Some heavy math in local vol calibratrion

Last week, I plotted the results we obtained for local vol by UnRisk's calibration engine.

To obtain such results, you need some serious mathematics. And this is exactly what we did.

In an abstract setting, we want to solve a nonlinear equation F (x) = y between Hilbert spaces X and Y. (Here: x is the parameter function of local volatility acting on two-dimenstional space (S,t). F is the operator that maps any local volatility function to the two-dimensional surface of call prices (K,T). The right hand side y itself (the call price function for the true local vol) cannot be measured but it is noisy (with noise level delta), and available only for certain points.

Instead of solving F(x) = y_delta, Tikhonov regularization solves

In the 1990's Engl, Kunisch, and Neubauer proved that, in this abstract setting, nonlinear Tikhonov regularization converges if
(a) the operator F is continuous,
(b) the operator F is weakly sequentially closed, and
(c) alpha > 0 and delta^2 / alpha -> 0.

Condition (b) might look unfamiliar. The reason for which we have to use the weak topology is that in infinite-dimensional settings, bounded sets in the strong topology need not have convergent subsequences.

For any parameter identification problem, where the solution is theoretically inifinite-dimensional, we have to analyse if the above conditions on the operator are satisfied. For local volaitility, this was carried out by Egger and Engl (Tikhonov Regularization Applied to the Inverse Problem of Option Pricing: Convergence Analysis and Rates, Inverse Problems, 2005).

At UnRisk, we do the math and the numerical implementation. This makes us deliver robust solutions. Can you say this for your system as well?