Term-Structure and Interest-Rate Models

Introduction and Context

The term structure of interest rates describes the relationship between the maturity of a default-free zero-coupon bond and its yield or continuously compounded spot rate. It is central to fixed-income valuation because every coupon-bearing bond, interest-rate derivative, and funding decision can be decomposed into cash flows occurring at different dates. A term structure is therefore more than a graph of yields: it is a pricing system that determines discount factors, forward rates, duration, scenario values, and no-arbitrage relationships.

Interest-rate models provide structured explanations of how rates evolve through time. They are used for bond pricing, yield-curve construction, risk measurement, hedging, mortgage valuation, callable-bond analysis, and derivatives pricing. Two broad traditions are especially important:

• Equilibrium models, such as Vasicek and Cox–Ingersoll–Ross (CIR), specify economic forces governing the short rate and derive bond prices from those assumptions.
• No-arbitrage models, such as Ho–Lee and Hull–White, are designed to reproduce today’s observed term structure exactly while specifying a risk-neutral evolution for interest rates.

The Heath–Jarrow–Morton (HJM) framework generalizes the no-arbitrage approach by modeling the entire forward-rate curve. Nelson–Siegel and Nelson–Siegel–Svensson are primarily curve-fitting models rather than complete stochastic pricing models, but they are widely used for yield-curve representation, forecasting, and calibration.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Distinguish spot rates, discount factors, forward rates, and instantaneous forward rates.
2. Explain the difference between equilibrium and no-arbitrage interest-rate models.
3. Derive and interpret the main bond-pricing results of the Vasicek and CIR models.
4. Explain the structure and applications of Ho–Lee and Hull–White models.
5. Construct and interpret short-rate binomial trees.
6. State the central no-arbitrage restriction in the HJM framework.
7. Explain Nelson–Siegel–Svensson curve modeling and practical calibration issues.
8. Select models appropriately for valuation, risk management, forecasting, and stress testing.

1. The Term Structure: Essential Building Blocks

A zero-coupon bond paying one monetary unit at time T has price P(0,T). The continuously compounded spot rate y(0,T) is defined by

where T is measured in years and y(0,T) is an annualized continuously compounded rate. Thus,

The discount factor P(0,T) is dimensionless. A spot rate is expressed as a rate per year, while the product y(0,T)T is dimensionless.

For two dates T₁ < T₂, the continuously compounded forward rate agreed at time 0 for the interval [T₁,T₂] is

Equivalently,

The instantaneous forward rate is the limiting forward rate over an infinitesimal interval:

The continuously compounded spot yield is the average instantaneous forward rate over [0,T]:

These relationships are identities, not assumptions about a particular stochastic model. They must hold whenever the curve is internally consistent.

Risk-Neutral Pricing

Under a risk-neutral measure Q, the price of a default-free claim paying X_T at time T is

where r_u is the instantaneous short rate and E^Q denotes expectation under Q. For a zero-coupon bond, X_T = 1, so

This equation explains why the distribution of the integrated short rate matters for bond pricing. It also shows why a model that describes the physical or real-world dynamics of rates is not automatically a pricing model: the appropriate risk premium or risk-neutral measure must be specified.

2. Equilibrium and No-Arbitrage Approaches

Equilibrium Models

An equilibrium model begins with assumptions about investor preferences, economic variables, and the stochastic behavior of the short rate. It derives bond prices and yields as implications of those assumptions. Vasicek and CIR are canonical one-factor equilibrium models.

Their strengths include economic interpretation, parsimonious dynamics, and analytical tractability. Their weakness is that the implied initial yield curve may differ materially from the market curve. A model can be economically coherent but poorly suited to pricing instruments whose values must be consistent with current market prices.

No-Arbitrage Models

A no-arbitrage model is calibrated so that the model reproduces the observed initial discount curve, usually exactly or to a controlled numerical tolerance. Ho–Lee and Hull–White are standard examples. Their drift terms are adjusted to match the initial term structure.

This approach is especially useful for derivative pricing and relative-value analysis. However, exact curve fit does not guarantee realistic future dynamics. A calibrated model may still generate implausible volatility, negative rates, poor mean reversion, or unstable extrapolation.

3. Vasicek Model

The Vasicek model specifies the short rate under the risk-neutral measure as

where r_t is the instantaneous short rate, a > 0 is the speed of mean reversion, b is the long-run mean level, σ > 0 is instantaneous volatility, and W_t^Q is a standard Brownian motion under Q.

The drift term a(b − r_t) pulls the rate toward b. If r_t < b, the expected change is positive; if r_t > b, it is negative. The diffusion term σ dW_t^Q introduces random shocks.

Conditional Distribution

Over a horizon h,

The conditional mean approaches b as h increases, while the conditional variance approaches σ²/(2a). The normal distribution implies that rates can become negative. This is not a mathematical error, but it may be economically undesirable in some applications.

Zero-Coupon Bond Pricing

The Vasicek bond price has the affine form

with τ = T − t and

Under the risk-neutral Vasicek specification above,

The formula assumes constant a, b, and σ and a continuously compounded short rate. It is valid under the risk-neutral dynamics stated above. In practice, the parameters under the physical measure and risk-neutral measure need not be identical.

The bond’s sensitivity to the short rate is

Thus, B(t,T) is a model-based measure of exposure to the short rate. It approaches 1/a for long maturities, reflecting mean reversion.

Interpretation and Limitations

Vasicek is analytically elegant and useful for understanding mean reversion and affine pricing. However:

• Rates may become negative because shocks are normally distributed.
• A single factor cannot reproduce all movements of a real yield curve.
• Constant volatility is often unrealistic.
• The equilibrium parameters may not fit today’s curve without modification.

4. CIR Model

The CIR model modifies the diffusion coefficient so that volatility depends on the level of the short rate:

The variables have the same general interpretation as in Vasicek, but σ√r_t is now the instantaneous volatility.

Positivity Condition

The CIR process remains strictly positive when the Feller condition holds:

If this condition holds and r₀ > 0, the process does not reach zero under standard regularity conditions. If the condition fails, the process may approach or reach zero, although it remains nonnegative in the standard CIR formulation. The condition is therefore a sufficient positivity condition, not merely a calibration preference.

Economic Interpretation

CIR embodies a volatility-level effect: rates have greater absolute volatility when rates are high and lower volatility near zero. This may be more realistic than the constant-volatility Vasicek model for certain markets.

The zero-coupon bond again has an affine form:

Define

Then

and

where τ = T − t. The formula assumes constant parameters and risk-neutral CIR dynamics.

Vasicek and CIR Compared

CIR avoids negative rates under suitable conditions and incorporates level-dependent volatility. Vasicek is simpler, often easier to estimate, and permits closed-form expressions that are computationally convenient. Neither model is automatically superior: the choice depends on the market regime, instrument set, and purpose of the analysis.

5. Ho–Lee and Hull–White No-Arbitrage Models

Ho–Lee Model

The Ho–Lee short-rate model is commonly written under Q as

where θ(t) is a deterministic time-dependent drift selected to fit the initial term structure, and σ is constant volatility.

In a bond-market implementation, the drift or tree parameters are chosen so that the present values of model-generated cash flows equal observed market discount factors. Ho–Lee is a direct no-arbitrage model, but it has no mean reversion. Consequently, long-horizon rate variance can grow substantially, and negative rates are possible.

A standard continuously compounded formulation with initial instantaneous forward curve f(0,t) uses a drift adjustment related to the derivative of the forward curve and the volatility convexity term. Exact expressions depend on whether the state variable is the short rate, a forward rate, or a discretized tree state; therefore calibration conventions must be stated explicitly rather than transferring formulas between implementations.

Hull–White Model

The one-factor Hull–White model extends Vasicek by making the mean level time-dependent:

where a > 0 is mean-reversion speed, σ is volatility, and θ(t) is selected to fit the initial curve. An equivalent form is

where α(t) is a time-varying target level.

The model retains Gaussian dynamics, so negative rates remain possible. Its major practical advantage is that the deterministic function θ(t) allows exact matching of the initial discount curve while a and σ govern future dynamics.

For constant a and σ, bond prices remain affine:

where

The function A(t,T) is determined by the initial discount curve and the volatility adjustment. In implementation, it is usually safer to calculate A from observed discount factors and the model’s bond-volatility function than to rely on an unqualified memorized expression, because alternative definitions of θ(t) produce different-looking but equivalent formulas.

Why Hull–White Is Widely Used

Hull–White is common in interest-rate derivatives because it offers:

• exact or near-exact initial curve fit;
• mean reversion;
• analytical or semi-analytical bond prices;
• efficient lattice construction;
• compatibility with caps, floors, swaptions, callable bonds, and mortgage-related products.

Its limitations include Gaussian negative rates, one-factor restrictions, parameter instability, and the difficulty of explaining a time-dependent drift economically.

6. Short-Rate Binomial Trees

A short-rate tree represents possible future short rates at discrete dates. At each node, the model supplies an up and down successor, with risk-neutral probabilities used for valuation. Trees are particularly useful for path-dependent or early-exercise securities, including callable bonds and American-style options.

For a recombining tree with time step Δt, a simple additive rate movement may be written as

and

where n indexes time, j indexes the state, μ is a local drift, and σ√Δt is the approximate one-step shock. The precise tree construction depends on the underlying continuous-time model.

For a claim with values V_u and V_d at the next nodes, the risk-neutral value before cash flow is

where p is the risk-neutral up probability. If a cash flow C_n is received at the node, the total value is V_n + C_n, subject to the security’s contractual timing convention.

Calibration Procedure

A practical tree calibration generally follows these steps:

1. Bootstrap market discount factors from deposits, futures, swaps, and bonds.
2. Choose a volatility and tree time grid.
3. Adjust node shifts or drift terms so that model bond prices equal market discount factors at every maturity.
4. Value securities by backward induction.
5. Test the tree against independent instruments and stress scenarios.

A tree that fits the initial curve but produces implausible option prices is not successful. Calibration must therefore include both fit diagnostics and economic plausibility checks.

7. Heath–Jarrow–Morton Framework

HJM models the instantaneous forward-rate curve directly. Under the risk-neutral measure, the forward rate f(t,T) evolves as

where T is the maturity of the forward rate, α(t,T) is its drift, σ(t,T) is a volatility vector, and W_t^Q is a vector Brownian motion.

The central HJM no-arbitrage condition is

provided the model is formulated under Q with the usual regularity conditions and the money-market account is the numeraire. In a one-factor model, the dot product becomes ordinary multiplication.

This restriction is crucial: the forward-rate drift cannot be selected independently of forward-rate volatility. Once volatility is specified, no-arbitrage determines the drift.

The associated bond price is

HJM is a framework rather than a single model. Different choices of σ(t,T) generate different models, including structures related to Ho–Lee and Hull–White. Its main advantage is conceptual consistency across maturities. Its challenges include high dimensionality, volatility-surface calibration, computational cost, and sensitivity to the specification of forward-rate volatility.

8. Nelson–Siegel–Svensson Curve Modeling

Nelson–Siegel–Svensson (NSS) is a parsimonious method for representing the cross-section of spot yields. For maturity τ, the NSS yield curve is

Here β₀ controls the long-run level, β₁ primarily controls the short-term slope, β₂ controls a medium-term curvature component, β₃ adds a second curvature or hump component, and λ₁ and λ₂ determine the locations of the curvature effects. τ and λ values must use the same time unit.

The Nelson–Siegel model is obtained by omitting the β₃ term. The factor loadings are dimensionless, so the β coefficients have the units of yield, usually decimal annual rates.

Calibration

Suppose observed yields y_i^obs exist at maturities τ_i. Parameters can be estimated by minimizing a weighted least-squares objective:

where w_i are nonnegative weights. If the objective uses yields, the residuals are yield differences. If it uses bond prices, price residuals should be scaled consistently because price and yield errors have different economic meanings.

For fixed λ₁ and λ₂, the model is linear in β₀, β₁, β₂, and β₃. This permits efficient variable projection: search numerically over λ values while solving the β coefficients by linear least squares at each trial. Constraints such as λ₁ > 0 and λ₂ > 0 are necessary. Additional ordering constraints may improve identification but should be justified rather than imposed mechanically.

Uses and Limitations

NSS is valuable for:

• summarizing a curve with a small number of interpretable factors;
• comparing curve movements across dates;
• forecasting and macro-financial analysis;
• generating smooth discount curves from irregular observations;
• providing starting values for more complex models.

NSS is not, by itself, a complete arbitrage-free stochastic model. A fitted yield curve may fail to reproduce individual bond prices exactly, generate unstable extrapolation, or imply problematic forward-rate behavior. Calibration should therefore examine fitted prices, spot rates, forward rates, parameter stability, and out-of-sample performance.

9. Worked Example: From Discount Factors to Forward Rates

Assume the observed discount factors are P(0,1) = 0.9700 and P(0,2) = 0.9320. The one-year and two-year continuously compounded spot rates are

and

The one-year forward rate beginning at year 1 is

The result is consistent with the identity

because 0.9700 × exp(−0.03996) is approximately 0.9320. The forward rate is higher than the one-year spot rate because the two-year discount factor declines more rapidly over the second year.

10. Illustrative Case: Calibrating a Callable-Bond Risk System

Background

A bank holds a portfolio of callable corporate bonds. The bonds contain issuer call rights, so their values depend on the future path of interest rates rather than only on today’s yield curve.

Problem

The existing valuation process discounts expected cash flows using a static curve. It cannot value the embedded call consistently, does not reproduce market swaption prices, and produces unstable hedge ratios when the curve changes shape.

Analysis

The bank first bootstraps discount factors from liquid government and swap instruments. It then estimates a one-factor Hull–White model using the initial curve and a volatility surface from caps and swaptions. A calibrated recombining short-rate tree is constructed. At each node, the callable bond is valued by comparing continuation value with the contractual call price.

The bank also fits an NSS curve for reporting and scenario generation. NSS is not used as the stochastic pricing engine; it provides a smooth representation of the observed curve and interpretable level, slope, and curvature shocks.

Approach and Solution

The implementation includes:

1. independent validation of discount-factor bootstrapping;
2. exact tree fit to benchmark zero-coupon prices;
3. calibration tests against liquid option instruments;
4. backward induction for the issuer’s optimal call decision;
5. sensitivities to parallel, slope, curvature, volatility, and mean-reversion shocks;
6. model-risk limits and challenger valuations using a second model.

Outcome

The bank obtains more stable hedge ratios and a valuation consistent with the embedded option. It discovers that the callable bond’s duration shortens sharply when rates fall because the probability of exercise rises. The analysis also reveals that results are highly sensitive to mean reversion and volatility assumptions, even though the initial curve fit is exact.

Learning Points

A current-curve fit is necessary but insufficient. Callable-bond valuation requires a dynamic model, a calibrated volatility structure, contractual exercise logic, and governance over model uncertainty. NSS and Hull–White serve different purposes: one summarizes the curve, while the other supplies a stochastic pricing framework.

11. Model Selection, Governance, and Contemporary Practice

Model selection should follow the instrument and decision problem. NSS is appropriate for curve description and factor analysis. Vasicek and CIR are useful for theoretical analysis and equilibrium interpretation. Hull–White and calibrated trees are often effective for callable and option-embedded securities. HJM is suitable when the dynamics of the entire forward curve and volatility structure are central.

Important governance practices include:

• documenting the pricing measure, compounding convention, day-count basis, and calendar assumptions;
• separating market-data errors from model errors;
• validating curve fit in both yields and prices;
• testing positivity, monotonicity, and absence of obvious forward-rate oscillations;
• assessing parameter stability through time;
• performing back-testing and out-of-sample validation;
• using challenger models and sensitivity analysis;
• controlling extrapolation beyond liquid maturities;
• retaining audit trails for calibration inputs and overrides.

Regulatory and accounting consequences may depend on the purpose of the valuation. Fair-value measurements, prudent valuation adjustments, capital calculations, and hedge-accounting documentation may require transparent assumptions and independent price verification. Artificial intelligence can assist with data cleaning, anomaly detection, parameter initialization, and regime classification, but it should not replace economic validation or model governance. Machine-learning curves may fit historical data extremely well while violating no-arbitrage relationships, producing unstable extrapolation, or becoming opaque during stressed markets.

Contemporary practice increasingly combines interpretable parametric curves, multi-curve frameworks for collateralized derivatives, stochastic volatility, regime-sensitive estimation, and machine-learning diagnostics. These developments improve flexibility but increase the need for controls, explainability, and validation.

Challenges and Limitations

No interest-rate model is universally adequate. Major limitations include:

• Parameter uncertainty: estimates of mean reversion and volatility are noisy and dependent on the sample period.
• Structural breaks: monetary-policy regimes, inflation dynamics, and market conventions change.
• One-factor restrictions: a single factor cannot fully represent independent movements in short, intermediate, and long maturities.
• Negative-rate behavior: Gaussian models permit negative rates, while square-root models may be too restrictive in markets where rates can approach zero.
• Liquidity and credit contamination: observed instruments may not be truly default-free.
• Curve-construction dependence: model conclusions inherit errors from bootstrapping and interpolation.
• Calibration instability: several parameter combinations may generate similar prices but different future dynamics.
• Volatility inconsistency: fitting the yield curve does not imply fitting caps, floors, and swaptions.

The correct response is not to seek a universally true model, but to align model complexity with material risks, validate assumptions, and communicate uncertainty clearly.

Chapter Summary

The term structure links discount factors, spot rates, forward rates, and bond prices. Equilibrium models such as Vasicek and CIR derive prices from assumed short-rate dynamics, while no-arbitrage models such as Ho–Lee and Hull–White are calibrated to reproduce the observed initial curve. Vasicek is Gaussian and mean reverting but permits negative rates. CIR introduces square-root volatility and can preserve positivity under the Feller condition. Ho–Lee provides a simple curve-fitting framework without mean reversion; Hull–White adds mean reversion and remains highly practical for derivative valuation.

Binomial trees discretize short-rate dynamics and support backward induction for securities with early exercise. HJM models the entire forward curve and imposes a no-arbitrage restriction linking forward-rate drift to volatility. NSS represents the yield curve with interpretable level, slope, and curvature factors, but it is a curve-fitting framework rather than a complete stochastic pricing model. Effective practice requires careful calibration, consistent conventions, independent validation, stress testing, and explicit model governance.

Key Terms

Term structure: Relationship between interest rates or yields and maturities.

Discount factor: Present value of one monetary unit paid at a specified future date.

Spot rate: Yield applying to a zero-coupon investment from today to a given maturity.

Forward rate: Rate agreed today for borrowing or investing over a future interval.

Instantaneous forward rate: Limiting forward rate over an infinitesimally short interval.

Short rate: Instantaneous risk-free interest rate used in continuous-time models.

Risk-neutral measure: Pricing probability measure under which discounted traded-asset prices are martingales.

Equilibrium model: Model deriving prices from economic assumptions about rates and investor behavior.

No-arbitrage model: Model calibrated to prevent pricing inconsistencies with observed market instruments.

Mean reversion: Tendency of a variable to move toward a long-run level.

Vasicek model: Gaussian mean-reverting short-rate model with constant volatility.

CIR model: Square-root short-rate model with level-dependent volatility.

Feller condition: Sufficient CIR condition, 2ab ≥ σ², for strict positivity.

Hull–White model: Time-dependent Gaussian mean-reverting model designed to fit the initial curve.

Short-rate tree: Discrete lattice representing possible future short-rate paths.

HJM framework: No-arbitrage framework modeling the evolution of forward rates across maturities.

Nelson–Siegel model: Parsimonious parametric representation of the yield curve.

Nelson–Siegel–Svensson model: Extended Nelson–Siegel curve with a second curvature factor.

Calibration: Estimation or adjustment of model parameters to market observations.

Model risk: Risk of loss caused by model error, inappropriate assumptions, or implementation failure.

Review Questions

Short Answer Questions

1. Define a discount factor and explain its relationship with a continuously compounded spot rate.
2. What is the principal difference between an equilibrium model and a no-arbitrage model?
3. Why can the Vasicek model generate negative interest rates?
4. State the Feller condition for the CIR model and explain its significance.
5. What is the central no-arbitrage restriction in the HJM framework?

Long Answer Questions

1. Compare the Vasicek and CIR models in terms of dynamics, positivity, volatility, tractability, and practical limitations.
2. Explain how Hull–White extends Vasicek and why it is useful for pricing interest-rate derivatives.
3. Describe the construction and calibration of a short-rate binomial tree.
4. Explain the structure, interpretation, and calibration of the Nelson–Siegel–Svensson model.
5. Discuss why fitting the initial yield curve does not guarantee reliable derivative pricing.

Analytical / Application Questions

1. Given P(0,1) = 0.9700 and P(0,2) = 0.9320, calculate the one-year and two-year continuously compounded spot rates and the one-year forward rate beginning in year 1.
2. A callable bond is valued using a static discount curve. Explain why this approach is inadequate and propose a dynamic valuation framework.
3. A calibrated model fits the current curve exactly but generates highly unstable long-term forward rates. Diagnose possible causes and recommend validation and governance actions.

Multiple Choice Questions with Answers

Question 1

Which expression defines the continuously compounded spot rate for maturity T?

A. y(0,T) = ln(P(0,T)) ÷ T

B. y(0,T) = −ln(P(0,T)) ÷ T

C. y(0,T) = P(0,T) ÷ T

D. y(0,T) = T ÷ P(0,T)

Correct Answer: B. y(0,T) = −ln(P(0,T)) ÷ T

Question 2

Which feature distinguishes an equilibrium model from a no-arbitrage model?

A. An equilibrium model must fit every observed bond price exactly.

B. A no-arbitrage model cannot contain stochastic interest rates.

C. An equilibrium model derives prices from economic assumptions, whereas a no-arbitrage model is calibrated to current market prices.

D. A no-arbitrage model does not require a pricing measure.

Correct Answer: C. An equilibrium model derives prices from economic assumptions, whereas a no-arbitrage model is calibrated to current market prices.

Question 3

Under the standard Vasicek model, what is the main implication of normally distributed short-rate shocks?

A. Rates can become negative.

B. Rates are guaranteed to remain positive.

C. Volatility is proportional to the square root of the rate.

D. The model has no mean reversion.

Correct Answer: A. Rates can become negative.

Question 4

Which condition is the standard Feller positivity condition for the CIR model?

A. a + b ≥ σ²

B. 2ab ≥ σ²

C. aσ ≥ 2b

D. b² ≥ 2aσ

Correct Answer: B. 2ab ≥ σ²

Question 5

What is the principal purpose of the time-dependent drift in the Hull–White model?

A. To eliminate all interest-rate volatility.

B. To guarantee that rates remain positive.

C. To fit the observed initial term structure.

D. To make the model independent of market data.

Correct Answer: C. To fit the observed initial term structure.

Question 6

In a risk-neutral short-rate tree, the value before the next cash flow is generally calculated as:

A. The arithmetic average of all future values without discounting.

B. The discounted risk-neutral expected value of successor-node values.

C. The maximum of the up and down successor values for every security.

D. The current short rate multiplied by maturity.

Correct Answer: B. The discounted risk-neutral expected value of successor-node values.

Question 7

What does the HJM no-arbitrage condition primarily determine?

A. The forward-rate drift from the forward-rate volatility structure.

B. The bond coupon from the short-rate level.

C. The number of maturities in the yield curve.

D. The credit spread from the risk-free rate.

Correct Answer: A. The forward-rate drift from the forward-rate volatility structure.

Question 8

In the Nelson–Siegel–Svensson model, β₀ primarily represents:

A. The long-maturity level of the curve.

B. The probability of default.

C. The short-rate volatility.

D. The tree’s risk-neutral probability.

Correct Answer: A. The long-maturity level of the curve.

Question 9

Using P(0,1) = 0.9700 and P(0,2) = 0.9320, the continuously compounded one-year forward rate for years 1 to 2 is approximately:

A. 1.96%

B. 3.05%

C. 4.00%

D. 6.80%

Correct Answer: C. 4.00%

Question 10

Why is NSS not, by itself, a complete stochastic interest-rate pricing model?

A. It cannot represent any yield-curve shapes.

B. It summarizes the cross-sectional curve but does not specify complete future stochastic dynamics and risk-neutral pricing rules.

C. It requires all rates to be negative.

D. It is defined only for one maturity.

Correct Answer: B. It summarizes the cross-sectional curve but does not specify complete future stochastic dynamics and risk-neutral pricing rules.