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Contrast the Vasicek and Gauss+ models in terms of their dynamics, characteristics and uses.

FRM® Part II · Market Risk Measurement and Management · The Vasicek and Gauss+ Models · FRM2.MR.16.b

The idea

The Vasicek model is a one-factor model, with the short rate following the process below. dr = k(θ − r) dt + σ dW. Here the short rate reverts at speed k toward one constant level θ, with constant volatility σ. Every rate on the curve is driven by that single shock, so all yields are perfectly correlated, the curve moves in a limited way, and the volatility of yields falls steadily as maturity rises. The model gives closed-form bond prices, which makes it a good teaching and benchmark model, but it cannot produce independent slope or curvature changes. Gauss+ has three factors in a cascade, and the short rate reverts to a moving target, the medium factor, rather than to a constant. That lets it reproduce non-parallel moves, so it suits curve risk and hedging. The common mistake is to say Vasicek has several sources of risk, when it has only one.

Formulas

Volatility of a T-year yield in the Vasicek model

YieldVol = σ × (1 − e^(−k × T)) ÷ (k × T)

σ
the short-rate volatility
k
the speed of mean reversion
T
the yield's maturity in years

One factor drives every yield, so all yields stay perfectly correlated even though long yields are less volatile.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which statement correctly contrasts the Vasicek and Gauss+ models?

  1. Vasicek has three factors, and Gauss+ has one factor with three maturities of bond prices derived from it.Why not: Reverses the contrast; Vasicek is the one-factor model.
  2. Both are one-factor models, and Gauss+ only adds a time-varying mean to Vasicek.Why not: Wrongly calls Gauss+ one-factor; its short, medium and long rates are separate factors.
  3. Vasicek has one factor and one source of risk, while Gauss+ has three rate factors in a cascade. - correct
  4. Vasicek has mean reversion and Gauss+ has none, since its three factors are random walks.Why not: Gauss+ is built on mean reversion at every level; Vasicek is not the only model with it.

Vasicek: dr = k(theta - r)dt + sigma dW, a single factor reverting to a constant. Gauss+ has r, m and l with their own mean reversion and shocks.

A harder one

After a shock lifts the medium-term expectation of the short rate, how does each model respond in terms of the level the short rate is pulled toward?

  1. Both models: the short rate's target rises, because each model has an adjustable long-run level that responds to the shock.Why not: Vasicek's long-run level theta is a fixed constant; only Gauss+ has a moving target.
  2. Gauss+: the target rises, as it is the medium factor. Vasicek: the target is the constant theta and does not move. - correct
  3. Vasicek: the target rises. Gauss+: the target is the constant long-run level and does not change.Why not: Swaps the models; in Gauss+ the short rate reverts to the medium factor, not to a constant.
  4. Neither model: both set the short rate's target to the long rate, which is unchanged by the shock to the medium factor.Why not: Neither model reverts the short rate to the long rate; in Gauss+ the target is the medium factor.

In Vasicek dr = k(theta - r)dt, so the target never moves. In Gauss+ dr = k_r(m - r)dt, so a change in m changes the pull on r.

The hardest tier

In a Vasicek model with sigma = 100 bp per year and k = 0.25, the volatility of a T-year zero-coupon yield is sigma x (1 - e^(-kT)) / (kT). What is the ratio of the 10-year yield volatility to the 1-year yield volatility, and what does it imply?

  1. About 0.41: long yields are much less volatile, and all yields share one shock, so they stay perfectly correlated. - correct
  2. About 0.09: the ratio of e^(-kT) at 10 years to e^(-kT) at 1 year, with no allowance for the 1/(kT) factor.Why not: Uses the exponential e^(-kT) alone instead of (1 - e^(-kT))/(kT) for the yield volatility.
  3. About 1.00: every yield moves with the same volatility of 100 bp because they share one factor.Why not: Confuses perfect correlation with equal volatility; yield volatility declines with maturity in Vasicek.
  4. About 0.41: but long and short yields are then only weakly correlated, since the volatility ratio is below one.Why not: The ratio is right but the implication is wrong; one factor means perfect correlation regardless of the volatility ratio.

1-year: 100 x (1 - e^(-0.25))/0.25 = 88.5 bp. 10-year: 100 x (1 - e^(-2.5))/2.5 = 36.7 bp. Ratio = 0.415. The yields differ in volatility but share one shock, hence perfect correlation.

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