3.9.6 Advanced Extensions

Basic Prerequisites: 05 · Failure Modes & Practice and 04 · Numeraire, HJM & Market Models.

1. Intuition & Practical Objective

This page is the launchpad for everything beyond the vanilla single-curve, flat-volatility world. The three directions, in one line each:

The smile in rates (BM Ch9–12). LFM’s lognormal vol is a property of the rate, not the strike - so one $v (T)$ cannot match two market caplet strikes. Fix: local-volatility, stochastic-volatility, or uncertain-parameter models that reproduce the strike-dependent implied-vol surface.
Multi-curve / basis risk. Post-2008 the discount curve (OIS) and the forward curve (LIBOR) diverged; single-curve pricing misvalues swaps, and the LFM’s forwards must be consistent with the discount curve.
Calibration (BM Ch7). Fitting LFM/SVM/UPM parameters to caps + swaptions is the practical art - two-step (vols→caps, correlation→swaptions) or full joint optimization.

Why these? These are exactly the steps where a working model meets the market. The vanilla LFM prices ATM caplets/swaptions correctly by construction; the residual - the smile, the basis, the correlation - is what an extensions page must put on a sound footing.

2. Mathematical Ground Truth & Derivations

2.1 The smile in the LFM (BM Ch9)

A T2-caplet resetting at T1 pays $τ (F (T_{1}; T_{1}, T_{2}) - K)^{+}$ and prices as $P (0, T_{2}) τ E^{Q^{T_{2}}} [(F - K)^{+}]$ . Under the LFM, $F$ is lognormal, so $v_{2} (T_{1}) = \int_{0}^{T_{1}} σ_{2}^{2} d t$ gives Black’s formula. Because LFM volatility is a property of the rate, not the strike, one $v$ cannot match two strikes - the market caplet prices imply a strike-dependent $v^{M K T} (T_{1}, K)$ , whose curve is the volatility smile (minimum near ATM underlying; skew = low strikes priced at higher vol, BM §9.1).

The implied density is recovered by Breeden–Litzenberger (BM eq. 9.4):

\frac{\partial ^{2}}{\partial K ^{2}} Cp l^{M K T} (0, T_{1}, T_{2}, K) = P (0, T_{2}) τ p_{2} (K),

so the smile is a statement about a non-lognormal implied density. Six families can reproduce it (BM §9.2): local-volatility (LVM), stochastic-volatility (SVM), jump-diffusion (JDM), market models of implied vol (MMIV), Lévy-driven (LDM), and uncertain-parameter models (UPM). Only LVM/SVM/UPM are treated in the book for tractability.

2.2 Local-volatility & shifted-lognormal (BM Ch10)

Shifted lognormal $F_{j} = X_{j} + α$ , $d F_{j} = β (t) (F_{j} - α) d W$ : caplet closed form $Cpl = τ N P (t, T_{j}) Bl (K - α, F_{j} (t) - α, U)$ - produces only skews (BM eq. 10.6).
Lognormal-mixture (LM) local vol $σ^{2} (t, y) = \sum_{i} Λ_{i} v_{i}^{2} / y^{2}$ : caplet is a convex combination of Black prices $\sum_{i} λ_{i} Bl (K, F_{j} (0), V_{i})$ with the implied-vol minimum exactly at the ATM strike and zero ATM slope (BM Prop 10.4.1, eqs 10.27–10.34).
CEV $d F_{j} = σ_{j} (t) F_{j}^{γ} d W$ , $0 < γ < 1$ : absorbing boundary at 0 for $γ < 1/2$ ; non-central- $χ^{2}$ caplet; LCEV fixes absorption (BM eqs 10.7–10.11).

2.3 Stochastic-volatility & SABR (BM Ch11)

General SVM: $d F_{j} = a_{j} ϕ (F_{j}) V^{γ} d Z_{j}$ , $d V = a_{V} d t + b_{V} d W$ . Zero rate-vol correlation ⇒ smile-shaped vols (min at ATM); a skew needs (i) non-zero correlation, (ii) displaced diffusion, or (iii) non-linear $ϕ$ (BM Ch11 opening). Heston-type (Wu–Zhang) uses correlated square-root vol priced by characteristic function + Fourier inversion. SABR (HKLW 2002): $d F = V F^{β} d Z$ , $d V = ε V d W$ - models a single forward asset (per swaption), with the Hagan implied-vol approximation separating beta skew $\propto - \frac{1}{2} (1 - β)$ and vanna skew $\propto \frac{1}{2} ρ λ$ (BM eqs 11.33–11.35).

2.4 Uncertain-parameter models (BM Ch12)

UPM = volatility is a random variable, not a diffusion: $d F_{t} = σ F_{t} d Z_{t}$ with $σ$ drawn $σ_{1} .. σ_{N}$ w.p. $λ_{1} .. λ_{N}$ just after time 0. The caplet is a mixture of adjusted Black prices (BM eq. 12.7):

Cpl = i \sum λ_{i} Bl (K + α_{j}^{i}, F_{j} (0) + α_{j}^{i}, V_{j}^{i}),

producing a smile with minimum at $F_{0}$ ; a shift $α$ adds skew. The swaption is likewise a mixture under the annuity measure (BM eq. 12.8).

2.5 Multi-curve & calibration (BM Ch1, Ch7)

Multi-curve: post-2008, discount at OIS but quote forwards off LIBOR; the two curves are separate. The swap rate and caplet formulas above must use the forward curve consistent with the discount curve - the single-curve relation $1 + τL = P (t, T) / P (t, S)$ is replaced by a spread between curves.

Calibration (BM Ch7): fit LFM vols to caps (fix average vols) and swaptions (fix correlation/terminal structure). Cascade calibration (CCA) recovers piecewise-constant vols one swaption at a time walking the swaption matrix; RCCAEI adds endogenous interpolation to remove negative/complex artifacts. Two-stage (vol→caps, correlation→swaptions) or joint optimization are the practical routes; instantaneous correlations are treated as exogenous during cascade fitting.

3. Computational Implementation - smile machinery in numbers

Stdlib only: (1) a UPM/lognormal-mixture caplet reproducing a smile with minimum at ATM, (2) shifted lognormal skew, (3) the SABR implied-vol approximation showing beta/vanna skew.

The UPM mixture produces a textbook smile: implied vol dips to its minimum at the ATM strike $F_{0} = 4.04%$ and rises away from it - the exact signature BM Ch10 (LM) proves analytically. SABR with $ρ < 0$ gives a downward skew through ATM with curvature from $ν$ .

4. Failure Modes & First-Principles Breakdowns

The under-determined extraction problem. Breeden–Litzenberger gives the implied density only up to the interpolation scheme on prices (BM §9.1). Two interpolations of the same quoted strikes give different smiles. The model does not “know” the true density.
LVM decorrelation caveat (BM §10.5.1). In any LVM the instantaneous corr $corr (d F, d ν^{2}) = + 1$ (vol is a function of $F$ ), yet the terminal corr with time-averaged squared vol is exactly 0 - an UVM projection artifact (BM §12.1.1). Do not read a local-vol model’s “correlation” as a stochastic-vol one.
SABR is not an LFM extension (BM Remark 11.4.1). It models a single forward asset; a genuine multi-rate LIBOR smile needs LVM/SVM/UPM with the full measure/correlation structure.
Over-parameterization. Lognormal-mixture/LMDM/HSDM have many free parameters (BM Ch10 §10.9: LM=5, LMDM=4, HSDM=5); joint UPM calibration can reach 243 free parameters (BM §12.9). Watch for instability and non-uniqueness.
UPM forward vols flatten. After the volatility “draw,” future implied vols lose the initial smile (BM §12 drawback); empirically mitigated by shifts, but a real limit for forward-start products.

5. References

Brigo–Mercurio, Interest Rate Models
Hull, Options, Futures, and Other Derivatives
Shreve, Stochastic Calculus for Finance I
Björk, Arbitrage Theory in Continuous Time

6. Connected Graph Bridges

Back: 05 · Failure Modes · Index Hub
Forward topic-folder pages: Heston & SABR · Implied Volatility Surfaces
Base: Stochastic Calculus & Itô · No-Arbitrage & Binomial Trees

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