Global optimisation of noisy grey-box functions with financial applications

Abstract

Financial derivatives of both plain vanilla and exotic type are at the core of today’s financial industry. For the valuation of these derivatives, mathematical pricing models are used that rely on different approaches such as (semi-)analytical transform methods, PDE approximations or Monte Carlo simulations. The calibration of the models to market prices, i.e. the estimation of appropriate model parameters, is a crucial procedure for making them applicable to real markets. Due to inherent complexity of the models, this typically results in a nonconvex optimisation problem that is hard to solve, thus requiring advanced techniques.

In this thesis, we study the general case of financial model calibration where model prices are approximated by standard Monte Carlo methods. We distinguish between two possibilities on how to employ Monte Carlo estimators along a calibration procedure: the simpler sample average approximation (SAA) strategy, which uses the same random sample for all function evaluations, and the more sophisticated variable sample average approximation (VSAA) strategy, which for each evaluation uses a different random sample with variable size. Both strategies have in common that they lead to (not prohibitively) expensive optimisation procedures providing approximating solutions to the original but unknown optimisation problem. Yet, whereas the former strategy results in a self-contained deterministic problem instance that may be fully solved by a suitable algorithm, the latter has to be considered together with a sequential sampling method that incorporates the strategy to select new evaluation points, which amounts to minimising a noisy objective function.

For both strategies, we initially establish essential convergence properties for the (optimal) estimators in the almost sure sense. Specifically, in the case of the SAA strategy, we complement the well-established strong consistency of optimal estimators with their almost sure rates of convergence. This, in turn, allows to draw several useful conclusions on their asymptotic bias and other notions of convergence. In the case of the VSAA strategy, we give conditions for the strong uniform consistency of the objective function estimators and provide corresponding uniform sample path bounds. Both results may be used to show convergence of a sequential sampling method adopting the VSAA scheme.

We then address the global optimisation within both considered procedures, and first present a novel modification to Gutmann’s radial basis function (RBF) method for expensive and deterministic objective functions that is more suited for deterministic calibration problems. This modification exploits the particular data-fitting structure of these problems and additionally enhances the inherent search mechanism of the original method by an extended local search. We show convergence of the modified method and demonstrate its effectiveness on relevant test problems and by calibrating the Hull-White interest rate model under the SAA strategy in a real-world setting. Moreover, as the method may be applied equally well to similar data-fitting problems that are not necessarily expensive, we also demonstrate its practicability by fitting the Nelson-Siegel and Svensson models to market zero rates.

Based on the RBF method, we further present a novel method for the global optimisation of expensive and noisy objective functions, where the level of noise is controlled by means of error bounds. The method uses a regularised least-squares criterion to construct suitable radial basis function approximants, which are then also used to determine new sample points in a similar manner as the original RBF method. We provide convergence of the method, albeit under some simplifying assumption on the error bounds, and evaluate its applicability on relevant test problems and by calibrating the Hull-White interest rate model under the VSAA strategy.

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Identifiers

ORCID for Joerg Fliege: orcid.org/0000-0002-4459-5419

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