Work out the short-rate change under the CIR and lognormal models, and characterise the basis point volatility they imply.

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

The idea

Both use ε, a standard normal draw. Over a short interval dt the CIR model gives a change written DrCIR = k(θ − r) × dt + σ√r × √dt × ε. Likewise the lognormal model gives the short-rate change written DrLog = a × r × dt + σ × r × √dt × ε. The annualised basis-point volatility is the coefficient on the shock before the square root of dt is applied. It is σ times the square root of r for CIR and σ × r for lognormal, versus a constant for normal models. With r at 4 percent and σ at 0.05 for CIR, the square root of 0.04 is 0.2, so volatility is 0.05 × 0.2, which is 1.0 percent, or 100 basis points a year. Doubling the rate multiplies CIR volatility by the square root of 2 but lognormal volatility by 2. Mistakes are entering r in percent instead of decimals, using the lognormal form for CIR, and forgetting the square root of dt for a monthly change. Exam point: compute the volatility coefficient first, then scale by the square root of dt.

Formulas

Short-rate change in the CIR model

DrCIR = k × (θ − r) × dt + σ × √r × √dt × ε

k: the speed of mean reversion
θ: the long-run rate
r: the current rate, as a decimal
dt: the interval in years
σ: the CIR volatility parameter
ε: a standard normal draw

Compute the volatility coefficient σ × √r first, then scale by √dt. Rates in decimals.

Short-rate change in the lognormal model

DrLog = a × r × dt + σ × r × √dt × ε

a: the drift parameter
r: the current rate, as a decimal
dt: the interval in years
σ: the proportional volatility
ε: a standard normal draw

Both the drift and the shock are proportional to the rate. Do not use this form for CIR, whose shock scales with √r.

A practice question

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

In a CIR model dr = k(θ - r) dt + σ√r dw with σ = 0.06 and r = 4 percent, the annualised basis-point volatility of the short rate is closest to:

24 basis points a yearWhy not: Uses σ r (lognormal) instead of σ√r.
120 basis points - correct
600 basis pointsWhy not: Uses σ alone, 6 percent, ignoring the √r factor.
1.2 basis pointsWhy not: Gets 1.2 percent but reports it as 1.2 basis points; 1 percent is 100 bp.

Basis-point volatility = σ√r = 0.06 x √0.04 = 0.06 x 0.2 = 1.2 percent = 120 bp a year.

A harder one

CIR: dr = k(θ - r) dt + σ√r dw with k = 0.30, θ = 5 percent, σ = 0.07 and r = 3 percent. Over one month (dt = 1/12) a standard normal draw of ε = -1.2 occurs. The change in the short rate is closest to:

-37 basis points - correct
-42 basis pointsWhy not: Omits the drift: shock only, 0.07 x √0.03 x √(1/12) x (-1.2).
-2 basis pointsWhy not: Uses the lognormal diffusion σ r instead of σ√r for the shock.
-140 basis pointsWhy not: Forgets to scale the shock by √dt, using σ√r x ε alone.

Drift = 0.30 x (0.05 - 0.03)/12 = 5 bp. Shock = 0.07 x √0.03 x √(1/12) x (-1.2) = -42 bp. Δr = 5 - 42 = -37 bp.

The hardest tier

A CIR model and a lognormal model are both calibrated to a 100 basis-point annual volatility at a 5 percent rate; the lognormal σ is 20 percent. If the short rate falls to 2 percent, the basis-point volatilities of the CIR model and the lognormal model respectively are closest to:

40 and 63 basis pointsWhy not: Swaps the two models: CIR volatility falls with √r (less sharply), lognormal falls proportionally.
63 and 40 basis points - correct
100 and 40 basis pointsWhy not: Treats CIR as a normal model with constant volatility, so it does not move with the rate.
40 and 40 basis pointsWhy not: Applies the proportional (lognormal) rule to both models.

CIR σ solves σ√0.05 = 0.01, so σ = 0.0447; at 2 percent: 0.0447 x √0.02 = 63 bp. Lognormal: 0.20 x 0.02 = 40 bp.

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