# Cox-Ingersoll-Ross (CIR) Model: The Interest Rate Model Explained Published: 2026-04-07 Author: Warren Team URL: https://www.heywarren.com/blog/cox-ingersoll-ross --- Interest rates don't wander randomly, they tend to revert to long-run average levels, and they can never go (sustainably) negative in practice. The Cox-Ingersoll-Ross model, developed in 1985, was one of the first mathematically elegant ways to capture these real-world interest rate behaviors. It remains foundational in fixed income derivatives pricing, bond valuation, and quantitative finance. Here's exactly how it works and why it matters. ## What Is the Cox-Ingersoll-Ross Model? The Cox-Ingersoll-Ross (CIR) model is a **mean-reverting stochastic process** for modeling the evolution of short-term interest rates over time. It was introduced by John C. Cox, Jonathan E. Ingersoll Jr., and Stephen A. Ross in their 1985 paper "A Theory of the Term Structure of Interest Rates." The model is used primarily for: - Pricing interest rate derivatives (caps, floors, swaptions) - Valuing bonds with embedded options - Generating interest rate scenarios for risk management - Pricing mortgages and mortgage-backed securities ## The CIR Equation The CIR model specifies that the short-term interest rate r follows the stochastic differential equation: ![The CIR stochastic differential equation splits into a mean-reversion drift term and a rate-dependent volatility term.](data:image/svg+xml,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20viewBox%3D%220%200%20875%20125%22%20width%3D%22875%22%20height%3D%22125%22%20role%3D%22img%22%3E%3Ctitle%3EFlow%20diagram%3C%2Ftitle%3E%3Crect%20width%3D%22100%25%22%20height%3D%22100%25%22%20fill%3D%22%23f8fafc%22%2F%3E%3Crect%20x%3D%2230%22%20y%3D%2225%22%20width%3D%22170%22%20height%3D%2275%22%20rx%3D%2210%22%20fill%3D%22white%22%20stroke%3D%22%232563eb%22%20stroke-width%3D%222%22%2F%3E%3Ctext%20x%3D%22115%22%20y%3D%2258.5%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3ECurrent%20Rate%20r%3C%2Ftext%3E%3Ctext%20x%3D%22115%22%20y%3D%2278.5%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2211%22%20fill%3D%22%2364748b%22%3EStarting%20point%3C%2Ftext%3E%3Cline%20x1%3D%22205%22%20y1%3D%2262.5%22%20x2%3D%22237%22%20y2%3D%2262.5%22%20stroke%3D%22%2364748b%22%20stroke-width%3D%222%22%2F%3E%3Cpolygon%20points%3D%22244%2C62.5%20235%2C57.5%20235%2C67.5%22%20fill%3D%22%2364748b%22%2F%3E%3Crect%20x%3D%22245%22%20y%3D%2225%22%20width%3D%22170%22%20height%3D%2275%22%20rx%3D%2210%22%20fill%3D%22white%22%20stroke%3D%22%232563eb%22%20stroke-width%3D%222%22%2F%3E%3Ctext%20x%3D%22330%22%20y%3D%2258.5%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3EDrift%3A%20%CE%BA%28%CE%B8%E2%88%92r%29dt%3C%2Ftext%3E%3Ctext%20x%3D%22330%22%20y%3D%2278.5%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2211%22%20fill%3D%22%2364748b%22%3EMean%20reversion%20pull%3C%2Ftext%3E%3Cline%20x1%3D%22420%22%20y1%3D%2262.5%22%20x2%3D%22452%22%20y2%3D%2262.5%22%20stroke%3D%22%2364748b%22%20stroke-width%3D%222%22%2F%3E%3Cpolygon%20points%3D%22459%2C62.5%20450%2C57.5%20450%2C67.5%22%20fill%3D%22%2364748b%22%2F%3E%3Crect%20x%3D%22460%22%20y%3D%2225%22%20width%3D%22170%22%20height%3D%2275%22%20rx%3D%2210%22%20fill%3D%22white%22%20stroke%3D%22%232563eb%22%20stroke-width%3D%222%22%2F%3E%3Ctext%20x%3D%22545%22%20y%3D%2258.5%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3ENoise%3A%20%CF%83%E2%88%9Ar%C2%B7dW%3C%2Ftext%3E%3Ctext%20x%3D%22545%22%20y%3D%2278.5%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2211%22%20fill%3D%22%2364748b%22%3ERate-scaled%20shock%3C%2Ftext%3E%3Cline%20x1%3D%22635%22%20y1%3D%2262.5%22%20x2%3D%22667%22%20y2%3D%2262.5%22%20stroke%3D%22%2364748b%22%20stroke-width%3D%222%22%2F%3E%3Cpolygon%20points%3D%22674%2C62.5%20665%2C57.5%20665%2C67.5%22%20fill%3D%22%2364748b%22%2F%3E%3Crect%20x%3D%22675%22%20y%3D%2225%22%20width%3D%22170%22%20height%3D%2275%22%20rx%3D%2210%22%20fill%3D%22white%22%20stroke%3D%22%232563eb%22%20stroke-width%3D%222%22%2F%3E%3Ctext%20x%3D%22760%22%20y%3D%2258.5%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3ENew%20Rate%20dr%3C%2Ftext%3E%3Ctext%20x%3D%22760%22%20y%3D%2278.5%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2211%22%20fill%3D%22%2364748b%22%3ECombined%20output%3C%2Ftext%3E%3C%2Fsvg%3E) *The CIR stochastic differential equation splits into a mean-reversion drift term and a rate-dependent volatility term.* **dr = κ(θ − r)dt + σ√r · dW** Where: - **r** = current short-term interest rate - **κ (kappa)** = speed of mean reversion (how quickly rates return to the long-run mean) - **θ (theta)** = long-run mean interest rate (where rates tend to revert toward) - **σ (sigma)** = volatility of interest rate movements - **dt** = small time increment - **dW** = Wiener process (random component) This equation has two components: 1. **κ(θ − r)dt**: The deterministic drift term, when r is below θ, this is positive (rates drift up); when r is above θ, this is negative (rates drift down). This is the mean reversion. 2. **σ√r · dW**: The stochastic volatility term, random shocks to the interest rate, scaled by the square root of the current rate level. ## Key Properties of the CIR Model ### Mean Reversion If current rates are above the long-run mean θ, the drift term κ(θ − r) is negative, pulling rates downward. If current rates are below θ, the drift is positive, rates trend upward. The speed κ determines how forcefully this reversion occurs. **Example**: If κ = 0.3, θ = 5%, and r = 8%: - Drift = 0.3 × (5% − 8%) = −0.9% per year - Rates are being pulled down by ~0.9 percentage points per year toward 5% ### Non-Negative Interest Rates The **√r** term in the volatility component ensures that when interest rates approach zero, the random component also approaches zero, making it increasingly unlikely (under certain parameter conditions) for rates to go negative. The technical condition for rates to remain strictly positive is **2κθ > σ²** (Feller condition). When this holds, rates cannot reach zero and remain non-negative. This was considered a crucial advantage over the earlier Vasicek model, which allowed negative rates, unrealistic for nominal interest rates (though negative rates did materialize in some economies post-2008, making this "advantage" more nuanced today). ### Closed-Form Bond Pricing One of the CIR model's major advantages is that [zero-coupon bond](/blog/zero-coupon-bond) prices have a **closed-form analytical solution**, no numerical simulation required. The price of a zero-coupon bond paying $1 at time T given current rate r is: **P(r, t, T) = A(t,T) · exp(−B(t,T) · r)** Where A and B are deterministic functions of the model parameters (κ, θ, σ) and the time to maturity. This makes the model computationally efficient compared to models requiring Monte Carlo simulation. ## CIR vs. Vasicek Model The Vasicek model (1977) is the CIR model's predecessor and uses: ![CIR volatility scales with the rate level (σ√r), while Vasicek uses constant volatility, making CIR more realistic at high and low rate regimes.](data:image/svg+xml,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20viewBox%3D%220%200%20800%20210%22%20width%3D%22800%22%20height%3D%22210%22%20role%3D%22img%22%3E%3Ctitle%3EComparison%3C%2Ftitle%3E%3Crect%20width%3D%22100%25%22%20height%3D%22100%25%22%20fill%3D%22%23f8fafc%22%2F%3E%3Ctext%20x%3D%22230%22%20y%3D%2257.5%22%20text-anchor%3D%22end%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3EVasicek%20%28constant%20%CF%83%29%3C%2Ftext%3E%3Crect%20x%3D%22240%22%20y%3D%2225%22%20width%3D%22160.71428571428572%22%20height%3D%2255%22%20rx%3D%226%22%20fill%3D%22%232563eb%22%2F%3E%3Ctext%20x%3D%22412.7142857142857%22%20y%3D%2257.5%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22700%22%20fill%3D%22%232563eb%22%3E%251.5%3C%2Ftext%3E%3Ctext%20x%3D%22230%22%20y%3D%22152.5%22%20text-anchor%3D%22end%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3ECIR%20at%20r%3D8%25%20%28%CF%83%E2%88%9Ar%29%3C%2Ftext%3E%3Crect%20x%3D%22240%22%20y%3D%22120%22%20width%3D%22450%22%20height%3D%2255%22%20rx%3D%226%22%20fill%3D%22%237c3aed%22%2F%3E%3Ctext%20x%3D%22702%22%20y%3D%22152.5%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22700%22%20fill%3D%22%237c3aed%22%3E%254.2%3C%2Ftext%3E%3C%2Fsvg%3E) *CIR volatility scales with the rate level (σ√r), while Vasicek uses constant volatility, making CIR more realistic at high and low rate regimes.* **dr = κ(θ − r)dt + σ · dW** The critical difference: Vasicek has constant volatility σ, while CIR has volatility σ√r that scales with the rate level. | Feature | Vasicek | CIR | |---|---|---| | Mean reversion | ✅ Yes | ✅ Yes | | Allows negative rates | ✅ Yes (problem) | ❌ No (under Feller condition) | | Volatility | Constant | Rate-dependent (σ√r) | | Bond pricing | Closed-form | Closed-form | | Calibration complexity | Simpler | Slightly more complex | | Distribution of rates | Normal | Chi-squared (non-central) | The rate-dependent volatility in CIR is empirically more realistic: interest rate volatility does tend to be higher when rates are high (1970s–80s US rates) and lower when rates are near zero. ## Calibrating the CIR Model To use the CIR model in practice, its three parameters (κ, θ, σ) must be estimated from market data. Common approaches: ### Historical Estimation (Time Series) Fit the model to historical interest rate data (e.g., 3-month Treasury bill rates) using maximum likelihood estimation or method of moments. This gives parameters reflecting past interest rate behavior. ### Risk-Neutral Calibration (Market Prices) Calibrate parameters to match observed market prices of liquid fixed income instruments (Treasury bonds, interest rate swaps, caps/floors). This gives "risk-neutral" parameters used for derivatives pricing. **Important**: Historical and risk-neutral parameters often differ. Pricing derivatives requires risk-neutral parameters; scenario analysis for risk management may use historical parameters. ## Applications in Finance ### Bond Pricing ![The CIR model's closed-form tractability makes it applicable across bond pricing, derivatives, and mortgage securities.](data:image/svg+xml,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20viewBox%3D%220%200%20760%20211%22%20width%3D%22760%22%20height%3D%22211%22%20role%3D%22img%22%3E%3Ctitle%3EHierarchy%3C%2Ftitle%3E%3Crect%20width%3D%22100%25%22%20height%3D%22100%25%22%20fill%3D%22%23f8fafc%22%2F%3E%3Crect%20x%3D%22300%22%20y%3D%2220%22%20width%3D%22160%22%20height%3D%2258%22%20rx%3D%228%22%20fill%3D%22%232563eb%22%2F%3E%3Ctext%20x%3D%22380%22%20y%3D%2254%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2214%22%20font-weight%3D%22700%22%20fill%3D%22white%22%3ECIR%20Model%3C%2Ftext%3E%3Cpath%20d%3D%22M%20380%2078%20L%20380%20105.5%20L%20110%20105.5%20L%20110%20133%22%20stroke%3D%22%23cbd5e1%22%20stroke-width%3D%222%22%20fill%3D%22none%22%2F%3E%3Crect%20x%3D%2230%22%20y%3D%22133%22%20width%3D%22160%22%20height%3D%2258%22%20rx%3D%228%22%20fill%3D%22white%22%20stroke%3D%22%230891b2%22%20stroke-width%3D%222%22%2F%3E%3Ctext%20x%3D%22110%22%20y%3D%22158%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2213%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3EBond%20Pricing%3C%2Ftext%3E%3Ctext%20x%3D%22110%22%20y%3D%22176%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2210%22%20fill%3D%22%2364748b%22%3EClosed-form%20solution%3C%2Ftext%3E%3Cpath%20d%3D%22M%20380%2078%20L%20380%20105.5%20L%20290%20105.5%20L%20290%20133%22%20stroke%3D%22%23cbd5e1%22%20stroke-width%3D%222%22%20fill%3D%22none%22%2F%3E%3Crect%20x%3D%22210%22%20y%3D%22133%22%20width%3D%22160%22%20height%3D%2258%22%20rx%3D%228%22%20fill%3D%22white%22%20stroke%3D%22%230891b2%22%20stroke-width%3D%222%22%2F%3E%3Ctext%20x%3D%22290%22%20y%3D%22158%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2213%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3ERate%20Derivatives%3C%2Ftext%3E%3Ctext%20x%3D%22290%22%20y%3D%22176%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2210%22%20fill%3D%22%2364748b%22%3ECaps%2C%20floors%2C%20swaptions%3C%2Ftext%3E%3Cpath%20d%3D%22M%20380%2078%20L%20380%20105.5%20L%20470%20105.5%20L%20470%20133%22%20stroke%3D%22%23cbd5e1%22%20stroke-width%3D%222%22%20fill%3D%22none%22%2F%3E%3Crect%20x%3D%22390%22%20y%3D%22133%22%20width%3D%22160%22%20height%3D%2258%22%20rx%3D%228%22%20fill%3D%22white%22%20stroke%3D%22%230891b2%22%20stroke-width%3D%222%22%2F%3E%3Ctext%20x%3D%22470%22%20y%3D%22158%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2213%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3EMortgage-Backed%3C%2Ftext%3E%3Ctext%20x%3D%22470%22%20y%3D%22176%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2210%22%20fill%3D%22%2364748b%22%3EPrepayment%20modeling%3C%2Ftext%3E%3Cpath%20d%3D%22M%20380%2078%20L%20380%20105.5%20L%20650%20105.5%20L%20650%20133%22%20stroke%3D%22%23cbd5e1%22%20stroke-width%3D%222%22%20fill%3D%22none%22%2F%3E%3Crect%20x%3D%22570%22%20y%3D%22133%22%20width%3D%22160%22%20height%3D%2258%22%20rx%3D%228%22%20fill%3D%22white%22%20stroke%3D%22%230891b2%22%20stroke-width%3D%222%22%2F%3E%3Ctext%20x%3D%22650%22%20y%3D%22158%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2213%22%20font-weight%3D%22600%22%20fill%3D%22%230f172a%22%3ETerm%20Structure%3C%2Ftext%3E%3Ctext%20x%3D%22650%22%20y%3D%22176%22%20text-anchor%3D%22middle%22%20font-family%3D%22system-ui%2C-apple-system%2Csans-serif%22%20font-size%3D%2210%22%20fill%3D%22%2364748b%22%3EYield%20curve%20shapes%3C%2Ftext%3E%3C%2Fsvg%3E) *The CIR model's closed-form tractability makes it applicable across bond pricing, derivatives, and mortgage securities.* Using the closed-form solution, the CIR model prices zero-coupon bonds analytically. Coupon bonds are priced as a sum of zero-coupon bonds, each discounted using the model. ### Interest Rate Derivatives Caps, floors, and swaptions can be priced using CIR dynamics, either analytically (for some instruments) or through numerical methods (finite differences or Monte Carlo simulation). ### Mortgage-Backed Securities Mortgage prepayment behavior depends heavily on interest rates. CIR (or extended models) generate interest rate paths for pricing mortgage cash flows under different rate scenarios. ### Term Structure Generation The CIR model implies a specific term structure (yield curve shape) as a function of the current short rate and parameters. This term structure can be: - **Normal (upward sloping)**: When short rates are below long-run mean - **Inverted**: When short rates are significantly above long-run mean - **Humped**: Under certain parameter configurations ## Limitations of the CIR Model **One-factor model**: The CIR model explains the entire yield curve with a single factor (the short rate). Real yield curves often exhibit complex multi-factor dynamics (level, slope, curvature) that single-factor models can't fully capture. **Parameter instability**: CIR parameters calibrated to one time period often perform poorly out-of-sample. The "true" values of κ, θ, and σ are not stable over time. **Does not fit all yield curve shapes**: The CIR model has difficulty fitting certain observed yield curve shapes, particularly complex humped or inverted shapes, without extensions. **Negative rates in practice**: Post-2008 negative nominal rates in Europe and Japan were impossible under the original CIR model, requiring modified versions. **Extensions**: These limitations motivated more complex models: - **Two-factor CIR**: Two mean-reverting factors driving the term structure - **CIR++**: Deterministic shift added to fit the current term structure perfectly - **Affine Term Structure Models**: A broader family including CIR as a special case ## Why CIR Matters for Investors Even if you never implement the CIR equations yourself, understanding the model's core concepts matters for: **Bond duration and convexity**: The rate-dependent volatility in CIR implies that long-duration bonds exhibit more interest rate risk than Vasicek suggests, relevant for portfolio risk management. **Options on bonds**: Pricing embedded options in callable bonds or mortgage securities relies on interest rate models. The model assumptions embedded in dealers' prices affect fair value assessments. **Stress testing**: Mean-reversion parameters from CIR inform how far and how fast interest rates can reasonably move in stress scenarios, relevant for duration hedging decisions. **Fixed income derivatives**: Any cap, floor, or swaption pricing by a dealer uses interest rate model assumptions. Understanding CIR helps you think critically about derivative valuations. ## Authoritative Sources For deeper background and primary-source data on this topic, the following authoritative sources are useful starting points: - [IRS](https://www.irs.gov/) - [SEC](https://www.sec.gov/) ## Conclusion The Cox-Ingersoll-Ross model represented a major advance in quantitative finance by combining mathematical tractability with economically realistic behavior, mean reversion toward a long-run equilibrium and rate-dependent volatility that prevents negative rates. While more sophisticated multi-factor models have largely replaced CIR for frontier derivatives pricing, the model's core concepts, mean reversion, rate-level-dependent volatility, and analytical bond pricing, remain foundational to understanding how interest rate models work. For related quantitative finance and fixed income topics, see our guides on [accrued interest](/blog/accrued-interest), [weighted average maturity](/blog/weighted-average-maturity), and [netting in finance](/blog/netting-in-finance). Warren at [heywarren.com](https://heywarren.com) applies rigorous quantitative analysis to give you institutional-quality investment insights. --- ## Related Reading **More from Warren**: - [Portfolio Management Service (PMS): What It Is and How It Works](/blog/portfolio-management-service) - [Mutual Exclusivity in Finance: Meaning, Capital Budgeting, and Decision Framework](/blog/mutual-exclusivity) - [ALCO Definition: What an Asset-Liability Committee Does and Why It Matters](/blog/alco-definition) **Authoritative sources**: - [SEC EDGAR, Company Filings](https://www.sec.gov/edgar/searchedgar/companysearch) - [Federal Reserve Economic Data (FRED)](https://fred.stlouisfed.org/) - [Bureau of Labor Statistics, Data Tools](https://www.bls.gov/data/)