FRM Exam Part II · The Vasicek and Gauss+ Models

Gauss+ Model: Calibration, Risk and Applications

Updated 11 October 2026 · Fact-checked

The Gauss+ model is a multifactor interest rate model built on mean-reverting Gaussian factors, with an added nonlinear term. You calibrate it to reprice today's curve, then match market volatilities. For hedging, you measure sensitivity to each factor and offset it with as many instruments as factors.

Understand Gauss+ Model: Calibration, Risk and Applications

Start with the problem. A one-factor model such as Vasicek moves every rate with one shock. That forces all rates to be perfectly correlated, so it cannot describe a curve that steepens, flattens or twists. Real curves move in more than one way, so you need more than one factor.

The Gauss+ model is a multifactor model. Its core is made of Gaussian (normally distributed) mean-reverting factors. Each factor has its own mean-reversion speed and volatility, and the factors can be correlated. The "+" refers to an added nonlinear term, which takes the model beyond a purely Gaussian one. Check your reading for the exact specification and for what it says about volatility, and do not assume more than it states. Remember the idea: Gaussian factors for flexible curve moves, plus a nonlinear extension.

Calibration means choosing parameters so the model matches the market. There are two parts. First, fit the observed curve: the model is set up so it reprices today's discount curve (for example, the USD swap curve). This is a no-arbitrage requirement, and the fit is exact by construction of the drift or shift term. Second, fit volatility: you choose the factor volatilities, mean-reversion speeds and correlation so model prices of caps, floors or swaptions are close to market prices or implied volatilities. This is a least-squares fit, and it usually leaves some residual error.

Uses. For risk measurement, you shock the factors and revalue the portfolio, or you use factor sensitivities in VaR. For hedging, you match the portfolio's exposure to each factor with traded instruments. For relative value, you compare model value with market price. A gap may signal a cheap or rich instrument, or it may signal model error. Always ask which one.

Versus Vasicek. Gauss+ gives richer curve moves because it has several factors. It is typically calibrated to the initial curve as well as to cap and swaption volatilities, and its nonlinear term goes beyond a purely Gaussian structure. Vasicek is an equilibrium model that does not fit today's curve exactly. It is simpler, has closed-form bond prices, and is easier to calibrate, but it has one factor and constant volatility. Gauss+ is harder to calibrate and has more model risk.

Key formulas to remember

Total volatility from two correlated factors: σ_total = √(σ₁² + σ₂² + 2ρσ₁σ₂); Applies when a rate is the sum of two factor shocks with equal weights. ρ is the correlation between the factors. With ρ = 0 it reduces to √(σ₁² + σ₂²).
Mean-reverting Gaussian factor (generic form): dx = −k·x·dt + σ·dW; k is the speed of mean reversion and σ is the volatility. Higher k makes shocks to that factor decay faster. The long-run (stationary) standard deviation of the factor is σ/√(2k), which falls as k rises. This is a generic form; check the exact specification in your reading.
Calibration objective: minimise Σ (model price − market price)² over the volatility parameters; The curve fit is exact by construction of the drift or shift term, so it is not part of this search. The objective applies to the volatility fit: parameters are chosen to match cap, floor or swaption prices or implied volatilities as closely as possible. This is a least-squares fit and usually leaves some residual error.
Multi-factor hedge conditions: Σ nᵢ × sensitivity of instrument i to factor j = − portfolio sensitivity to factor j, for every factor j; You need at least as many hedge instruments as factors. Solve all the equations together.
Relative value signal: Rich/cheap = market price − model price; A positive value means the instrument looks rich versus the model. It is only a signal if the model is well calibrated.

How to solve Gauss+ Model: Calibration, Risk and Applications questions

Use this order for most Gauss+ questions, whether they ask about calibration, risk or applications.

1Identify what is asked: curve fitting, volatility calibration, risk measurement, hedging or relative value.
2Recall the structure: several mean-reverting Gaussian factors, factor correlation, and the added nonlinear volatility feature.
3For calibration, separate the two jobs. The curve is fitted exactly. Volatility parameters are fitted approximately to caps or swaptions.
4For risk, compute the exposure to each factor, not to one parallel shift. Then apply the factor volatilities and correlation to get the P&L spread.
5For hedging, write one equation per factor and solve for the hedge notionals. Check the sign: a long exposure needs a short hedge.
6For relative value, compare model and market prices. Then ask whether the gap could come from poor calibration, liquidity or model error.
7Compare with Vasicek if asked: more factors and fit to the curve, against more parameters and more model risk.
8Check units (basis points, USD) and the sign before choosing an answer.

Quickest way: Fast elimination for Gauss+ questions

When to use it: Use this when you have about two minutes and the options mix theory statements with simple arithmetic.

Be careful with any option that says the model reprices the curve only approximately. Curve fit is normally designed to be exact, but check what the question or reading states before you reject it.
Reject any option that says one factor is enough to hedge a twist. Several factors need several hedge instruments.
For volatility of two factors, use √(σ₁² + σ₂² + 2ρσ₁σ₂). Test ρ = 0 first for a quick estimate.
For a hedge, solve the factor equations in order. Use the simplest equation to express one notional in terms of the other.
If asked about a gap between model and market prices, prefer the answer that mentions both a possible opportunity and possible model error.

Common mistakes in Gauss+ Model: Calibration, Risk and Applications

Saying Gauss+ calibration fits the curve and volatilities equally well.
Both are called calibration, so they get lumped together.
Fix: Remember that the curve fit is normally designed to be exact, through the model's drift or shift terms as set out in your reading. Volatilities are fitted by minimising errors, and some error remains.
Hedging a multifactor model with a single DV01 hedge.
DV01 hedging is familiar from earlier chapters and assumes a parallel shift.
Fix: Hedge each factor separately. A DV01 hedge can leave slope or curvature risk if the factors are not all offset.
Adding factor volatilities directly (80 + 60 = 140bp).
Volatilities feel additive.
Fix: Combine them with variances and correlation: √(σ₁² + σ₂² + 2ρσ₁σ₂). Only with ρ = 1 does the sum equal 140bp.
Treating a relative value gap as a sure profit.
The model price feels like the true value.
Fix: The gap may be a calibration error or a missing risk. Check the calibration quality, liquidity and model assumptions before acting.
Claiming Gauss+ is always better than Vasicek.
More factors sounds like a strict improvement.
Fix: Say it is richer but harder to calibrate and carries more model risk. Vasicek is simpler and has closed-form bond prices.
Getting the hedge sign wrong.
Students solve the equations but forget that the hedge offsets the portfolio.
Fix: Set the hedge sensitivity equal to the negative of the portfolio's. Long exposure means selling the hedge instruments.

Worked examples

Example 1

A two-factor Gauss-type model has factor volatilities of 80bp and 60bp a year for a swap rate, with correlation 0 between the factors. The market normal implied volatility for that swap rate is 90bp. (a) What is the model's total volatility? (b) By what common factor must you scale both factor volatilities to match the market?

Show the solution

Total volatility = √(σ₁² + σ₂² + 2ρσ₁σ₂) with ρ = 0.
σ₁² = 6,400 and σ₂² = 3,600, so the sum is 10,000.
√10,000 = 100bp.
Target is 90bp, so scale = 90 ÷ 100 = 0.9.
Check: 80 × 0.9 = 72 and 60 × 0.9 = 54. Then 72² = 5,184 and 54² = 2,916, which sum to 8,100, and √8,100 = 90.

Answer: (a) The model volatility is 100bp. (b) Scale both factor volatilities by 0.9, to 72bp and 54bp, to match 90bp.

Example 2

A USD portfolio has sensitivity to factor 1 of +5,000 and to factor 2 of −400 (USD change per unit factor shock). Per USD 1 million notional, Bond A has sensitivities of 400 to factor 1 and 100 to factor 2. Bond B has 300 to factor 1 and −200 to factor 2. What positions in A and B hedge both factors?

Show the solution

Let a and b be the notionals in USD millions. The hedge must offset the portfolio, so the hedge sensitivities are −5,000 and +400.
Factor 1: 400a + 300b = −5,000.
Factor 2: 100a − 200b = +400.
From factor 2: a = 4 + 2b.
Substitute: 400(4 + 2b) + 300b = −5,000, so 1,600 + 800b + 300b = −5,000.
1,100b = −6,600, so b = −6.
Then a = 4 + 2(−6) = −8.
Check factor 1: 400(−8) + 300(−6) = −3,200 − 1,800 = −5,000. Check factor 2: 100(−8) − 200(−6) = −800 + 1,200 = +400.

Answer: Sell USD 8 million of Bond A and sell USD 6 million of Bond B. This offsets both factor exposures.

Exam tips

Calibration questions often test the split: exact curve fit, approximate volatility fit. Keep that line ready.
In hedging questions, count factors and count instruments. If instruments are fewer than factors, some risk stays unhedged.
Comparison with Vasicek is common. Give one benefit (richer curve moves, fit to the curve) and one cost (more parameters, more model risk).
For relative value questions, pick the answer that acknowledges calibration or model risk, not the one that promises arbitrage.
Do the arithmetic with variances, not volatilities. Check the correlation term sign.

Practice questions from The Vasicek and Gauss+ Models

Gauss+ Model: Calibration, Risk and Applications: frequently asked questions

How is the Gauss+ model calibrated to market data?

You do two things. First, you set the model so it reprices the observed discount curve; this fit is normally designed to be exact, but check your reading's specification. Second, you choose factor volatilities, mean-reversion speeds and correlation so model prices of caps, floors or swaptions are as close as possible to market prices or implied volatilities.

How do you hedge interest rate risk with the Gauss+ model?

You compute the portfolio's sensitivity to each model factor. Then you choose hedge instruments whose factor sensitivities add up to the negative of the portfolio's. You need at least as many instruments as factors, and you solve the equations together.

What are the advantages of Gauss+ over Vasicek?

Gauss+ has several factors, so the curve can change shape rather than only move in one way. It is typically calibrated to the initial curve and to cap or swaption volatilities, and it adds a nonlinear term beyond a purely Gaussian structure. Vasicek is an equilibrium model that does not fit today's curve exactly. The cost of Gauss+ is more parameters, harder calibration and more model risk.

Does a gap between Gauss+ price and market price mean a trade opportunity?

Not necessarily. The gap can reflect poor calibration, liquidity effects or risks the model omits. Treat it as a signal to investigate, not as a guaranteed profit.