Economic Modelling · Models of the term structure of interest rates

Calibration and Practical Use of Interest Rate Models

Updated 11 October 2026 · Fact-checked

Calibration means choosing model parameters so the model reproduces observed market prices, such as the yield curve or cap volatilities. You then use the fitted model to value derivatives and guarantees. Arbitrage-free models fit today's curve exactly. Equilibrium models fit it only approximately, through a few parameters.

Understand Calibration and Practical Use of Interest Rate Models

A short rate model describes how the interest rate moves over time. Before you use it for pricing, its parameters must be set. Calibration is that setting step. You pick parameters so that model prices match prices you can see in the market.

There are two broad families. Equilibrium models (Vasicek, Cox-Ingersoll-Ross) have a few constant parameters. They produce a yield curve as an output. That curve will not match today's market curve exactly, because you have too few parameters to fit every maturity. Arbitrage-free models (Hull-White, Ho-Lee, Black-Karasinski) add a time-dependent term, often written θ(t) or a function of t. You choose this term so the model reproduces today's yield curve exactly. So the initial curve is an input, not an output.

Two kinds of data are used. The yield curve (zero-coupon bond prices) fixes the drift or the level. Option prices, such as caps, floors and swaptions, give implied volatilities. These fix the volatility and mean reversion parameters. Volatility cannot be read from the yield curve alone.

Practical use follows. You calibrate to instruments that are similar to what you want to value. To value an interest rate cap, calibrate to caps. To value a maturity guarantee on a policy, you need long-term rates and volatilities, so you use long-dated instruments. Under a risk-neutral approach, you value by discounting expected payoffs at the risk-free rate. Under a real-world approach, you use the model to project scenarios for risk management.

Calibration has limits. Parameters can be unstable over time. A model that fits one set of options may misprice others. Fitting more parameters improves the fit but can lead to overfitting. Always check the fitted model against instruments you did not use to fit it.

Key rules to remember

Vasicek short rate model: dr = a(b − r)dt + σ dW; a is the speed of mean reversion, b the long-run mean, σ the volatility. Rates can be negative. Equilibrium model with constant parameters.
Hull-White (extended Vasicek) model: dr = [θ(t) − a r]dt + σ dW; θ(t) is chosen to fit the initial yield curve exactly. This is what makes it arbitrage-free.
Ho-Lee model: dr = θ(t)dt + σ dW; No mean reversion. θ(t) fits the initial curve.
CIR model: dr = a(b − r)dt + σ√r dW; Volatility rises with the rate. Rates stay non-negative if 2ab ≥ σ².
Zero-coupon bond price: P(0,T) = exp(−∫₀ᵀ f(0,s) ds) = exp(−T y(0,T)); f is the instantaneous forward rate and y the continuously compounded spot rate. This is the target the calibrated model must match.
Risk-neutral price of a derivative: V(0) = E_Q[ exp(−∫₀ᵀ r(s) ds) × payoff ]; Expectation is under the risk-neutral measure Q, using the model calibrated to market prices.
Calibration objective: minimise Σ (model price − market price)²; Least squares over chosen instruments. Weighting by vega or using relative errors is also common.

How to solve Calibration and Practical Use of Interest Rate Models questions

Use this order for any calibration or application question.

1State the purpose: what must be valued, and over what term.
2Choose the model type and give a reason. Use an arbitrage-free model if the question needs an exact fit to today's curve. Use an equilibrium model if you want simplicity or a long-term view of rates.
3List the parameters to find and the market data available: yield curve, cap or swaption volatilities, bond prices.
4Use the yield curve to fix the drift or θ(t). Use option volatilities to fix σ and the mean reversion speed a.
5Set up the fitting criterion, such as least squares between model and market prices, and say what you are minimising over.
6Value the instrument under the risk-neutral measure using the calibrated model, by formula or simulation.
7Check the result against instruments not used in the fit, and comment on parameter stability and model risk.
8State your assumptions clearly, such as risk-neutral pricing, no transaction costs and the form of volatility.

Quickest way: Match, fit, price

When to use it: Use this for short written parts and MCQs asking which model or data to use.

Ask: does the model need to match today's yield curve exactly? If yes, choose an arbitrage-free model with θ(t).
Ask: what carries information about volatility? Option prices, not the yield curve.
Match the calibration instruments to the product: caps for caps, long-dated swaptions for long guarantees.
Price by risk-neutral discounting, then add one line on limitations.

Common mistakes in Calibration and Practical Use of Interest Rate Models

Saying an equilibrium model such as Vasicek fits today's yield curve exactly.
Students mix up equilibrium and arbitrage-free models.
Fix: Constant-parameter equilibrium models generate their own curve. Only models with a time-dependent term, like Hull-White, are fitted exactly.
Trying to find volatility from the yield curve alone.
The curve is the most visible market data.
Fix: Volatility comes from option prices or from historical rate changes. The curve gives levels, not variability.
Calibrating to instruments unlike the product being valued.
Students use whatever data is given.
Fix: Match the instruments to the product's maturity and type. A 20-year guarantee needs long-dated data.
Using real-world parameters to price a derivative under the risk-neutral measure.
The two measures are confused.
Fix: For market-consistent pricing, calibrate to market prices and discount at the risk-free rate. Use real-world parameters for projecting risk.
Treating a perfect fit as proof of a good model.
A close fit looks convincing.
Fix: Mention overfitting, unstable parameters and testing on instruments outside the fitting set.
Forgetting that Vasicek and Hull-White can give negative rates.
Students remember only the mean reversion feature.
Fix: State the negative rate possibility as a limitation, and note that CIR avoids it under a condition on its parameters.

Worked examples

Example 1

A company wants to value a 10-year cap on 1-year rates. It has today's zero-coupon yield curve and quoted cap volatilities. Explain how you would calibrate a Hull-White model and why you would choose it over Vasicek.

Show the solution

Purpose: price a cap, so the model must be consistent with market prices of similar caps and with today's discount curve.
Model choice: Hull-White has dr = [θ(t) − a r]dt + σ dW. θ(t) is a free function, so the model can reproduce today's zero-coupon bond prices exactly. Vasicek with constant parameters cannot.
Step 1: choose θ(t) so that model P(0,T) equals the market P(0,T) for all T. This uses the yield curve.
Step 2: choose a and σ to minimise the sum of squared differences between model cap prices and market cap prices, or between implied volatilities. The cap volatilities carry the volatility information.
Step 3: value the cap as a sum of caplets, each a call on the rate, using the fitted model under the risk-neutral measure.
Step 4: test the model on caps or swaptions not used in the fit, and comment on stability of a and σ.

Answer: Fit θ(t) to the yield curve, fit a and σ to cap volatilities by least squares, then price the cap risk-neutrally. Hull-White is chosen because it fits today's curve exactly, which Vasicek cannot.

Example 2

Under a Vasicek model dr = a(b − r)dt + σ dW, a = 0.25, b = 8% and r(0) = 6%. Find the expected short rate at time 4 under the model. Give the result to 4 decimal places and explain what this shows about fitting.

Show the solution

For Vasicek, E[r(t)] = b + (r(0) − b)e^(−at).
Substitute: E[r(4)] = 0.08 + (0.06 − 0.08) × e^(−0.25 × 4).
e^(−1) = 0.367879.
(0.06 − 0.08) = −0.02, and −0.02 × 0.367879 = −0.0073576.
E[r(4)] = 0.08 − 0.0073576 = 0.0726424.
So the expected rate is 7.26% to two decimal places in percent, or 0.0726 as a decimal.

Answer: E[r(4)] = 0.0726 (7.26%). The model's expected path moves from 6% towards 8% at a speed set by a. With only a, b and σ, the model cannot match an arbitrary market curve exactly, which is why Hull-White adds θ(t).

Exam tips

When asked to compare models, always say which one fits today's yield curve exactly and why: θ(t) is time-dependent.
Name the data for each parameter: the curve for drift or level, options for volatility and mean reversion.
For guarantees and derivatives, say that valuation is risk-neutral and uses market-consistent calibration. Mention that long-dated products need long-dated data.
In written answers, finish with one or two limitations: negative rates, parameter instability, overfitting and model risk.
Practise Vasicek expectations. E[r(t)] = b + (r(0) − b)e^(−at) is a common short calculation.

Practice questions from Models of the term structure of interest rates

Calibration and Practical Use of Interest Rate Models: frequently asked questions

What is the difference between equilibrium and arbitrage-free interest rate models?

Equilibrium models have constant parameters and generate a yield curve as an output, so they match the market curve only roughly. Arbitrage-free models have a time-dependent term that is fitted so the model reproduces today's curve exactly. The second type is preferred for pricing derivatives.

How do you calibrate an interest rate model to market data?

Choose parameters so that model prices match market prices for chosen instruments. Use the yield curve to fix the drift or θ(t) and option prices such as caps and swaptions to fix volatility and mean reversion. A common method is least squares on the price or volatility differences.

Which interest rate model is used for valuing insurance guarantees?

Any model that can be calibrated to market prices can be used. Arbitrage-free models are common for market-consistent valuation. The model should be calibrated to long-dated instruments that match the term of the guarantee.

Why can't the yield curve alone give the volatility parameter?

The yield curve shows rate levels at one date and says little about how rates move. Volatility is found from option prices, which depend on future rate variability, or from historical rate changes.