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Online Local Volatility Calibration by Convex Regularization with Morozov's Principle and Convergence Rates | | Vinicius V.L. Albani
; Jorge P. Zubelli
; | | Date: |
1 Nov 2012 | | Abstract: | In this article we address the regularization of the ill-posed problem of
determining the local volatility surface (as a function of time to maturity and
price) from market given option prices. We integrate the ever-increasing flow
of option price information into the well-accepted local volatility model of
Dupire. This leads to considering both the local volatility surfaces and their
corresponding prices as indexed by the observed underlying stock price as time
goes by in appropriate function spaces.
The parameter to data map consists of a nonlinear operator that maps the
(variable) diffusion coefficient of a parabolic initial value problem into its
solutions evaluated at certain sets. We tackle the inverse problem by convex
regularization techniques in appropriate Bochner-Sobolev spaces.
As a preparation, we prove key regularity properties that enable us to apply
convex regularization techniques. This forward framework is then used to build
a calibration technique that combines online methods with convex Tikhonov
regularization tools. Such procedure is used to solve the inverse problem of
local volatility identification. As a result, we prove convergence rates with
respect to noise and a corresponding Morozov discrepancy principle for the
regularization parameter. We conclude by illustrating and validating the
theoretical results by means of numerical tests with synthetic as well as real
data. | | Source: | arXiv, 1211.0170 | | Services: | Forum | Review | PDF | Favorites | |
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