# Ill-posed problem: Local volatility calibration. Regularization vs Smoothing

I have asked my question on Mathematics site of Stack Exchange but maybe I will get the answer rather here.

I am working on inverse problem - calibration of local volatility (financial application). This inverse problem is ill-posed. The optimization algorithm takes a $\sigma$ surface as input and solves a PDE to calculate the cost function for each input. After inputs are evolved following genetic algorithm rules. There are several authors proposing next approaches to regularize the input surface:

1. Regularization by means of Tikhonov
2. Smoothing using B-splines
3. Regularization using multiscale approach

Some authors use on of the first two approaches. However, the author of my article uses both of them(Main Article). For example the smoothness norm is introduced as:

$\|\sigma\|^2 = \sum_{i}^{T} \int_{-A}^{A} dx \int_{0}^{T}|\frac{d\sigma}{dT}|^2 + |\frac{d^2\sigma}{dx^2}|^2 dx$

or in matrix form:

$\|\sigma\|^2 = \theta A \theta^{T}$

where

$\theta$ is a matrix of control points in B-Spline representation and

$\sigma(T_{i},\cdot) = \sum_{m=1}^{M} \theta(i,m)\phi_{m}(\cdot)$

$\sigma(t,x) = \frac{T_{i+1}-t}{T_{i+1}-T_{i}}\sigma(T_{i},x)+\frac{t-T_{i}}{T_{i+1}-T_{i}}\sigma(T_{i+1},x)$

are spline and linear interpolations by $x$ and $T$ axis respectively. After the authors use Cholesky decomposition to obtain matrix $B$ such that $A = BB^{T}$. Finally they use genetic optimization algorithm to perform random changes of control points: $\theta_{new} = \theta + B\epsilon_{i}$.

The PDE is solved again with new candidate solutions $\theta_{new}$ until the stopping criteria is not achieved.

Basically I do not understand the difference between regularization and smoothing.

What I do understand:

• B-Splines allow us to obtain a smooth representation of a function of interest

• Regularization is putting some convenient constraint on the function to be optimized (including the constraint of smoothness).

Isn't it enough to introduce a spline representation of the surface? (won't it be already smooth - as I understand it will).

Why do not we apply the random change directly on $\theta$? kind of $\theta_{new} = \theta+\epsilon_{i}$

The role of smoothness norm knowing that B-Spline function is apriori smooth is not clear for me. Thanks!

• Usually practitioners calibrate the implied volatility surface and then compute on the fly the local volatility using Dupire's formula. I am curious as to why you want to calibrate the local volatility surface directly. – Antoine Conze Nov 10 '17 at 7:27
• Apologies if I'm misinterpreting your question or saying something obvious. Something that's quite standard in polynomial curve fitting is to fit a higher order polynomial (to have a more flexible functional form) but then use regularization to penalize overfitting and push the fitted polynomial curve smoother in some aesthetic sense (rather than a precise mathematical, continuous $k$th derivative sense). – Matthew Gunn Nov 10 '17 at 15:31
• @MatthewGunn So basically B-spline smoothing is already a kind of regularization? I mean the term smoothing is equivalent to regularization? Because as you said, the B-Spline by definition does not pass by all control points, preventing the overfitting, but creates a smooth curve in aesthetic sense – Maksym Bondarenko Nov 12 '17 at 23:32
• @AntoineConze Rama Cont in his article claims why Dupire formula is unstable: Dupire (1994) presents a formula for reconstructing local volatility functions from a continuum of call option prices; however, this formula involves taking derivatives from discrete data and is numerically unstable. A discretized version of the Dupire formula is the implied tree method of Derman et al (1996), which is prone to similar instabilities leading to “negative probabilities” – Maksym Bondarenko Nov 13 '17 at 1:23
• The standard approach is to build a smooth implied volatility surface from option prices and then to apply the Dupire formula to the implied volatility surface. If you calibrate directly a local volatility surface you will have a hard time computing vegas. – Antoine Conze Nov 13 '17 at 7:32