FRM Part II · FRM Exam Part II

The Art of Term Structure Models: Volatility and Distribution: formula sheet

Full chapter guide

Key formulas

Short-rate dynamics: dr = λ(t)dt + σ(t)dw; λ is drift, σ is volatility, dw is a normal shock with variance dt.
Rate change over a step: Δr ~ Normal(λΔt, σ²Δt); Standard deviation of the change is σ√Δt. Valid for normal (constant or given-σ) models.
Distribution of the rate in Model 1 (zero drift): r(T) ~ Normal(r₀, σ²T); Standard deviation grows with √T.
Constant drift normal model: r(T) ~ Normal(r₀ + λT, σ²T); Drift shifts the mean; volatility drives the spread.
Lognormal volatility: dr = λ(t)r dt + σ r dw; Absolute rate volatility equals σ × r, so it scales with the rate level.
CIR volatility: dr = k(θ − r)dt + σ√r dw; Absolute volatility is σ√r. Rate-dependent and keeps rates nonnegative under the usual conditions.
Annualising volatility: σ_annual = σ_daily × √252 (or σ_monthly × √12); Assumes independent changes. Convert basis points consistently.
Model 1 dynamics: dr = σ dw; No drift term. σ is the annual volatility of rate changes, in basis points per year. Model 2 would add + λ dt.
Distribution of the rate: r(T) ~ Normal(mean = r0, standard deviation = σ√T); Mean stays at r0 because drift is zero. Standard deviation grows with the square root of time.
Tree step size: Up or down move per step = σ√dt; Use dt in years. For quarterly steps dt = 0.25 and √dt = 0.5. For monthly steps dt = 1/12.
Node rates: r(n, k) = r0 + (n − 2k) × σ√dt; n = number of steps, k = number of down moves. Gives n + 1 nodes at step n.
Node probabilities: Each branch probability = 0.5; P(k downs in n steps) = C(n, k) ÷ 2ⁿ; Use for the chance of reaching a specific node.
Backward induction: V(node) = 0.5 × [V(up) + V(down)] ÷ (1 + r(node) × dt); Discount with the rate at the current node, over one step. Cash flows at that node are added after discounting.
Probability of a negative rate: P(r(T) < 0) = Φ(−r0 ÷ (σ√T)); Φ is the standard normal cumulative distribution. Keep r0 and σ in the same units.
Ho-Lee short rate process: dr = λ(t)dt + σdw; σ is constant. λ(t) varies with time. Risk-neutral dynamics. Rates are normal, so negative rates are possible.
Continuous-time drift (calibration result): λ(t) = ∂f(0,t)/∂t + σ²t; f(0,t) is today's instantaneous forward rate for date t. The first term is the forward curve slope. The second is the convexity term.
Expected short rate: E[r(t)] = f(0,t) + σ²t²/2; Risk-neutral expectation, equal to r(0) plus the integral of λ from 0 to t when r(0) = f(0,0).
Short rate variance: Var[r(t)] = σ²t; Grows without limit as t increases because there is no mean reversion.
Binomial tree step: r(next) = r + λ(t)dt ± σ√dt; Up and down moves have risk-neutral probability 0.5 each. The drift per step is set to fit the bond price.
Vasicek short-rate process: dr = k(θ − r)dt + σ dW; k is the speed of mean reversion, θ the long-run rate, σ the constant volatility. Under the risk-neutral measure, θ is the risk-neutral long-run level.
Expected rate over a short step: E[Δr] = k(θ − r) × dt; Positive if r < θ, negative if r > θ.
Expected rate at horizon T: E[r(T)] = θ + (r(0) − θ) × e^(−kT); The gap to θ shrinks by the factor e^(−kT).
Variance of the short rate at T: Var[r(T)] = σ² × (1 − e^(−2kT)) ÷ (2k); For k > 0, this tends to σ² ÷ (2k) as T grows, so long-horizon variance is bounded.
Volatility of the T-year zero-coupon yield: Yield volatility = σ × (1 − e^(−kT)) ÷ (kT); This is the volatility of the T-year zero-coupon yield (the T-year spot rate). It equals σ × B(T) ÷ T, where B(T) = (1 − e^(−kT)) ÷ k. It does not apply to the short rate itself, whose volatility stays σ. It falls as T increases, so the volatility term structure slopes downward. Under Ho-Lee, B(T) = T, so the zero-coupon yield volatility is σ × T ÷ T = σ at every maturity.
Half-life of a shock: Half-life = ln(2) ÷ k; Time for the gap to θ to halve.
Limit k → 0: k(θ − r)dt → 0, so dr = σ dW; As k goes to zero the drift term vanishes. The model becomes a driftless, constant-volatility normal model. Mean reversion disappears and long-rate volatility equals short-rate volatility.
Model 3 dynamics: dr = λ(t)dt + σ(t)dw, with σ(t) = σ·e^(−αt); Volatility of rate changes declines at exponential rate α > 0. λ(t) is fitted to the initial curve. No mean reversion.
Model 3 short-rate standard deviation at horizon T: s.d.(r_T) = σ·√[(1 − e^(−2αT)) ÷ (2α)]; Variance is the integral of σ²e^(−2αs) from 0 to T. As T grows it approaches σ ÷ √(2α).
Constant-volatility comparison: s.d.(r_T) = σ·√T; Ho-Lee and Model 1. Grows without bound, so it is larger than the Model 3 value for the same σ.
Hull-White dynamics: dr = (θ(t) − a·r)dt + σ·dw; a is the mean-reversion speed. θ(t) is time-dependent and fitted to today's curve. Vasicek is the case with constant θ.
Hull-White short-rate standard deviation: s.d.(r_T) = σ·√[(1 − e^(−2aT)) ÷ (2a)]; Same form as Model 3 with a in place of α. Long-run limit is σ ÷ √(2a).
Hull-White zero-coupon bond price: P(t,T) = A(t,T)·e^(−B(t,T)·r), B(t,T) = (1 − e^(−a(T−t))) ÷ a; B is the sensitivity of ln P to the short rate. For a given Δr, ΔP ÷ P ≈ −B·Δr. B is less than T−t when a > 0.
CIR short-rate process: dr = k(θ − r)dt + σ√r dw; k = reversion speed, θ = long-run mean, σ = scaling constant (not in bp).
CIR basis-point volatility: Volatility of r = σ√r; Annual standard deviation of rate changes, in decimal terms (0.02 = 200 bp). Multiply by √dt for a short step.
Lognormal model: dr = a·r·dt + σ·r·dw; Basis-point volatility = σ·r. Rate stays positive.
Vasicek process (for contrast): dr = k(θ − r)dt + σ dw; Constant volatility; negative rates possible.
Feller condition: 2kθ ≥ σ²; CIR rates are non-negative because volatility shrinks near zero and the drift at r = 0 is positive (kθ > 0). The Feller condition ensures the rate never reaches zero, so it stays strictly positive. If it is violated, the rate can hit zero but is reflected and remains non-negative.
Expected rate change over a short step: E[Δr] = k(θ − r)dt; Random shock has standard deviation σ√r × √dt.
Salomon Brothers (lognormal) model: dr = a·r·dt + σ·r·dw; Drift and volatility are both proportional to r. The rate stays positive. There is no mean reversion in this form.
Black-Karasinski model: d(ln r) = k(t)·[ln θ(t) − ln r]·dt + σ(t)·dw; Mean-reverting lognormal model. ln r reverts to ln θ at speed k. It needs a tree or numerical calibration.
Basis-point volatility: Annual bp vol = σ × r; σ is a percentage volatility, for example 20%. Convert to bps by multiplying by the rate level.
Tree node spacing: r(up) = r(down) × e^(2σ√dt); Applies to adjacent nodes on the same date. Use the same σ and dt for that step.
Expected rate with proportional drift: E[r(T)] = r(0) × e^(aT); Applies to dr = a·r·dt + σ·r·dw. The drift multiplies the rate, not a basis-point amount.
Log-rate dynamics: ln r is normally distributed, so r > 0; This is why lognormal models cannot produce negative rates.
Model 1 (normal, constant volatility): dr = λ dt + σ dw; Rates are normal, change volatility is σ at all rate levels and maturities. Flat volatility term structure.
Ho-Lee: dr = λ(t) dt + σ dw; Time-dependent drift fits the initial curve. Volatility stays constant.
Vasicek: dr = k(θ − r) dt + σ dw; Mean reversion at speed k. A higher k makes long-maturity rate volatility decay faster.
Vasicek volatility of the instantaneous forward rate at T: σ(T) = σ × e^(−kT); This is the volatility of the instantaneous forward rate at T, not of the T-year spot rate. It decays exponentially with maturity.
Vasicek volatility of the T-year spot rate: σ × (1 − e^(−kT)) ÷ (kT); Use this when the question asks for the volatility of the T-year rate. It also falls as T and k rise.
Time-dependent volatility (Model 3 / Hull-White): dr = λ(t) dt + σ(t) dw; σ(t) can be set to match a humped or sloping market volatility curve.
CIR: dr = k(θ − r) dt + σ√r dw; Volatility rises with √r. Rates stay non-negative under the standard condition on parameters.
Plain lognormal model: dr = a r dt + σ r dw; Percentage volatility is constant. Rates are positive and right-skewed. Black-Karasinski is a different model: lognormal rates with mean reversion in ln r.

Quick revision

Model 1: dr = σdw, constant volatility, no drift, normally distributed rates.
Ho-Lee: dr = λ(t)dt + σdw, a time-dependent drift fits the initial curve, volatility is constant.
Vasicek: dr = k(θ − r)dt + σdw, the rate reverts toward θ at speed k.
Mean reversion lowers the volatility of long-term rates relative to short-term rates.
Normal models can produce negative rates, because the distribution is normal.
Model 3 has a time-dependent drift and time-dependent volatility but no mean reversion. Hull-White adds mean reversion to a time-dependent drift and time-dependent volatility.
CIR: volatility is proportional to √r, so basis-point volatility rises with the rate level.
Lognormal models (the basic lognormal model and Black-Karasinski) keep rates positive, and basis-point volatility is proportional to the rate.
The basic lognormal model has constant proportional volatility and no mean reversion, while Black-Karasinski combines lognormal rates with mean reversion.
Match the model to its volatility term structure first, and then check the distribution and sign of rates.
Constant versus time-dependent versus rate-dependent volatility is the main distinction across the models.

Common mistakes

Scaling volatility by T instead of √T. Fix: Variance scales with T. Standard deviation scales with √T.
Mixing basis points and percent. Fix: Convert first: 120 bp = 1.20%. Write units beside every number.
Using σ as the step size instead of σ√dt. Fix: Always multiply σ by √dt first. For quarterly steps the move is σ × 0.5. For monthly steps it is σ ÷ √12.
Mixing basis points and percent, for example subtracting 120 from 3%. Fix: Convert σ to percent first: 120 bp = 1.20%. Write units next to every number.
Saying the drift is zero when the forward curve is flat Fix: Always add σ²t. A flat curve gives λ(t) = σ²t, which is positive.
Using σ in percent when squaring Fix: Convert to decimals first: σ = 1% = 0.01, so σ² = 0.0001.
Treating θ as the current rate or as the expected rate at all horizons. Fix: θ is only the level toward which rates are pulled. The expected rate at T is θ + (r − θ)e^(−kT), which sits between r and θ.
Getting the drift sign wrong. Fix: Always write k(θ − r). If r is below θ, drift is positive.
Saying Model 3 has mean reversion. Fix: Model 3 has no pull-back term. Only the volatility σ(t) declines. Mean reversion needs −a·r in the drift.
Using σ√T for the horizon standard deviation in Model 3 or Hull-White. Fix: Use σ√((1 − e^(−2kT)) ÷ (2k)). Check that your answer is below σ√T.

Exam tips

Always check whether the question gives σ in absolute terms (normal) or proportional terms (lognormal).
Expect conceptual items on why volatility lowers long-term spot rates through convexity.
Convert time to years before using √T.
Link model choice to a feature: negative rates, rate-dependent volatility, or fit to the volatility term structure.
Eliminate options that confuse drift with volatility or scale volatility by T.
Read the units first. Many wrong answers come from bp versus percent and from forgetting that dt is in years.
Questions often ask what the model implies, not for a number. Know the three facts: zero drift, constant volatility, negative rates possible.
On tree questions, compute σ√dt once and write it down. Every node then follows by adding or subtracting.