Why Local Volatility Calibration Uses Both Splines and Regularization
Summary
The document describes a question about calibrating a local volatility surface through an inverse problem. It outlines a representation that uses B-splines across one dimension and linear interpolation across time, then penalizes variation in time and curvature across the other dimension. The penalty is expressed as a quadratic form in the spline control points. A Cholesky factor of that form is used to shape random changes to the control points during genetic optimization, with the pricing PDE recalculated for candidate surfaces.
The author asks how this penalty differs from the smoothness already provided by splines, and why mutations should be transformed by the penalty matrix rather than applied directly. The document poses these questions but gives no answer, calibration results, or comparison of methods. It is therefore useful for identifying a modeling distinction: a smooth basis limits the kinds of surfaces that can be represented, while a penalty can further favor less variable surfaces or guide optimization steps. The effectiveness of either choice depends on the calibration setup and is not established here.
Key ideas
- B-spline basis functions provide a smooth parameterization of the local volatility surface.
- A separate quadratic penalty can discourage temporal variation and spatial curvature within that parameterization.
- The described genetic search transforms random control-point changes using a factor of the penalty matrix.
- The document raises, but does not resolve, how basis smoothness and explicit regularization differ in practice.
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# Ill-posed problem: Local volatility calibration. Regularization vs Smoothing
# 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:
- Regularization by means of Tikhonov
- Smoothing using B-splines
- 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!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.