Convex Regularization of Local Volatility Estimation

Abstract

We apply convex regularization techniques to the problem of calibrating Dupire’s local volatility surface model taking into account the practical requirement of discrete grids and noisy data. Such requirements are the consequence of bid and ask spreads, quantization of the quoted prices and lack of liquidity of option prices for strikes far away from the at-the-money level. We obtain convergence rates and results comparable to those obtained in the idealized continuous setting. Our results allow us to take into account separately the uncertainties due to the price noise and those due to discretization errors, thus, allowing estimating better discretization levels both in the domain and in the image of the parameter to solution operator by a Morozov’s discrepancy principle. We illustrate the results with simulated as well as real market data. We also validate the results by comparing the implied volatility prices of market data with the computed prices of the calibrated model.

A connection between uniqueness of minimizers in Tikhonov-type regularization and Morozov-like discrepancy principles

“…One of the consequences of the use of discrepancy-based choices appears in the proof of convergence-rate results, where no assumption on the asymptotic behavior of α = α(δ, u δ ) when δ → 0 is needed. See [5,6,4,3].…”

“…Another possibility is by imposing that b − a is in some fixed finite-dimensional subspace of H 1+ε (D). We choose the second option since it has a strong connection with the approach presented in [4,3], where the authors propose a simultaneous discrepancy-based choice for the level of discretization in the domain and the regularization parameter in Tikhonov-type regularization. Therefore, we can state the following results: Under the hypotheses of Proposition 5.3, Theorem 4.1 holds, and there exists a unique local volatility surface minimizing the Tikhonov functional (3), whenever the objective set and the noise level in price data are small enough.…”

“…Moreover, the discrepancy in (5) can also be used in the simultaneous selection of the discretization level in the domain and the regularization parameter. See [4,3].…”

Simultaneous identification of volatility and interest rate functions-a two-parameter regularization approach

“…Introduction. Many authors with focus on analysis and numerics have considered the inverse problem arising in financial markets of recovering local volatility surfaces from option price data in the past years; see, e.g., [1,2,3,4,10,11,12,13,15,23,34]. In most cases, tools from regularization theory have been incorporated in the treatment of the inverse problem in order to overcome or at least to suppress the occurring ill-posedness phenomena.…”