AUTOMATED OPTION PRICING: NUMERICAL METHODS

2013 ◽  
Vol 16 (08) ◽  
pp. 1350042 ◽  
Author(s):  
PIERRE HENRY-LABORDÈRE
Keyword(s):  
Large Scale ◽  
Prior Probability ◽  
Interior Point Algorithm ◽  
Option Prices ◽  
Large Scale Problems ◽  
Convex Problem ◽  
Primal Dual ◽  
Model Independent ◽  
Quasi Newton ◽  
Infinite Linear

In this paper, we investigate model-independent bounds for option prices given a set of market instruments. This super-replication problem can be written as a semi-infinite linear programing problem. As these super-replication prices can be large and the densities ℚ which achieve the upper bounds quite singular, we restrict ℚ to be close in the entropy sense to a prior probability measure at a next stage. This leads to our risk-neutral weighted Monte Carlo approach which is connected to a constrained convex problem. We explain how to solve efficiently these large-scale problems using a primal-dual interior-point algorithm within the cutting-plane method and a quasi-Newton algorithm. Various examples illustrate the efficiency of these algorithms and the large range of applicability.

Multivariate Monotone Inclusions in Saddle Form

2021 ◽  
Author(s):  
Minh N. Bùi ◽  
Patrick L. Combettes
Keyword(s):  
Large Scale ◽  
Operator Splitting ◽  
Monotone Operators ◽  
Inclusion Problem ◽  
Minimization Problems ◽  
Large Scale Problems ◽  
Novel Approach ◽  
Primal Dual ◽  
Monotone Inclusion Problem ◽  
Monotonicity Preserving

We propose a novel approach to monotone operator splitting based on the notion of a saddle operator. Under investigation is a highly structured multivariate monotone inclusion problem involving a mix of set-valued, cocoercive, and Lipschitzian monotone operators, as well as various monotonicity-preserving operations among them. This model encompasses most formulations found in the literature. A limitation of existing primal-dual algorithms is that they operate in a product space that is too small to achieve full splitting of our problem in the sense that each operator is used individually. To circumvent this difficulty, we recast the problem as that of finding a zero of a saddle operator that acts on a bigger space. This leads to an algorithm of unprecedented flexibility, which achieves full splitting, exploits the specific attributes of each operator, is asynchronous, and requires to activate only blocks of operators at each iteration, as opposed to activating all of them. The latter feature is of critical importance in large-scale problems. The weak convergence of the main algorithm is established, as well as the strong convergence of a variant. Various applications are discussed, and instantiations of the proposed framework in the context of variational inequalities and minimization problems are presented.

scholarly journals Variable Smoothing for Convex Optimization Problems Using Stochastic Gradients

2020 ◽  
Vol 85 (2) ◽  
Author(s):  
Radu Ioan Boţ ◽  
Axel Böhm
Keyword(s):  
Convex Optimization ◽  
Linear Operator ◽  
Large Scale ◽  
Optimization Problems ◽  
Convex Optimization Problem ◽  
Nonsmooth Function ◽  
Large Scale Problems ◽  
Structured Convex Optimization ◽  
Primal Dual ◽  
Novel Algorithms

AbstractWe aim to solve a structured convex optimization problem, where a nonsmooth function is composed with a linear operator. When opting for full splitting schemes, usually, primal–dual type methods are employed as they are effective and also well studied. However, under the additional assumption of Lipschitz continuity of the nonsmooth function which is composed with the linear operator we can derive novel algorithms through regularization via the Moreau envelope. Furthermore, we tackle large scale problems by means of stochastic oracle calls, very similar to stochastic gradient techniques. Applications to total variational denoising and deblurring, and matrix factorization are provided.

scholarly journals Investigation of Improved Cooperative Coevolution for Large-Scale Global Optimization Problems

Algorithms ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 146
Author(s):  
Aleksei Vakhnin ◽  
Evgenii Sopov
Keyword(s):  
Global Optimization ◽  
Evolutionary Algorithms ◽  
Numerical Experiments ◽  
Large Scale ◽  
Optimization Problems ◽  
State Of The Art ◽  
Fixed Number ◽  
High Dimensional ◽  
Cooperative Coevolution ◽  
Large Scale Problems

Modern real-valued optimization problems are complex and high-dimensional, and they are known as “large-scale global optimization (LSGO)” problems. Classic evolutionary algorithms (EAs) perform poorly on this class of problems because of the curse of dimensionality. Cooperative Coevolution (CC) is a high-performed framework for performing the decomposition of large-scale problems into smaller and easier subproblems by grouping objective variables. The efficiency of CC strongly depends on the size of groups and the grouping approach. In this study, an improved CC (iCC) approach for solving LSGO problems has been proposed and investigated. iCC changes the number of variables in subcomponents dynamically during the optimization process. The SHADE algorithm is used as a subcomponent optimizer. We have investigated the performance of iCC-SHADE and CC-SHADE on fifteen problems from the LSGO CEC’13 benchmark set provided by the IEEE Congress of Evolutionary Computation. The results of numerical experiments have shown that iCC-SHADE outperforms, on average, CC-SHADE with a fixed number of subcomponents. Also, we have compared iCC-SHADE with some state-of-the-art LSGO metaheuristics. The experimental results have shown that the proposed algorithm is competitive with other efficient metaheuristics.

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