Brownian Bridge™

1. The weighted NLS calibration objective is $\mathcal{L}(\theta) = \frac{1}{2}\|Wr(\theta)\|^2$ where $r_i(\theta) = y_i - f_i(\theta)$ . What is the gradient $\nabla_\theta \mathcal{L}$ ?

−J(θ)ᵀ W² r(θ)+J(θ)ᵀ W² r(θ)−J(θ)ᵀ W r(θ)J(θ)ᵀ J(θ) θ

2. The Gauss-Newton approximation to the Hessian discards the second-order term $\sum_i w_i^2 r_i \nabla^2_\theta f_i$ . Under what two conditions is this approximation valid?

Residuals small at solution OR model nearly linear in θJacobian has full rank AND step size is smallWeights are uniform AND problem is convexN > p (overdetermined) AND parameters are bounded

3. For Heston calibration to 40 market implied vols with 5 model parameters, the Jacobian $J$ has dimensions:

40 × 55 × 405 × 540 × 40

4. A calibration in **price space** rather than **implied vol space** would systematically:

Up-weight expensive ITM options and down-weight cheap OTM optionsUp-weight OTM options because they carry more model sensitivityProduce unbiased weights across all strikesOnly affect the convergence speed, not the fitted parameters

5. The normal equations for the Gauss-Newton step are $(J^\top W^2 J)\,\delta\theta = J^\top W^2 r$ . The system is singular when:

J has rank less than p (fewer linearly independent columns than parameters)N > p (more instruments than parameters)The residuals r(θ) are largeThe weights W are non-uniform

6. The condition number $\kappa(J^\top J)$ of the calibration problem is $10^5$ . A 0.1% relative perturbation in market data would cause approximately what relative change in calibrated parameters?

Up to 10% (amplified by the condition number)Exactly 0.1% (no amplification)Less than 0.1% (condition number damps perturbations)Impossible to determine without knowing the specific perturbation

7. A vol desk calibrates Heston to today's market and reports $v_0 = 0.03$ (spot variance). The ATM implied vol for a 1-month option is $\sqrt{0.03} \approx 17.3\%$ . Tomorrow the 1-month ATM vol jumps to 22%. The recalibration will primarily adjust which parameter?

v_0

(initial variance), as it directly controls the current spot vol level

\bar{v}

(long-run variance), as the whole term structure shifted

\kappa

(mean reversion speed), as the reverting force changed

\xi

(vol of vol), as the vol change indicates higher vol uncertainty

8. A desk quant is asked to calibrate a 5-parameter model to 3 market quotes. What can you immediately say about the calibration problem?

It is under-determined: the 3×5 Jacobian cannot have full column rank, so infinitely many parameter sets fit the data exactlyIt is over-determined: more parameters than instruments means a unique solution existsIt is exactly determined: Gauss-Newton will converge in one stepNothing can be said without knowing the specific model

Quiz: Model Calibration as a Non-Linear Least Squares Problem

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