Interest Rate Models (Vasicek, CIR, Nelson-Siegel)
Interest Rate Term-Structure Models (Vasicek, CIR, Nelson-Siegel) · Also known as: term structure models, short-rate models, yield curve models, Vasicek model, CIR model, Nelson-Siegel model, Faiz Oranı Modelleri (Vasicek, CIR, Nelson-Siegel)
Interest rate models are structural models that describe how interest rates evolve over time within a stochastic differential equation framework. The family covers Vasicek's normal short-rate process (1977), the CIR square-root process, the adjustable Hull-White extension, and the Nelson-Siegel approach to fitting the yield curve (1987).
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When to use it
Use interest rate models when you have a continuous interest-rate or yield time series of reasonable length (at least about 60 observations) and want to model term-structure dynamics, forecast rates, or price interest-rate-sensitive instruments. Pick the model to match your need: Vasicek for analytic tractability when negative rates are acceptable, CIR when positivity must be enforced via the Feller condition, Hull-White when the model must reprice market instruments exactly, and Nelson-Siegel when the goal is to fit and summarise the shape of an observed yield curve. Calibration can be done to historical data or to current market prices.
Strengths & limitations
- Provides a coherent stochastic framework for the dynamics of interest rates rather than treating each maturity in isolation.
- Offers a model for every need: Vasicek gives analytic tractability, CIR guarantees positive rates under the Feller condition, Hull-White matches market prices exactly, and Nelson-Siegel parsimoniously fits the yield curve with three interpretable factors.
- Calibration can use either historical rate series or current market prices, supporting both forecasting and pricing applications.
- The Vasicek model can produce negative interest rates, which may be unrealistic in some regimes.
- Single-factor short-rate models impose a rigid structure and may not fit the entire observed yield curve well.
- Calibration is sensitive to the data window and to the choice of the market price of risk λ, and Nelson-Siegel is a descriptive curve fit rather than a no-arbitrage dynamic model.
Frequently asked
What is the difference between Vasicek and CIR?
Both are mean-reverting short-rate models, but Vasicek uses a constant diffusion term and can therefore produce negative rates, while CIR scales the diffusion by the square root of the rate so that, under the Feller condition, the rate is guaranteed to stay positive.
What does the market price of risk λ represent?
It captures the difference between the physical (real-world) probability measure and the risk-neutral measure used for pricing. You must account for λ explicitly when translating model dynamics estimated on historical data into prices.
How does Nelson-Siegel differ from the short-rate models?
Nelson-Siegel is a parsimonious curve-fitting model: it describes the shape of the yield curve through three factors — level, slope, and curvature — rather than specifying a stochastic differential equation for how the rate evolves. It is descriptive rather than a no-arbitrage dynamic model.
How much data do I need to calibrate these models?
A reasonable length of interest-rate time series is required — at least about 60 observations. Calibration can be performed on a historical series or on a cross-section of current market prices, depending on whether you want real-world or risk-neutral parameters.
Sources
- Vasicek, O. (1977). An Equilibrium Characterization of the Term Structure. Journal of Financial Economics, 5(2), 177–188. DOI: 10.1016/0304-405X(77)90016-2 ↗
- Nelson, C. R. & Siegel, A. F. (1987). Parsimonious Modeling of Yield Curves. Journal of Business, 60(4), 473–489. DOI: 10.1086/296409 ↗
How to cite this page
ScholarGate. (2026, June 1). Interest Rate Term-Structure Models (Vasicek, CIR, Nelson-Siegel). ScholarGate. https://scholargate.app/en/finance/interest-rate-models
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