The applications of partial integro-differential equations related to adaptive wavelet collocation methods for viscosity solutions to jump-diffusion models

2014 ◽  
Vol 246 ◽  
pp. 316-335 ◽  
Author(s):  
Hua Li ◽  
Lan Di ◽  
Antony Ware ◽  
George Yuan
Keyword(s):  
Differential Equations ◽  
Viscosity Solutions ◽  
Jump Diffusion ◽  
Diffusion Models ◽  
Collocation Methods ◽  
Adaptive Wavelet

scholarly journals Quadratic Spline Wavelets for Sparse Discretization of Jump–Diffusion Models

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 999
Author(s):  
Dana Černá
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Jump Diffusion ◽  
Diffusion Models ◽  
Dirichlet Boundary Conditions ◽  
Quadratic Spline ◽  
Vanishing Moments ◽  
Bounded Interval ◽  
Spline Wavelets ◽  
The Galerkin Method

This paper is concerned with a construction of new quadratic spline wavelets on a bounded interval satisfying homogeneous Dirichlet boundary conditions. The inner wavelets are translations and dilations of four generators. Two of them are symmetrical and two anti-symmetrical. The wavelets have three vanishing moments and the basis is well-conditioned. Furthermore, wavelets at levels i and j where i - j > 2 are orthogonal. Thus, matrices arising from discretization by the Galerkin method with this basis have O 1 nonzero entries in each column for various types of differential equations, which is not the case for most other wavelet bases. To illustrate applicability, the constructed bases are used for option pricing under jump–diffusion models, which are represented by partial integro-differential equations. Due to the orthogonality property and decay of entries of matrices corresponding to the integral term, the Crank–Nicolson method with Richardson extrapolation combined with the wavelet–Galerkin method also leads to matrices that can be approximated by matrices with O 1 nonzero entries in each column. Numerical experiments are provided for European options under the Merton model.

scholarly journals Geometric step options and Lévy models: duality, PIDEs, and semi-analytical pricing

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Walter Farkas ◽  
Ludovic Mathys
Keyword(s):  
Differential Equations ◽  
Option Pricing ◽  
Present Article ◽  
Jump Diffusion ◽  
Diffusion Models ◽  
Double Barrier ◽  
Early Exercise ◽  
Duality Relations ◽  
The Impact ◽  
Early Exercise Premium

<p style='text-indent:20px;'>The present article studies geometric step options in exponential Lévy markets. Our contribution is manifold and extends several aspects of the geometric step option pricing literature. First, we provide symmetry and duality relations and derive various characterizations for both European-type and American-type geometric double barrier step options. In particular, we are able to obtain a jump-diffusion disentanglement for the early exercise premium of American-type geometric double barrier step contracts and its maturity-randomized equivalent as well as to characterize the diffusion and jump contributions to these early exercise premiums separately by means of partial integro-differential equations and ordinary integro-differential equations. As an application of our characterizations, we derive semi-analytical pricing results for (regular) European-type and American-type geometric down-and-out step call options under hyper-exponential jump-diffusion models. Lastly, we use the latter results to discuss the early exercise structure of geometric step options once jumps are added and to subsequently provide an analysis of the impact of jumps on the price and hedging parameters of (European-type and American-type) geometric step contracts.</p>

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