measure, write:

dr_t = κ(θ−r_t)dt + σ√r_tdW_t.

The parameters have distinct jobs:

• κ > 0: mean-reversion speed;
• θ > 0: long-run mean level under that measure;
• σ > 0: volatility scale;
• W_t: Brownian motion.

The drift pulls the short rate toward θ. The diffusion term σ√r makes volatility state-dependent.

2. Why the square root matters

In a Gaussian short-rate model with constant diffusion, a large negative shock can move the rate below zero regardless of its current level.

In CIR, the instantaneous diffusion magnitude is:

σ√r.

As r → 0, that diffusion term approaches zero. The process becomes less noisy near the boundary while the drift approaches κθ > 0, pushing upward.

This creates nonnegative dynamics under the continuous-time model.

3. Conditional mean shows the mean-reversion mechanism

The conditional expectation over horizon Δ is:

E[r_t+Δ|r_t] = θ + (r_t−θ)e^−κΔ.

If the current rate is above θ, the expected future rate falls toward θ. If it is below θ, the expected rate rises.

The characteristic mean-reversion half-life is:

t_1/2 = ln 2 / κ.

This is useful for interpreting κ: doubling κ roughly halves the time needed for an expected deviation from the long-run level to decay by 50%.

4. Conditional variance is state dependent

A standard conditional-variance formula is:

Var(r_t+Δ|r_t) = r_tσ²e^−κΔ(1−e^−κΔ)/κ + θσ²(1−e^−κΔ)²/(2κ).

Unlike a Gaussian model, uncertainty depends partly on the current rate level. High-rate states generate more instantaneous diffusion than low-rate states.

5. The Feller condition

The famous boundary condition is:

2κθ ≥ σ².

When the strict form holds and the process starts positive, zero is inaccessible in the standard CIR diffusion. The rate remains strictly positive.

If the condition is violated, the process remains nonnegative but can reach zero. What happens at the boundary depends on the square-root diffusion’s boundary classification.

This distinction matters. It is inaccurate to say that violating the Feller condition automatically makes CIR negative. The continuous process still has a nonnegative state space; what changes is whether zero can be attained.

6. Feller is a model property, not a calibration target at any cost

A modeller may be tempted to force 2κθ ≥ σ² even when market data prefer parameters outside that region.

That can worsen fit or distort parameter interpretation.

A better workflow treats the condition as a diagnostic and then asks whether the intended numerical scheme and product set can tolerate boundary contact.

Evidence against fit: if enforcing Feller causes systematic pricing errors while relaxing it materially improves fit, the simple one-factor CIR specification may be too restrictive for the market rather than the optimiser being wrong.

7. The exact transition law

CIR has an unusually valuable property: its finite-horizon transition distribution is known exactly.

For Δ > 0 under constant parameters:

r_t+Δ = cY,

where Y follows a noncentral chi-square distribution:

Y ~ χ′²_d(λ).

The scale is:

c = σ²(1−e^−κΔ)/(4κ).

The degrees of freedom are:

d = 4κθ/σ².

The noncentrality parameter is:

λ = 4κe^−κΔr_t / [σ²(1−e^−κΔ)].

This exact distribution is central to both simulation and likelihood-based estimation.

8. Exact simulation algorithm

1. Read current rate r_t.
2. Choose time step Δ.
3. Compute c, d and λ.
4. Draw Y from the noncentral chi-square distribution.
5. Set r_t+Δ = cY.
6. Repeat recursively for the path.

The sampled rate is automatically nonnegative because the noncentral chi-square variable is nonnegative.

This avoids a major numerical weakness of naive Euler simulation.

9. Why naive Euler can create impossible negative rates

An Euler–Maruyama step would be:

r_t+Δ ≈ r_t + κ(θ−r_t)Δ + σ√r_t√Δ Z,

with Z ~ N(0,1).

The Gaussian shock is unbounded. A sufficiently negative Z can push the discrete approximation below zero even though the true continuous CIR process is nonnegative.

Once r is negative, √r is undefined in real arithmetic.

This is numerical failure, not evidence that the CIR SDE itself went negative.

10. Common Euler repairs change the algorithm

Implementations sometimes use:

• absorption: replace negative values with zero;
• reflection: replace r with |r|;
• partial truncation;
• full truncation: use max(r,0) inside drift/diffusion formulas.

These can improve practical simulation but introduce discretisation bias.

Falsifier: compare moments, option prices or path statistics from the repaired Euler scheme against exact-transition sampling. If the difference remains material at the chosen step size, the approximation is not validated.

11. Exact likelihood estimation

Because the transition density is known, historical short-rate data can be estimated by maximum likelihood without approximating each transition as Gaussian.

For observations r₀, …, r_T, construct:

log L(κ,θ,σ) = Σ log f(r_t+1|r_t; κ,θ,σ),

where f is the scaled noncentral chi-square density.

Pearson and Sun’s empirical work is a classic example of using the exact conditional density for estimating equilibrium term-structure models.

12. Physical and risk-neutral parameters are different objects

Historical time-series estimation occurs under the physical probability measure P. Bond pricing occurs under the risk-neutral measure Q.

The market price of interest-rate risk changes the drift when moving between measures.

Therefore:

historically estimated κ and θ are not automatically the risk-neutral κ and θ used for bond pricing.

A calibration engine must state which parameters belong to which measure and how the risk premium connects them.

13. Zero-coupon bond prices are exponential-affine

Under risk-neutral CIR dynamics:

dr = κ(θ−r)dt + σ√r dW^Q,

the price at time t of a zero-coupon bond paying one at T has the form:

P(t,T) = A(τ)e^−B(τ)r_t,

where τ = T−t.

Define:

γ = √(κ² + 2σ²).

Then:

B(τ) = 2(e^γτ−1) / [(γ+κ)(e^γτ−1)+2γ].

And:

A(τ) = {2γe^(κ+γ)τ/2 / [(γ+κ)(e^γτ−1)+2γ]}^2κθ/σ².

The entire bond curve is therefore generated from one state variable r_t and three risk-neutral parameters.

14. Yield and forward rates follow from bond prices

The continuously compounded zero yield is:

y(t,T) = −ln P(t,T)/(T−t).

Forward rates can be derived from maturity derivatives of ln P(t,T).

The affine structure makes CIR analytically elegant and computationally cheap.

It also creates a limitation: one factor and a few parameters cannot fit every observed yield-curve shape exactly.

15. Calibration to a market yield curve

A simple risk-neutral calibration can choose κ, θ, σ and possibly the current state r₀ to minimize errors between model and market bond prices or yields.

A weighted objective might be:

min Σ w_i[P_model(T_i)−P_market(T_i)]².

Alternatively, errors can be measured in yields, log prices or durations.

The metric matters. A small price error on a short bond is not economically equivalent to the same price error on a long-duration bond.

16. Identifiability can be weak

Different combinations of κ, θ and σ can produce similar yield curves over a limited maturity range.

That means an optimiser can find a visually good curve fit while parameter estimates are unstable.

Diagnostics:

• restart optimisation from multiple initial guesses;
• inspect the Hessian or profile likelihood;
• bootstrap input data;
• compare parameter stability across dates;
• check whether small market changes cause huge parameter jumps.

17. The long-run yield is not simply θ

The short-rate mean θ is not automatically equal to the asymptotic zero yield. Convexity and volatility enter long-maturity bond pricing through the affine coefficients.

This is a common conceptual mistake: the long-run expectation of the state and the long-run bond yield are related but not identical objects.

18. CIR cannot naturally represent negative rates

Basic CIR keeps the short rate nonnegative. That was historically attractive, but modern interest-rate markets have experienced negative policy and government yields.

If the market curve itself contains negative rates, unshifted CIR may be structurally incapable of fitting it.

Falsifier: if an observed discount curve requires negative instantaneous forwards or short rates in regions the model forbids, the specification is wrong for that use case.

19. Shifted CIR / CIR++

A common extension writes:

r_t = x_t + φ(t),

where x_t follows a CIR process and φ(t) is a deterministic shift chosen to fit the initial term structure exactly.

This separates:

• stochastic positive square-root factor;
• deterministic curve-fitting adjustment.

The extension improves initial-curve fit but sacrifices the simple interpretation that the observed short rate itself is a pure CIR state.

20. Inputs and outputs

Inputs can include:

• current short rate;
• κ, θ, σ under the chosen measure;
• time horizon and step;
• yield-curve observations;
• market-price-of-risk specification;
• calibration weights;
• simulation seed;
• exact or approximate transition method;
• shift function for CIR++ if used.

Outputs can include:

• conditional mean and variance;
• exact transition distribution;
• simulated short-rate paths;
• zero-coupon bond prices;
• zero yields and forwards;
• calibration residuals;
• Feller-condition diagnostic;
• parameter uncertainty;
• out-of-sample pricing errors.

21. Evidence polarity

Evidence for confidence includes exact-transition simulation reproducing theoretical moments, bond prices satisfying P(T,T)=1, stable calibration from multiple starting points, residuals without strong maturity patterns, historical transitions compatible with the fitted density, and out-of-sample rate movements broadly consistent with the model’s conditional distribution.

Evidence against confidence includes parameter instability, persistent yield-curve residual shapes, negative rates in the market that the model cannot represent, Feller violations that cause numerical approximations to misbehave, poor serial diagnostics in historical residuals, or accurate short-maturity fit combined with systematic long-maturity mispricing.

22. Counterexample: Euler creates negative paths

A naive Euler implementation generates r = −0.002.

This does not falsify the CIR SDE; it falsifies the discretisation for that step size and parameter regime.

Repair test: switch to exact noncentral-chi-square sampling and confirm the negative values disappear.

23. Counterexample: the Feller condition fails but the exact process remains nonnegative

Suppose 2κθ < σ². Zero can become attainable.

A programmer may incorrectly reject the model as “allowing negative rates.” That conclusion is wrong.

Correct diagnostic: distinguish boundary attainability from sign. Use an exact or positivity-preserving scheme and inspect how much probability mass accumulates near zero.

24. Counterexample: perfect curve fit through over-interpretation

A single-day yield curve is fitted well with one parameter vector. The modeller then interprets θ as a reliable economic forecast of the future policy-rate level.

That leap is unsupported because risk-neutral parameters encode pricing and risk premia, not simply physical expectations.

Falsifier: estimate the physical dynamics separately and compare.

25. Counterexample: one-factor shape failure

One factor tends to move the whole curve through a restricted family of shapes.

Real curves show level, slope and curvature movements that cannot always be represented by one short-rate state.

Falsifier: run principal-component analysis of yield changes. If several independent factors are needed for material risk, one-factor CIR is structurally incomplete for that task.

26. Counterexample: exact bond pricing, poor option pricing

CIR can fit bonds reasonably but still fail to reproduce observed option volatilities because one square-root factor imposes a restricted distribution of future rates.

Falsifier: calibrate to bond prices first, then test swaptions or caps excluded from calibration. Systematic smile or maturity residuals indicate missing volatility structure.

27. Counterexample: calibration depends on the error metric

Minimising raw bond-price errors may prioritize high-price instruments. Minimising yield errors weights maturities differently. Minimising duration-normalised errors changes the objective again.

Diagnostic: compare parameter estimates under several defensible weighting schemes. If parameters swing dramatically, the data do not identify the model robustly.

28. Historical tests have found real limitations

Empirical work on CIR-family term-structure models has long shown a mixed picture. Pearson and Sun used exact conditional densities and found meaningful evidence about parameter restrictions and fit, while other studies found that simple one-factor specifications can leave serial structure or maturity-dependent pricing errors.

The lesson is not that CIR is obsolete. It is that analytical elegance must still face empirical falsification.

29. Alternatives

Vasicek uses Gaussian mean reversion and admits negative rates.

Hull–White adds a time-dependent drift so the initial yield curve can be fitted exactly while retaining Gaussian tractability.

CIR++ adds a deterministic shift to the CIR factor.

Two-factor CIR models add another square-root state to improve curve and volatility dynamics.

HJM models the forward curve directly and imposes a no-arbitrage drift restriction.

Market models model selected forward rates rather than the instantaneous short rate.

30. Connections to the Bukit Timah Tutor knowledge estate

Hull–White calibration algorithms provide the nearest Gaussian short-rate comparison: easier exact curve fit, but no positivity structure.

Heath–Jarrow–Morton algorithms reverse the modelling viewpoint by starting from forward-rate dynamics.

Yield-curve construction algorithms explain how the market discount curve used for calibration is built from instruments.

Yield-curve PCA algorithms provide an empirical test of whether one factor is enough.

The lane hub is Finance & Banking Algorithms | Applied Mathematics in Real Financial Systems.

31. What would falsify confidence?

Confidence should be withdrawn if the exact transition law does not reproduce theoretical moments; if numerical paths become negative only because of discretisation; if calibration parameters are unstable; if residuals show systematic maturity patterns; if the market requires negative rates the model forbids; if one-factor dynamics cannot reproduce observed curve movements; or if out-of-sample bond/option prices show persistent structure the model cannot explain.

32. Verification and update triggers

Preserve the probability measure, parameter set, Feller diagnostic, calibration instruments, error metric, optimiser settings, exact-transition implementation, simulation seed and benchmark calculations for every release.

Trigger revalidation when the yield curve moves into negative-rate territory, the model’s Feller status changes materially, volatility regimes shift, exact-simulation libraries change, parameter estimates jump, option-market residuals grow, a second empirical curve factor becomes material, or the intended use expands beyond the products originally validated.

33. Primary and high-quality references

• John C. Cox, Jonathan E. Ingersoll Jr. and Stephen A. Ross, A Theory of the Term Structure of Interest Rates, Econometrica, 1985.
• John C. Cox, Jonathan E. Ingersoll Jr. and Stephen A. Ross, publication record and related materials via RePEc.
• Neil D. Pearson and Tong-Sheng Sun, Exploiting the Conditional Density in Estimating the Term Structure: An Application to the Cox, Ingersoll, and Ross Model, Journal of Finance, 1994.
• Michael R. Gibbons and Krishna Ramaswamy, empirical tests of the CIR term-structure model, A Test of the Cox, Ingersoll, and Ross Model of the Term Structure, Review of Financial Studies.
• Federal Reserve Board and OCC, SR 11-7 — Guidance on Model Risk Management, for benchmarking, validation and model limitations.

Educational boundary: CIR is a tractable model with powerful exact mathematics. Positivity, exact simulation and closed-form bond prices do not guarantee that one square-root factor is a sufficient empirical description of the interest-rate market.