Brownian Bridge™

1. In the SABR model, $\beta$ controls the relationship between the ATM vol and the forward rate. In a negative interest rate environment (e.g., EUR post-2016), which value of $\beta$ is standard and why?

β = 0 (normal SABR): forward moves are absolute, allowing F to go negativeβ = 1 (lognormal SABR): vol is proportional to the forward, keeping F > 0β = 0.5: a compromise that handles both positive and negative ratesβ is calibrated from market data in all rate environments

2. Why is $\beta$ fixed rather than calibrated in SABR, even though it is a model parameter?

β and ρ have degenerate effects on skew: fitting both simultaneously produces an ill-conditioned Jacobian with multiple equivalent solutionsβ can only take integer values (0 or 1), so it cannot be optimised by gradient-based methodsβ is determined by the payoff structure of the option being pricedCalibrating β is computationally too expensive for real-time use

3. For SABR per-expiry calibration, the ATM implied vol formula is used to initialise $\alpha$ . For $\beta = 0.5$ , $F = 0.04$ , $T = 1$ , ignoring the correction term, the initial estimate is $\alpha_0 \approx$ :

σ_ATM × F^(1−β) = σ_ATM × F^0.5 = σ_ATM × 0.2σ_ATM / F = 25 × σ_ATMσ_ATM × F = 0.04 × σ_ATMσ_ATM directly (β=0.5 cancels the F dependence)

4. The Rebonato (1999) swaption approximation freezes the swaption weights $w_n$ at $t=0$ . This makes the swaption implied vol a quadratic function of the LMM vol parameters $\sigma_n$ . The main consequence is:

The approximation is fast and enables analytic gradient computation, but has O(v²) relative accuracy errorsThe approximation is exact and no numerical error is introducedThe approximation is only valid for at-the-money swaptionsThe approximation requires flat forward rates to be valid

5. An LMM with $N = 20$ forward rates has how many free parameters in its (unconstrained) correlation matrix, and how many with the Rebonato rank-1 exponential parametrisation $\rho_{mn} = e^{-\lambda|m-n|}$ ?

190 unconstrained vs 1 with Rebonato rank-1400 unconstrained vs 20 with rank-120 unconstrained vs 1 with rank-1190 unconstrained vs 20 with rank-1

6. Cascade calibration proceeds from short to long maturities. You are calibrating forward rate vol $\sigma_3$ using the 3-year co-terminal swaption, given already-calibrated $\sigma_1, \sigma_2$ . A quant points out that $\sigma_1$ has a 5% error. How does this affect $\sigma_3$ ?

Errors in σ₁ propagate and accumulate into σ₃: the cascade converts individual errors into compounding biasNo effect: each cascade step is independent by designThe error in σ₁ is corrected when σ₂ is calibrated to the 2y swaptionThe Rebonato approximation absorbs the σ₁ error in the swaption weighting

7. A SABR calibration to 1-year EUR swaption strikes produces negative implied vols at very low strikes (near 0% strike for a 4% forward). The most likely cause is:

The Hagan approximation breaks down for large log-moneyness ln(F/K): the expansion fails for very OTM optionsThe Feller condition is violated for EUR SABRNormal SABR (β=0) was used instead of log-normalThe vol of vol ν is too small to generate a smile at low strikes

8. A desk calibrates LMM separately to caplets (constraining only individual forward vols $\sigma_n$ ) and then observes that the calibrated model misprices swaptions by 3 vol points. The root cause is:

Caplets constrain only the diagonal: each σₙ independently. Swaptions also depend on correlations ρₘₙ, which are unconstrained by caplet-only calibrationThe Rebonato approximation is not valid for this marketThe forward rates F₁,…,Fₙ are not lognormal under this LMMSwaptions require a different numeraire than caplets

Quiz: SABR and LMM Calibration — Multi-Instrument Fitting

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