Brownian Bridge™

Setup

The SABR and Libor Market Model (LMM) are the workhorses of rates vol calibration. SABR is calibrated per expiry-tenor pair to a few strikes; LMM is calibrated to the full swaption cube (a 3D array of expiries, tenors, and strikes). Both present calibration challenges qualitatively different from the Heston setting:

SABR: very few parameters (3 per slice after fixing $\beta$ ) but a highly non-linear dependence on the ATM vol. The calibration is fast but requires special treatment of the near-ATM region.
LMM: many parameters (a full correlation matrix and one vol per rate), but the problem decomposes into tractable sub-problems via rank reduction and cascade calibration.

INSIGHT

Financial Insight. On a rates desk, SABR is run every morning on the swaption cube: for each expiry-tenor pair, a SABR model is calibrated to 5–7 strike quotes. The calibrated $(\alpha, \rho, \nu)$ per slice defines the "SABR surface". LMM is calibrated less frequently (typically weekly or when the model is used for exotic pricing), as it requires the full swaption cube and is more computationally intensive. The practical skill is knowing which model to use for which task and how to calibrate each efficiently.

Notation and conventions:

SABR: forward rate $F > 0$ under the forward measure for a given expiry-tenor. $\beta \in [0,1]$ fixed. Calibrate $(\alpha, \rho, \nu)$ where $\alpha > 0$ (initial vol), $\rho \in (-1, 1)$ (skew parameter), $\nu > 0$ (vol of vol).
LMM (BGM model): $N$ forward rates $F_1(t), \ldots, F_N(t)$ for consecutive tenors. Each $F_n$ has its own piecewise-constant vol $\sigma_n$ and a common correlation matrix $\rho \in \mathbb{R}^{N \times N}$ .
All vols are Black-Scholes normal or log-normal, annualised, in decimal.
Swaption strikes quoted as spreads to the ATM forward swap rate.

Theory

SABR Per-Expiry Calibration

Fixing $\beta$ : The $\beta$ parameter controls whether the ATM vol is stable (normal dynamics, $\beta = 0$ ) or inversely related to the forward rate (log-normal dynamics, $\beta = 1$ ). In the EUR post-2016 negative rates environment, $\beta = 0$ (normal SABR) is standard for caps/floors. For GBP and USD swaptions, $\beta = 0.5$ or $\beta = 1$ is common. $\beta$ is not calibrated — it is set by market convention and the risk quant.

REMARK

Remark. $\beta$ and $\rho$ have degenerate effects on the smile skew. Fitting both simultaneously produces an ill-conditioned Jacobian. Fix $\beta$ by market convention; calibrate only $(\alpha, \rho, \nu)$ .

The Hagan SABR implied vol formula (for $\beta$ fixed, $K \neq F$ ):

$\sigma_N^{\text{SABR}}(K, F; \alpha, \rho, \nu) \approx \frac{\alpha\,(FK)^{(\beta-1)/2}}{1 + \frac{(1-\beta)^2}{24}\ln^2\!\tfrac{F}{K} + \frac{(1-\beta)^4}{1920}\ln^4\!\tfrac{F}{K}} \cdot \frac{z}{\chi(z)} \cdot \left[1 + \left(\frac{(1-\beta)^2 \alpha^2}{24(FK)^{1-\beta}} + \frac{\rho\beta\nu\alpha}{4(FK)^{(1-\beta)/2}} + \frac{(2 - 3\rho^2)\nu^2}{24}\right)T\right]$

where $z = \frac{\nu}{\alpha}(FK)^{(1-\beta)/2}\ln\tfrac{F}{K}$ and $\chi(z) = \ln\!\frac{\sqrt{1-2\rho z + z^2} + z - \rho}{1 - \rho}$ .

At the money ( $K = F$ ), the formula simplifies to:

THEOREM

Theorem 5.1 (SABR ATM Implied Vol). At $K = F$ (ATM), the Hagan formula reduces to: $\sigma_{\text{ATM}}^{\text{SABR}} \;=\; \alpha\,F^{\beta-1} \left[1 + \left(\frac{(1-\beta)^2\alpha^2}{24 F^{2(1-\beta)}} + \frac{\rho\beta\nu\alpha}{4 F^{1-\beta}} + \frac{(2-3\rho^2)\nu^2}{24}\right)T\right]$

For $\beta = 1$ (log-normal): $\sigma_{\text{ATM}} = \alpha\left[1 + \left(\frac{\rho\nu\alpha}{4} + \frac{(2-3\rho^2)\nu^2}{24}\right)T\right]$

This ATM formula provides a near-explicit equation for $\alpha$ given $\sigma_{\text{ATM}}$ (it is quadratic in $\alpha$ for $\beta = 1$ ): solve the quadratic to initialise $\alpha$ , then refine via NLS on all quotes.

SABR calibration procedure (per expiry-tenor slice):

Fix $\beta$ (market convention).
Estimate $\alpha$ from the ATM implied vol by solving the ATM formula above.
Set initial $\rho \approx$ ATM skew slope estimate; $\nu \approx$ smile curvature estimate.
Run LM with 3 parameters $(\alpha, \rho, \nu)$ and $N_K = 5$ – $7$ quoted strikes.
Repeat for each expiry-tenor pair in the swaption cube.

Since each per-slice calibration involves only 3 parameters and 5–7 instruments, it converges in 5–15 LM iterations.

WARNING

Warning — Hagan Formula Validity. The Hagan formula is an asymptotic expansion in $\nu^2 T$ (vol of vol times time). For long-dated expiries ( $T > 10$ ) or high vol of vol ( $\nu > 1$ ), the approximation breaks down and the formula can produce negative vols or arbitrage violations. Use the Antonov-Konikov-Spector exact formula or the effective approximation of Obloj (2008) for long dates. Verify calibrated smiles are positive and monotone for all strikes before using for pricing.

LMM: The Model and Its Parameters

DEFINITION

Definition 5.1 (LIBOR Market Model — Brace-Gatarek-Musiela 1997). Under the spot LIBOR measure, the $n$ -th forward rate $F_n(t)$ for the period $[T_{n-1}, T_n]$ evolves as: $dF_n(t) = \mu_n(t)\,dt + \sigma_n(t)\,F_n(t)\,dW_n(t), \quad t \le T_{n-1}$ where $W_n$ are correlated Brownian motions with $d\langle W_m, W_n\rangle_t = \rho_{mn}\,dt$ , $\sigma_n(t)$ is the deterministic vol function for $F_n$ , and $\mu_n$ is the drift (uniquely determined by no-arbitrage under the spot measure).

For calibration to caplets, the key formula is:

$C_n = \tau_n \cdot \text{Black}(F_n(0), K, v_n \sqrt{T_{n-1}})$

where $v_n^2 = \frac{1}{T_{n-1}} \int_0^{T_{n-1}} \sigma_n^2(t)\,dt$ is the root mean square vol of $F_n$ .

For calibration to swaptions, the Rebonato (1999) approximation gives:

THEOREM

Theorem 5.2 (Rebonato Swaption Approximation). The implied vol $\Sigma_{\alpha\beta}$ of a payer swaption on the swap $[T_\alpha, T_\beta]$ with notional payment at each $T_n$ , $n=\alpha+1,\ldots,\beta$ , is approximately:

$\Sigma_{\alpha\beta}^2 T_\alpha \;\approx\; \frac{1}{S_{\alpha\beta}^2} \sum_{n,m=\alpha+1}^{\beta} w_n\,w_m\,\rho_{nm}\,F_n(0)\,F_m(0) \int_0^{T_\alpha} \sigma_n(t)\,\sigma_m(t)\,dt$

where $S_{\alpha\beta} = \sum_{n=\alpha+1}^{\beta} \tau_n D_n F_n(0)$ is the forward swap rate (with $D_n$ discount factors) and $w_n = \tau_n D_n F_n(0) / S_{\alpha\beta}$ are the swaption weights.

This approximation converts the LMM swaption vol into a quadratic function of the vol parameters — enabling fast gradient-based calibration.

Rank Reduction for the Correlation Matrix

The LMM correlation matrix $\rho \in \mathbb{R}^{N\times N}$ has $N(N-1)/2$ free parameters — far too many to calibrate from a finite swaption cube. Rank reduction restricts the correlation matrix to have a specific parametric form.

DEFINITION

Definition 5.2 (Rank- $k$ Approximation). Write $\rho = B B^\top$ where $B \in \mathbb{R}^{N \times k}$ has normalised rows ( $\|b_n\| = 1$ ). This ensures $\rho$ is PSD with unit diagonal. For $k$ factors, there are $kN - k(k+1)/2$ free parameters (after removing rotational freedom).

Rebonato parametrisation (1-factor): $\rho_{mn} = \exp(-\lambda|n - m|)$ for a single parameter $\lambda > 0$ . This is a separable exponential form — one parameter controls all pairwise correlations.

For calibration with limited swaption data, $k = 2$ or $k = 3$ factors is standard:

$k = 2$ : captures the overall level and the short-rate/long-rate spread direction.
$k = 3$ : adds curvature of the correlation structure.

Cascade (Sequential) Calibration

Rather than calibrating all LMM parameters simultaneously (which is a large, ill-conditioned NLS problem), cascade calibration proceeds sequentially:

DEFINITION

Definition 5.3 (Cascade Calibration). For $n = 1, 2, \ldots, N-1$ (in order of increasing expiry):

Calibrate $\sigma_n(t)$ on $[T_{n-1}, T_n)$ to match the co-terminal swaption $[T_n, T_\beta]$ given the already-calibrated $\sigma_1, \ldots, \sigma_{n-1}$ .
Each calibration step involves a single 1D root-finding problem (or a small NLS with few parameters).

This "peeling" approach converts a large joint optimisation into $N-1$ small problems.

Limitation of cascade calibration: the sequential approach may not produce the globally optimal joint solution. Specifically, the Rebonato swaption approximation introduces approximation errors that accumulate through the cascade. In practice, cascade calibration is followed by a global LM polish to reduce the joint residuals.

EXAMPLE

Example 5.1 (3-Rate LMM Calibration). With $N = 3$ forward rates ( $F_1, F_2, F_3$ ) at quarterly intervals, the correlation matrix is: $\rho = \begin{pmatrix} 1 & \rho_{12} & \rho_{13} \\ \rho_{12} & 1 & \rho_{23} \\ \rho_{13} & \rho_{23} & 1 \end{pmatrix}$ Using the Rebonato parametrisation $\rho_{mn} = e^{-\lambda|m-n|}$ reduces this to one parameter $\lambda$ . With flat vols $\sigma_n = \sigma$ for all $n$ , calibration to 3 caplet vols gives $\sigma$ from caplets and $\lambda$ from the ratio of the 2-year swaption vol to the 1-year caplet vol.

SABR-LMM: Stochastic Vol for Each Rate

The combined SABR-LMM model assigns SABR dynamics to each forward rate $F_n$ , giving each a time-varying vol $\alpha_n(t)$ and introducing additional correlations between vol processes. This enables fitting to the swaption smile (not just ATM) but at the cost of many more parameters. Full SABR-LMM calibration is a research-grade task; the standard simplification is to calibrate SABR per-expiry-tenor (for smile) and LMM for the correlation structure (for exotics sensitivity) separately.

Validation

The companion notebook implements SABR and a simplified 3-rate LMM calibration:

SABR Hagan formula for given $(\alpha, \beta, \rho, \nu)$ evaluated across a strike grid.
ATM formula to initialise $\alpha$ from market ATM vol — verified against the full formula.
Per-expiry SABR calibration to synthetic market quotes using scipy.optimize.least_squares.
Recovery of the true parameters to within $10^{-3}$ .
Rebonato swaption approximation for a 3-rate LMM — compared against the exact result.
Rank-1 Rebonato correlation parametrisation — fitting $\lambda$ to a caplet/swaption ratio.

PRACTICE

Before opening the notebook: For SABR with $\beta = 0.5$ , $F = 0.04$ , $T = 1$ , and quotes $\sigma_{\text{ATM}} = 0.25$ , $\sigma(K = 0.02) = 0.30$ , $\sigma(K = 0.06) = 0.22$ : (a) What sign of $\rho$ do you expect (the smile is negatively skewed)? (b) Estimate $\alpha$ using the ATM formula (ignore the correction term for a first estimate). (c) Which of $\rho$ or $\nu$ primarily controls the skew vs. the curvature?

Limitations

WARNING

Warning — SABR Arbitrage at Extreme Strikes. The Hagan approximation produces implied vols that correspond to negative risk-neutral densities at very low or very high strikes. This is a defect of the approximation, not the true SABR model. The issue manifests as negative butterfly spreads. In production, always verify that the calibrated smile is free of butterfly arbitrage (second derivative of call price with respect to strike $\ge 0$ ) before using it for pricing. Use the Antonov-Konikov-Spector interpolation or the free boundary SABR (for negative rates) as alternatives.

WARNING

Warning — LMM Dimensionality. A 20-rate LMM has a $20\times 20$ correlation matrix (190 free entries) plus 20 vol parameters. Without rank reduction, the calibration is grossly under-determined by the swaption cube (which has at most $20 \times 20 \times 5 = 2000$ quotes, but many of these are illiquid). Rank reduction to $k = 3$ factors reduces the correlation parameters to $\approx 3 \times 20 - 6 = 54$ — manageable but still large. The choice of $k$ is a model risk decision.

Other limitations:

Cascade calibration ordering: the cascade proceeds from short to long maturities. Errors in early-calibrated vols propagate and amplify into later calibrations. The cascade is an approximation to the joint problem; global LM refinement is necessary.
SABR parameter migration: SABR parameters calibrated today may differ significantly from yesterday's if the vol surface has moved. The SABR model is not dynamically consistent (i.e., there is no SABR dynamics for $\alpha$ itself) — re-calibration is not a hedge. For dynamic hedging, need the SABR-LMM extension.
Rebonato approximation accuracy: the Rebonato approximation has errors of $O(v^2 T)$ where $v$ is the vol level. For high-vol or long-dated swaptions ( $T > 10$ years), the approximation can be off by more than 1 vol point. Use exact Monte Carlo or PDE benchmarks to validate.
Correlation instability: the LMM correlation matrix must be PSD. After calibration, a matrix assembled from independently fitted entries may not be PSD — project onto the nearest PSD matrix (eigenvalue clipping) if needed.

Interview Angle

PRACTICE

L1 (Junior) — Typical questions:

What is the SABR model and what are its parameters? Expected: $dF = \alpha F^\beta dW^1$ , $d\alpha = \nu\alpha dW^2$ , $dW^1 dW^2 = \rho dt$ . Parameters: $\alpha$ (initial vol), $\beta$ (CEV exponent, usually fixed), $\rho$ (skew), $\nu$ (vol of vol, curvature).
Why is $\beta$ fixed rather than calibrated in SABR? Expected: $\beta$ and $\rho$ have degenerate effects on the skew — fitting both simultaneously is ill-conditioned. Market convention fixes $\beta$ by rate regime (normal for negative rates, log-normal for positive rates).
What is the Libor Market Model (LMM)? Expected: directly models a set of forward LIBOR/SOFR rates as log-normal processes (under BGM / BG). Consistent with Black's formula for caplets. Used for pricing path-dependent interest rate products (Bermudans, CMS, etc.).

PRACTICE

L2 (Senior) — Typical questions:

Describe the SABR per-expiry calibration procedure. What is the key simplification at ATM? Expected: fix $\beta$ ; use ATM formula to analytically pin $\alpha$ ; run LM on $(\rho, \nu)$ (or $(\alpha, \rho, \nu)$ with $\alpha$ from ATM). ATM formula is approximately linear in $\alpha$ → good initialisation. Per slice: 3 parameters, 5–7 instruments, fast convergence.
Explain rank reduction of the LMM correlation matrix. Why is it necessary? Expected: $N(N-1)/2$ free parameters in $\rho$ — too many for the data. Rank- $k$ factorisation $\rho = BB^\top$ reduces to $kN - k(k+1)/2$ parameters. $k=2$ or 3 is standard. State the Rebonato 1-parameter form $\rho_{mn} = e^{-\lambda|m-n|}$ .
What is cascade calibration? What are its weaknesses? Expected: sequential calibration from short to long expiries; each step is a small root-finding problem. Weaknesses: errors propagate, approximation quality degrades for long dates, not globally optimal. Mitigated by global LM polish afterward.

PRACTICE

L3 (Researcher) — Typical questions:

Under what conditions does the Hagan SABR approximation produce arbitrage-free smiles? How would you fix the problem in practice? Expected: approximation produces negative densities at very low strikes (near zero) and very high strikes. Condition: approximation is valid for $\nu^2 T \ll 1$ and $\alpha \ll 1$ . Fix: use the exact SABR density (numerically from the 2D Kolmogorov PDE), or use the free-boundary SABR, or the Antonov-Konikov-Spector closed form. For low strikes in negative rate regimes: shifted SABR or normal SABR.
Derive the Rebonato swaption approximation formula. What assumptions does it make? Expected: swap rate is approximated as a weighted sum of forward rates; the weight $w_n \approx \tau_n D_n F_n(0)/S_{\alpha\beta}$ is frozen at $t=0$ (the "freezing" approximation). The log-normal dynamics of the frozen approximation then give a log-normal swaption vol via the co-variance integral. Accuracy: $O(v^2)$ relative error in vol.
Compare calibrating the LMM to caplets only vs. to swaptions. What additional information do swaptions provide? Expected: caplets constrain each $\sigma_n$ independently (diagonal information). Swaptions constrain the integrated covariance $\sum_{n,m} \rho_{nm} \sigma_n \sigma_m$ (off-diagonal information). Calibrating only to caplets leaves $\rho$ unconstrained — any correlation matrix fits. Swaptions uniquely determine the correlation structure under the Rebonato approximation. Joint calibration to caplets + swaptions determines both vols and correlations.