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Home / Financial Tools / Short-Rate Term Structure Model Suite: Vasicek, CIR & Hull-White

Graduate-Level Modeling · Term Structure & Fixed Income Derivatives

Short-Rate Term Structure Model Suite: Vasicek, CIR & Hull-White

Three models, one short rate. Vasicek and CIR approximate the market curve. Hull-White is built to match it exactly, and this tool proves it, basis point by basis point.

How To Use This Model

Reading This Tool

Calibrate and compare three short-rate models side by side: Vasicek, CIR, and Hull-White, each with its own mean reversion, volatility and simulated rate paths.

Vasicek and CIR only approximately fit an observed yield curve. Hull-White is built specifically to fit it exactly, by solving for a time-varying drift. This tool proves that difference directly: it prices zero-coupon bonds under all three models against the same input market curve, runs full Monte Carlo path simulation, and prices a bond option in closed form using the Jamshidian formula.

Model Selection & Shared Settings

Vasicek Parameters

CIR Parameters

Hull-White Parameters

Input Market Curve (Nelson-Siegel, For Hull-White Calibration)

Bond Option (Jamshidian, Vasicek/Hull-White)

CIR requires the Feller condition 2κθ ≥ σ² to guarantee the short rate never hits zero, this tool checks it live and flags a violation.

Simulated Short-Rate Distribution At Horizon

Vasicek

Mean Rate At Horizon

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Std. Dev. At Horizon

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5th / 95th Percentile

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% Of Paths Below Zero

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Simulated Short-Rate Paths (Fan Chart, 5th–95th Percentile Band)

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Model-Implied Zero Curves vs. The Input Market Curve

Market Curve (Nelson-Siegel) Vasicek-Implied CIR-Implied Hull-White-Implied

Fit Error vs. Market Curve (Basis Points)

Zero-Coupon Bond & Option Pricing

Bond Price (Selected Model)

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Implied Zero Yield

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Bond Option Value (Notional)

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Forward Bond Price At Option Expiry

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Bond Price At This Maturity, Across All Three Models

Full Term Structure Of Bond Prices & Yields (Selected Model)

Why Hull-White Fits The Curve And The Others Don't

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What Mean Reversion Speed Is Actually Controlling

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Why CIR Exists At All Given Vasicek Already Works

Vasicek's short rate is Normally distributed, which means it can go negative with positive probability, mathematically convenient but economically awkward for a rate that used to be assumed non-negative. CIR's square-root diffusion term keeps the process non-negative (given the Feller condition holds), at the cost of losing Vasicek's simple closed-form path distribution. Since rates went negative in several real economies last decade, this "problem" CIR was built to solve is less universally a problem than it once seemed, but the mathematics is still foundational.

Reading The Bond Option Price

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Term Structure Modeling & Fixed Income Derivatives

The Core Formulas

Vasicek: dr = κ(θ−r)dt + σdW, P(t,T)=A·e−Br, B=(1−e−κτ)/κ
CIR: dr = κ(θ−r)dt + σ√r·dW, γ=√(κ²+2σ²)
Hull-White: dr = [θ(t) − κr]dt + σdW, θ(t) = ∂f(0,t)/∂t + κf(0,t) + σ²/(2κ)·(1−e−2κt)
Jamshidian Bond Option: Call = P(0,S)N(d₁) − K·P(0,T)N(d₂), σP = σB(T,S)√[(1−e−2κT)/(2κ)]

Hull-White's θ(t) is derived directly, in closed form, from the Nelson-Siegel input curve's instantaneous forward rate and its time-derivative, which is exactly what makes the model reproduce that curve exactly rather than merely approximately.

When To Actually Use Each Model

  • Vasicek: teaching the foundational affine term structure model, simplest closed forms, tractable but allows negative rates.
  • CIR: modeling a rate process that must stay strictly positive, subject to the Feller condition holding.
  • Hull-White: pricing interest rate derivatives (caps, floors, swaptions, bond options) consistently with today's observed market curve, the industry-standard no-arbitrage short-rate model for exactly this reason.

Key Assumptions & Limitations

  • This tool uses a single-factor short-rate model for all three specifications, real trading desks often use two-factor or three-factor extensions to capture curve twists a single factor can't.
  • The "market curve" here is a smooth, four-parameter Nelson-Siegel form for tractability, real calibration fits directly to observed bond and swap prices with far more curve points.
  • CIR bond options require a non-central chi-squared distribution rather than the Gaussian-based Jamshidian formula shown here, that pricing is not implemented in this tool.

Foundational References

Vasicek, O. (1977). An Equilibrium Characterization of the Term Structure. Journal of Financial Economics, 5(2), 177-188.

Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985). A Theory of the Term Structure of Interest Rates. Econometrica, 53(2), 385-407.

Hull, J., & White, A. (1990). Pricing Interest-Rate-Derivative Securities. Review of Financial Studies, 3(4), 573-592.

Jamshidian, F. (1989). An Exact Bond Option Formula. The Journal of Finance, 44(1), 205-209.

Want the empirical curve-fitting side of this too?

Open The Nelson-Siegel Calibrator →