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Weigh up how well time-dependent volatility models work.

FRM® Part II · Market Risk Measurement and Management · The Art of Term Structure Models: Volatility and Distribution · FRM2.MR.15.c

The idea

Time-dependent volatility is a useful but limited refinement. Its strength is flexibility. Choosing σ(t) lets a normal model match the market term structure of volatility, for example cap or swaption volatilities across expiries, in addition to the term structure of rates fitted by the drift. Its weaknesses are structural. Volatility is a deterministic function of calendar time, so the model implies that tomorrow's volatility curve is just the remaining part of today's. The market rarely delivers that, and parameters must be recalibrated often. Volatility cannot depend on the rate level or be random, and there is one volatility per expiry, so strike skew is not captured. Rates remain normal, so negative rates are possible. Fitted values can also be inconsistent. If implied cumulative variance falls with horizon, no real σ(t) exists without mean reversion. Exam point: good for fitting today's volatility curve, poor at describing how volatility will behave.

A practice question

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

Which is a principal weakness of a time-dependent volatility model such as dr = λ(t) dt + σ(t) dw?

  1. It cannot be fitted to the current term structure of interest rates.Why not: The drift λ(t) is normally chosen precisely to fit the current term structure.
  2. It forces rates to be non-negative, which contradicts markets with negative rates.Why not: A normal model permits negative rates; it does not force them to be positive.
  3. It requires a separate Monte Carlo for every strike.Why not: Nothing in the specification calls for strike-by-strike simulation; the weakness is in how volatility is modelled.
  4. Volatility is a predetermined function of calendar time, independent of the level of rates. - correct

The model treats volatility as a known schedule in calendar time. It cannot let volatility rise with rates or be random, which is the main criticism.

A harder one

A desk calibrates σ(t) to cap volatilities for the next ten years. One year later it notes that the market's volatility curve, quoted by time to expiry, has a very different shape from the one the calibrated model implies for that date. Which statement best explains the problem?

  1. The future volatility curve is just the remaining part of today's schedule, so any market shift forces recalibration. - correct
  2. The model implies future volatility is random, so mismatches are expected and need no recalibration of the schedule.Why not: The volatility in the model is deterministic; the mismatch signals misspecification, which is why recalibration is needed.
  3. The mismatch arises because mean reversion was omitted from the model; adding it would make the schedule stable.Why not: Adding mean reversion does not fix the dependence on calendar time; it is the deterministic schedule that is at issue.
  4. The mismatch arises because the drift λ(t) has been fitted to the rate curve instead of the volatility curve.Why not: Drift and volatility are fitted separately; the issue is the predetermined volatility schedule.

In a time-dependent model today's calibration fixes volatility at each future date. If, as is common, the market re-quotes volatility by time to expiry, the model's implied curve at that date will not match and must be recalibrated.

The hardest tier

A bank calibrates a model without mean reversion to normal caplet volatilities (annualised) of 100 basis points for a one-year expiry and 80 basis points for a two-year expiry. The cumulative variance to expiry T equals the squared caplet volatility times T. The implied short-rate volatility for year 2 alone is closest to:

  1. 60 basis pointsWhy not: Extrapolates linearly: 2 x 80 - 100, instead of working with variances.
  2. 53 basis points - correct
  3. 80 basis pointsWhy not: Uses the two-year volatility as the year-2 volatility, ignoring that it is an average over two years.
  4. 113 basis pointsWhy not: Takes the cumulative standard deviation to year 2, 80 x √2, as the volatility for year 2.

Cumulative variance: year 1 = 100² x 1 = 10,000; to year 2 = 80² x 2 = 12,800. Year-2 variance = 2,800, so the volatility is √2,800 = 52.9 bp.

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