Inverse Problems and Imaging (IPI)

Data driven recovery of local volatility surfaces

Pages: 799 - 823, Volume 11, Issue 5, October 2017      doi:10.3934/ipi.2017038

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Vinicius Albani - Dept. of Mathematics, UFSC, Florianopolis, Brazil (email)
Uri M. Ascher - Dept. of Computer Science, University of British Columbia, Canada (email)
Xu Yang - IMPA, Rio de Janeiro, Brazil (email)
Jorge P. Zubelli - IMPA, Rio de Janeiro, Brazil (email)

Abstract: This paper examines issues of data completion and location uncertainty, popular in many practical PDE-based inverse problems, in the context of option calibration via recovery of local volatility surfaces. While real data is usually more accessible for this application than for many others, the data is often given only at a restricted set of locations. We show that attempts to “complete missing data” by approximation or interpolation, proposed and applied in the literature, may produce results that are inferior to treating the data as scarce. Furthermore, model uncertainties may arise which translate to uncertainty in data locations, and we show how a model-based adjustment of the asset price may prove advantageous in such situations. We further compare a carefully calibrated Tikhonov-type regularization approach against a similarly adapted EnKF method, in an attempt to fine-tune the data assimilation process. The EnKF method offers reassurance as a different method for assessing the solution in a problem where information about the true solution is difficult to come by. However, additional advantage in the latter approach turns out to be limited in our context.

Keywords:  Data science, local volatility calibration, inverse problem, Tikhonov-type regularization, ensemble Kalman filter.
Mathematics Subject Classification:  Primary: 45Q05, 97M30, 65R32.

Received: September 2016;      Revised: May 2017;      Available Online: June 2017.