Regularisation and Stability in Calibration

1. In the SVD of the Jacobian $J = U\Sigma V^\top$, the unregularised least-squares update is $\delta\theta_{\mathrm{LS}} = \sum_j \frac{u_j^\top \delta\hat{\sigma}}{\sigma_j} v_j$. Why do small singular values $\sigma_j \approx 0$ cause calibration instability?

2. Tikhonov regularisation adds a penalty $\lambda\|\theta - \theta_0\|^2$ to the calibration objective. In terms of the SVD filter factors $f_j(\lambda) = \sigma_j^2/(\sigma_j^2 + \lambda)$, what happens to parameter directions with large singular values ($\sigma_j \gg \sqrt{\lambda}$)?

3. The Morozov discrepancy principle selects $\lambda$ such that $\|r(\theta_\lambda)\| \approx \epsilon\sqrt{N}$. In a Heston calibration with $N = 50$ instruments and a bid-ask half-spread of $0.2$ vol points ($\epsilon = 0.002$), what is the target residual norm?

4. Regularising the Heston calibration by adding a Tikhonov penalty $\lambda\|\theta - \theta_{\mathrm{prev}}\|^2$ (where $\theta_{\mathrm{prev}}$ is yesterday's calibration) eliminates calibration bias on days when the market makes a large move.

5. In the L-curve method, the optimal regularisation parameter $\lambda^*$ is identified at the corner of the L-shaped curve plotting residual norm versus regularisation norm. What characterises a point on the 'horizontal arm' of the L-curve (small $\lambda$)?

6. Penalising surface curvature in local vol calibration via $\lambda_1 \|\partial_{kk} \sigma_{\mathrm{loc}}\|^2$ suppresses oscillations across strikes. What is the key failure mode when $\lambda_1$ is chosen too large?