Pricing early-exercise and discrete barrier options by Shannon wavelet expansions

We present a pricing method based on Shannon wavelet expansions for early-exercise and discretely-monitored barrier options under exponential Lévy asset dynamics. Shannon wavelets are smooth, and thus approximate the densities that occur in finance well, resulting in exponential convergence. Application of the Fast Fourier Transform yields an efficient implementation and since wavelets give local approximations, the domain boundary errors can be naturally resolved, which is the main improvement over existing methods

Niching an estimation-of-distribution algorithm by hierarchical Gaussian mixture learning

Estimation-of-Distribution Algorithms (EDAs) have been applied with quite some success when solving real-valued optimization problems, especially in the case of Black Box Optimization (BBO). Generally, the performance of an EDA depends on the match between its driving probability distribution and the landscape of the problem being solved. Because most well-known EDAs, including CMA-ES, NES, and AMaLGaM, use a uni-modal search distribution, they have a high risk of getting trapped in local optima when a problem is multi-modal with a (moderate) number of relatively comparable modes. This risk could potentially be mitigated using niching methods that define multiple regions of interest where separate search distributions govern sub-populations. However, a key question is how to determine a suitable number of niches, especially in BBO. In this paper, we present a novel, adaptive niching approach that determines the niches through hierarchical clustering based on the correlation between the probability densities and fitness values of solutions. We test the performance of a combination of this niching approach with AMaLGaM on both new and well-known niching benchmark problems and ind that the new approach properly identifies multiple landscape modes, leading to much beter performance on multi-modal problems than with a non-niched, uni-modal EDA

Ensuring smoothly navigable approximation sets by Bézier curve parameterizations in evolutionary bi-objective optimization

The aim of bi-objective optimization is to obtain an approximation set of (near) Pareto optimal solutions. A decision maker then navigates this set to select a final desired solution, often using a visualization of the approximation front. The front provides a navigational ordering of solutions to traverse, but this ordering does not necessarily map to a smooth trajectory through decision space. This forces the decision maker to inspect the decision variables of each solution individually, potentially making navigation of the approximation set unintuitive. In this work, we aim to improve approximation set navigability by enforcing a form of smoothness or continuity between solutions in terms of their decision variables. Imposing smoothness as a restriction upon common domination-based multi-objective evolutionary algorithms is not straightforward. Therefore, we use the recently introduced uncrowded hypervolume (UHV) to reformulate the multi-objective optimization problem as a single-objective problem in which parameterized approximation sets are directly optimized. We study here the case of parameterizing approximation sets as smooth Bézier curves in decision space. We approach the resulting single-objective problem with the gene-pool optimal mixing evolutionary algorithm (GOMEA), and we call the resulting algorithm BezEA. We analyze the behavior of BezEA and compare it to optimization of the UHV with GOMEA as well as the domination-based multi-objective GOMEA. We show that high-quality approximation sets can be obtained with BezEA, sometimes even outperforming the domination- and UHV-based algorithms, while smoothness of the navigation trajectory through decision space is guaranteed

Multi-objective optimization by uncrowded hypervolume gradient ascent

Evolutionary algorithms (EAs) are the preferred method for solving black-box multi-objective optimization problems, but when gradients of the objective functions are available, it is not straightforward to exploit these efficiently. By contrast, gradient-based optimization is well-established for single-objective optimization. A single-objective reformulation of the multi-objective problem could therefore offer a solution. Of particular interest to this end is the recently introduced uncrowded hypervolume (UHV) indicator, which is Pareto compliant and also takes into account dominated solutions. In this work, we show that the gradient of the UHV can often be computed, which allows for a direct application of gradient ascent algorithms. We compare this new approach with two EAs for UHV optimization as well as with one gradient-based algorithm for optimizing the well-established hypervolume. On several bi-objective benchmarks, we find that gradient-based algorithms outperform the tested EAs by obtaining a better hypervolume with fewer evaluations whenever exact gradients of the multiple objective functions are available and in case of small evaluation budgets. For larger budgets, however, EAs perform similarly or better. We further find that, when finite differences are used to approximate the gradients of the multiple objectives, our new gradient-based algorithm is s

Uncrowded-hypervolume based multi-objective optimization

This repository contains implementations (C++) of different uncrowded-hypervolume based methods for (gradient-free) multi-objective optimization. Using the (uncrowded) hypervolume, multi-objective optimization problems can be formulated as (high-dimensional) single-objective optimization problems, in the hypervolume of a solution sets is optimized. For details, see the corresponding publications/preprints

Fourier and wavelet option pricing methods

In this overview chapter, we will discuss the use of exponentially converging option pricing techniques for option valuation. We will focus on the pricing of European options, and they are the basic instruments within a calibration procedure when fitting the parameters in asset dynamics. The numerical solution is governed by the solution of the discounted expectation of the pay-off function. For the computation of the expectation, we require knowledge about the corresponding probability density function, which is typically not available for relevant stochastic asset price processes. Many publications regarding highly efficient pricing of these contracts are available, where computation often takes place in the Fourier space. Methods based on quadrature and the Fast Fourier Transform (FFT) [1-3] and methods based on Fourier cosine expansions [4,5] have therefore been developed because for relevant log-asset price processes, the characteristic function appears t

Two-Phase Real-Valued Multimodal Optimization with the Hill-Valley Evolutionary Algorithm

The aim of multimodal optimization (MMO) is to obtain all global optima of an optimization problem. In this chapter, we introduce a general framework for two-phase MMO evolutionary algorithms (EAs), in which different high-fitness regions (niches) are located in the first phase via clustering, and each of the located niches is separately optimized with a core search algorithm in the second phase. One such two-phase MMO EA is the Hill-Valley Evolutionary Algorithm (HillVall-EA). In HillVallEA, the remarkably simple hill-valley clustering method is used. The idea behind hill-valley clustering is that two solutions belong to the same niche (valley) when there is no hill in between them, which can be easily tested by performing additional function evaluations. We compare hill-valley clustering to two other recently introduced fitness-informed clustering methods: nearest-better clustering and hierarchical Gaussian mixture learning. We show how these clustering methods, as well as different core search algorithms, influence the resulting optimization performance of the two-phase MMO framework on the commonly used CEC 2013 niching benchmark suite. Our results show that HillVallEA, equipped with the core search algorithm Adapted Maximum-Likelihood Gaussian Model Univariate (AMu) as core search algorithm, outperforms all other MMO EAs, both within the limited benchmark budget, and in the long run. HillVallEA-AMu was the winner of the GECCO niching competition in 2018 and 2019, and is currently, to the best of our knowledge, the best performing algorithm on this benchmark suite

Evaluation of bi-objective treatment planning for high-dose-rate prostate brachytherapy—A retrospective observer study

Purpose: Bi-objective treatment planning for high-dose-rate prostate brachytherapy is a novel treatment planning method with two separate objectives that represent target coverage and organ-at-risk sparing. In this study, we investigated the feasibility and plan quality of this method by means of a retrospective observer study.Methods and Materials: Current planning sessions were recorded to configure a bi-objective optimization model and to assess its applicability to our clinical practice. Optimization software, GOMEA, was then used to automatically generate a large set of plans with different trade-offs in the two objectives for each of 18 patients treated with high-dose-rate prostate brachytherapy. From this set, five plans per patient were selected for comparison to the clinical plan in terms of satisfaction of planning criteria and in a retrospective observer study. Three brachytherapists were asked to evaluate the blinded plans and select the preferred one.Results: Recordings demonstrated applicability of the bi-objective optimization model to our clinical practice. For 14/18 patients, GOMEA plans satisfied all planning criteria, compared with 4/18 clinical plans. In the observer study, in 53/54 cases, a GOMEA plan was preferred over the clinical plan. When asked for consensus among observers, this ratio was 17/18 patients. Observers highly appreciated the insight gained from comparing multiple plans with different trade-offs simultaneously.Conclusions: The bi-objective optimization model adapted well t

Towards artificial intelligence-based automated treatment planning in clinical practice: A prospective study of the first clinical experiences in high-dose-rate prostate brachytherapy

Purpose: This prospective study evaluates our first clinical experiences with the novel ‘‘BRachytherapy via artificial Intelligent GOMEA-Heuristic based Treatment planning'' (BRIGHT) applied to high-dose-rate prostate brachytherapy. MethodS AND MATERIALs: Between March 2020 and October 2021, 14 prostate cancer patients were treated in our center with a 15Gy HDR-brachytherapy boost. BRIGHT was used for bi-objective treatment plan optimization and selection of the most desirable plans from a coverage-sparing trade-off curve. Selected BRIGHT plans were imported into the commercial treatment planning system Oncentra Brachy. In Oncentra Brachy a dose distribution comparison was performed for clinical plan choice, followed by manual fine-tuning of the preferred BRIGHT plan when deemed necessary. The reasons for plan selection, clinical plan choice, and fine-tuning, as well as process speed were monitored. For each patient, the dose-volume parameters of the (fine-tuned) clinical plan were evaluated. Results: In all patients, BRIGHT provided solutions satisfying all protocol values for coverage and sparing. In four patients not all dose-volume criteria of the clinical plan were satisfied after manual fine-tuning. Detailed information on tumour coverage, dose-distribution, dwell time pattern, and insight provided by the patient-specific trade-off curve, were used for clinical plan choice. Median time spent on treatment planning was 42 min, consisting of 16 min plan optimization and selection, and 26 min undesirable process steps. ConclusionS: BRIGHT is implemented in our clinic and provides automated prostate high-dose-rate brachytherapy planning with trade-off based plan selection. Based on our experience, additional optimization aims need to be implemented to further improve direct clinical applicability of treatment plans and process efficiency