Abstract
We address the inverse problem of local volatility surface calibration 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 resulting parameter to data map is defined in appropriate Bochner-Sobolev spaces. Under this framework, we prove key regularity properties. This enable us 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 discrepancy-based choice for the regularization parameter. We conclude by illustrating the theoretical results by means of numerical tests.
Highlights
A number of interesting problems in nonlinear analysis are motivated by questions from mathematical finance
The analysis presented in [5, 6, 7, 10] was based on an a priori choice of the regularization parameter with convex regularization tools
Making use of convex regularization tools, we provide some convergence rates with respect to the noise level
Summary
A number of interesting problems in nonlinear analysis are motivated by questions from mathematical finance. We prove that the so-called direct problem is well-posed, i.e., the forward operator satisfies key regularity properties This framework generalizes in a nontrivial way the structure used in previous works [5, 6, 7, 10] since it requires the introduction of more tools, in particular that of Bochner spaces. This is done in Theorem 1 and Propositions 4, 5, 6 and 7.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
