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Describe how term structure models can cope with short rates possibly going below zero.

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

The idea

A normal short-rate model has no lower bound, so every future date carries a positive probability of a negative rate. For a zero-drift model, the distance below zero, in standard deviations, is Z = −r ÷ (σ × √T). The probability of a negative rate is N(Z). It is small when the starting rate is high and volatility low, but grows with the horizon. Models cope in several ways. They can accept the small probability. They can floor the tree at zero, which is simple but makes the rate non-normal and lifts the expected rate above the unfloored tree's. Or they can change the volatility specification. A lognormal model, with volatility proportional to the rate, or a square-root model, with volatility proportional to the square root of the rate, keeps rates positive, at the cost that basis point volatility now moves with the rate level. Mean reversion alone does not remove negative rates. Exam point: rate-dependent volatility keeps rates positive, whereas drift or mean reversion alone does not.

Formulas

Probability of a negative short rate in a zero-drift normal model

Prob = N(−r ÷ (σ × √T))

N
the standard normal cumulative distribution function
r
today's short rate
σ
the annual basis-point volatility
T
the horizon in years

Assumes normal rates and no drift. The probability grows with the horizon and with volatility; mean reversion changes the mean and spread but does not make it zero.

A practice question

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

Which modelling choice can keep the short rate from going negative?

  1. Adding mean reversion to a normal model, as in Vasicek, which guarantees rates stay above zero.Why not: Mean reversion pulls rates toward a mean but a normal shock can still push the rate below zero.
  2. Choosing the Ho-Lee drift large enough to offset every possible downward move.Why not: A time-dependent drift cannot offset unbounded normal shocks; negative rates remain possible.
  3. Specifying volatility in proportion to the rate, as in a lognormal model, so volatility shrinks as the rate approaches zero. - correct
  4. Raising the volatility so the tree spans a wider range of rates.Why not: Higher volatility widens the distribution and increases, not decreases, the chance of negative rates.

Normal models have constant basis point volatility, so a large enough down move always crosses zero. A lognormal model (or a square-root model) scales volatility with the rate, so shocks shrink near zero and rates stay positive.

A harder one

A normal model is calibrated with a basis point volatility of 100 bp at a short rate of 5.00%. It is replaced by a lognormal model with the same volatility at 5.00%, which is 20% of the rate. When the short rate falls to 0.50%, the basis point volatility of the lognormal model is closest to:

  1. 100 bpWhy not: Keeps volatility constant, which is the normal model's behaviour.
  2. 10 bp - correct
  3. 32 bpWhy not: Scales volatility with the square root of the rate (100 x sqrt(0.5/5)), the square-root model's behaviour.
  4. 20 bpWhy not: Treats the 20% proportional volatility as 20 basis points.

Lognormal volatility is proportional to the rate: 20% x 0.50% = 0.10% = 10 bp. The normal model stays at 100 bp, and a square-root model would give about 32 bp.

The hardest tier

A zero-drift tree has an initial rate of 0.25% and each step moves the rate up or down by 0.50% with equal probability. At every node any negative rate is replaced by zero before the next step. The expected rate after two steps is closest to:

  1. 0.500% - correct
  2. 0.250%Why not: Ignores the floor; the unfloored zero-drift tree has an expected rate of 0.25%.
  3. 0.375%Why not: Floors the first step but lets the zero node move down into negative rates again at step two.
  4. 0.438%Why not: Floors only the final-date nodes of an unfloored tree, ignoring that the floor applies at the intermediate node too.

After step one: 0.75% and 0% (from -0.25% floored). Step two from 0.75%: 1.25% and 0.25%. From 0%: 0.50% and -0.50% floored to 0%. Average = (1.25 + 0.25 + 0.50 + 0) / 4 = 0.500%, above the unfloored 0.25%.

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