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Characterise how well the Vasicek Model performs.

FRM® Part II · Market Risk Measurement and Management · The Art of Term Structure Models: Drift · FRM2.MR.14.h

The idea

The Vasicek model does well on one dimension and poorly on others. Mean reversion bounds long-horizon dispersion. Here k is the speed of mean reversion, σ is the annual volatility, and SD is the standard deviation of the rate at horizon T. SD = σ × √((1 − e^(−2kT)) ÷ (2k)). As T grows this tends to a ceiling of σ divided by the square root of 2k, instead of growing with the square root of T. Long-maturity yields are also less volatile than short ones, as in the data. The weaknesses are these. With constant parameters the model can generate only a limited family of curve shapes, so it does not fit the observed term structure and cannot price relative to market bonds. Normal rates can still go negative, volatility is constant, and one factor moves all rates together. Letting θ depend on time, as in the extended Vasicek or Hull-White model, repairs the fit. Exam point: realistic rate dynamics, but not arbitrage-free.

Formulas

Standard deviation of the Vasicek short rate at horizon T

SD = σ × √((1 − e^(−2 × k × T)) ÷ (2 × k))

σ
the annual volatility
k
the speed of mean reversion
T
the horizon in years

As T grows it tends to σ ÷ √(2 × k), whereas with no mean reversion it would grow as σ × √T.

A practice question

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

Which feature is the main strength of the Vasicek model compared with a zero-drift or constant-drift normal model?

  1. It guarantees non-negative interest rates.Why not: Vasicek uses normal shocks, so negative rates are still possible.
  2. It matches the observed term structure exactly through a time-dependent drift.Why not: That is the arbitrage-free feature of Ho-Lee; Vasicek has constant parameters.
  3. Mean reversion, which bounds long-horizon rate dispersion and makes long-maturity yields less volatile than short ones. - correct
  4. It lets volatility itself vary randomly over time.Why not: Vasicek volatility is constant; stochastic volatility is a different model.

The drift k x (theta - r) pulls the rate back toward theta, so long-horizon dispersion is bounded and long yields move less than short ones, consistent with observed data.

A harder one

In a Vasicek model with k = 0.40 and annual volatility of 100 bp, the standard deviation of the short rate three years ahead is sigma x sqrt((1 - exp(-2kT)) / (2k)). The value is closest to:

  1. 173 bpWhy not: Uses sigma x sqrt(T), the zero-drift result, ignoring mean reversion.
  2. 107 bp - correct
  3. 151 bpWhy not: Divides by k instead of 2k inside the square root.
  4. 100 bpWhy not: Uses the annual volatility as the standard deviation at every horizon.

(1 - exp(-2.4)) / 0.80 = 1.1366, whose square root is 1.0661. Standard deviation = 106.6 bp, versus 173 bp with no mean reversion.

The hardest tier

A bank fits a Vasicek model to six on-the-run swap rates. The model reproduces the broad level of the curve, but the residuals are systematic: it overprices some maturities and underprices others in the same way every day. What is the best interpretation and remedy?

  1. Mean reversion is too weak; raising k enough will remove the residuals at every maturity.Why not: One speed parameter cannot reshape an arbitrary curve; the limitation is the number of free parameters.
  2. Constant parameters can generate only a limited family of curve shapes; a time-dependent theta (extended Vasicek or Hull-White) is the standard remedy. - correct
  3. The residuals show that volatility is stochastic; the model needs a random volatility factor.Why not: Stochastic volatility concerns option prices, not systematic curve-fit residuals in a constant-parameter model.
  4. Residuals of this kind are noise, and the fit would be perfect with more data.Why not: The pattern repeats every day, so it reflects the model's structure, not sampling noise.

With four constant parameters the model produces only simple curve shapes, so systematic misfits are expected. Making theta a function of time lets the model fit the curve exactly, at the cost of some of the model's equilibrium interpretation.

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