Literature Review: High-Dimensional Option Pricing Problems with Sparse Grid and PCA (original) (raw)

Laplace transformation method for the Black-Scholes equation

2009

In this paper we apply the innovative Laplace transformation method introduced by Sheen, Sloan, and Thom\'ee (IMA J. Numer. Anal., 2003) to solve the Black-Scholes equation. The algorithm is of arbitrary high convergence rate and naturally parallelizable. It is shown that the method is very efficient for calculating various options. Existence and uniqueness properties of the Laplace transformed Black-Scholes equation are analyzed. Also a transparent boundary condition associated with the Laplace transformation method is proposed. Several numerical results for various options under various situations confirm the efficiency, convergence and parallelization property of the proposed scheme.

Journalof Computational Finance

The most important problem in stock market prediction is the accuracy. Because there is inherent complexity for the case, most of the models have restrictions in this regard. As it is mentioned before, the studies have further discussed the next step price prediction and less attention is paid to direction prediction (see Tables 1&2). The first group includes models that predict stock price in the next time and criteria like MSE, RMSE, MAE and MAPE are used to evaluation (see Tables 1&2). The second group contains models which predict stock direction for the next time and criteria like direct, hit ratio and accuracy are used to appraise them (see Tables 1&2). The results of model that deals with price prediction and the first group criteria only concerned do not suffice to make decision and trade in the real world because a model may be practically used in which MAPE criterion, is suitable, but trade leads to loss. In order to prevent the mentioned problem and practically use the results, it is required stock price prediction happen in the next time interval considering stock movement direction prediction [54]. Further, this is explained by an example.

On an improved computational solution for the 3D HCIR PDE in finance

Analele Universitatii "Ovidius" Constanta - Seria Matematica

The aim of this work is to tackle the three–dimensional (3D) Heston– Cox–Ingersoll–Ross (HCIR) time–dependent partial differential equation (PDE) computationally by employing a non–uniform discretization and gathering the finite difference (FD) weighting coe cients into differentiation matrices. In fact, a non–uniform discretization of the 3D computational domain is employed to achieve the second–order of accuracy for all the spatial variables. It is contributed that under what conditions the proposed procedure is stable. This stability bound is novel in literature for solving this model. Several financial experiments are worked out along with computation of the hedging quantities Delta and Gamma.

The SINC way: a fast and accurate approach to Fourier pricing

Quantitative Finance, 2021

The goal of this paper is to investigate the method outlined by one of us (PR) in Cherubini et al. (2009) to compute option prices. We name it the SINC approach. While the COS method by Fang and Osterlee (2009) leverages the Fourier-cosine expansion of truncated densities, the SINC approach builds on the Shannon Sampling Theorem revisited for functions with bounded support. We provide several results which were missing in the early derivation: i) a rigorous proof of the convergence of the SINC formula to the correct option price when the support grows and the number of Fourier frequencies increases; ii) ready to implement formulas for put, Cash-or-Nothing, and Asset-or-Nothing options; iii) a systematic comparison with the COS formula for several log-price models; iv) a numerical challenge against alternative Fast Fourier specifications, such as Carr and Madan (1999) and Lewis (2000); v) an extensive pricing exercise under the rough Heston model of Jaisson and Rosenbaum (2015); vi) ...

An Operator Splitting Method for Pricing American Options

Computational Methods in Applied Sciences, 2008

Pricing American options using partial (integro-)differential equation based methods leads to linear complementarity problems (LCPs). The numerical solution of these problems resulting from the Black-Scholes model, Kou’s jump-diffusion model, and Heston’s stochastic volatility model are considered. The finite difference discretization is described. The solutions of the discrete LCPs are approximated using an operator splitting method which separates the linear problem