scholarly journals The Adjoint Method for the Inverse Problem of Option Pricing

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Shou-Lei Wang ◽  
Yu-Fei Yang ◽  
Yu-Hua Zeng
Keyword(s):  
Inverse Problem ◽  
Option Pricing ◽  
Implied Volatility ◽  
Adjoint Method ◽  
Numerical Examples ◽  
Limited Memory ◽  
Minimization Procedure ◽  
Volatility Function ◽  
The Cost ◽  
Quasi Newton

The estimation of implied volatility is a typical PDE inverse problem. In this paper, we propose theTV-L1model for identifying the implied volatility. The optimal volatility function is found by minimizing the cost functional measuring the discrepancy. The gradient is computed via the adjoint method which provides us with an exact value of the gradient needed for the minimization procedure. We use the limited memory quasi-Newton algorithm (L-BFGS) to find the optimal and numerical examples shows the effectiveness of the presented method.

A limited-memory quasi-Newton inversion for 1D magnetotellurics

Geophysics ◽  
2006 ◽  
Vol 71 (5) ◽  
pp. G191-G196 ◽  
Author(s):  
Anna Avdeeva ◽  
Dmitry Avdeev
Keyword(s):  
Inverse Problem ◽  
Large Scale ◽  
Convergence Rates ◽  
Earth Model ◽  
Problem Solution ◽  
Data Set ◽  
Electromagnetic Problems ◽  
Limited Memory ◽  
Conductivity Structure ◽  
Quasi Newton

We apply a limited-memory quasi-Newton (QN) method to the 1D magnetotelluric (MT) inverse problem. Using this method we invert a realistic synthetic MT impedance data set calculated for a layered earth model. The calculation of gradients based on the adjoint method speeds up the inverse problem solution many times. In addition, regularization stabilizes the QN inversion result and a few correction pairs are sufficient to produce reasonable results. Comparison with the L-BFGS-B algorithm shows similar convergence rates. This study is a first step towards the solution of large-scale electromagnetic problems, with a full treatment of the 3D conductivity structure of the earth.

Two-dimensional anisotropic magnetotelluric inversion using a limited-memory quasi-Newton method

Geophysics ◽  
2021 ◽  
pp. 1-71
Author(s):  
Guo Yu ◽  
Colin G. Farquharson ◽  
Qibin Xiao ◽  
Man Li
Keyword(s):  
Inverse Problem ◽  
Objective Function ◽  
Hessian Matrix ◽  
Two Dimensional ◽  
Inversion Algorithm ◽  
Anisotropic Models ◽  
Limited Memory ◽  
Iterative Minimization ◽  
Special Cases ◽  
Quasi Newton

We have developed a two-dimensional (2D) anisotropic magnetotelluric (MT) inversion algorithm that uses a limited-memory quasi-Newton (QN) method for bounds-constrained optimization. This algorithm solves the inverse problem, which is non-linear, by iterative minimization of linearized approximations of the classical Tikhonov regularized objective function. The QN approximation for the Hessian matrix is only implemented for the data-misfit term of the objective function; the part of the Hessian matrix for the regularization is explicitly computed. This adjustment results in a better approximation for the data-misfit term in particular. The inversion algorithm considers arbitrary anisotropy, and is extended for special cases including azimuthal and vertical anisotropy. The algorithm is shown to be stable and converges rapidly for several simple anisotropic models. These synthetic tests also confirm that the anisotropic inversion produces a correct anomaly with different but equivalent anisotropic parameters. We also consider a complex 2D anisotropic model; the successful results for this model further confirm that the inversion algorithm presented here, which uses the novel modified limited-memory QN approach, is capable of solving the 2D anisotropic magnetotelluric inverse problem. Finally, we present a practical application on MT data collected in northern Tibet to demonstrate the effectiveness and stability of our algorithm.

scholarly journals On preconditioner updates for sequences of saddle-point linear systems

2018 ◽  
Vol 9 (1) ◽  
pp. 35-41
Author(s):  
Valentina De Simone ◽  
Daniela di Serafino ◽  
Benedetta Morini
Keyword(s):  
Saddle Point ◽  
Linear Systems ◽  
Solution Process ◽  
Newton Methods ◽  
Krylov Methods ◽  
Limited Memory ◽  
Short Survey ◽  
The Cost ◽  
Quasi Newton

Abstract Updating preconditioners for the solution of sequences of large and sparse saddle- point linear systems via Krylov methods has received increasing attention in the last few years, because it allows to reduce the cost of preconditioning while keeping the efficiency of the overall solution process. This paper provides a short survey of the two approaches proposed in the literature for this problem: updating the factors of a preconditioner available in a block LDLT form, and updating a preconditioner via a limited-memory technique inspired by quasi-Newton methods.

scholarly journals An Inverse Problem Involving a Viscous Eikonal Equation with Applications in Electrophysiology

2021 ◽  
Author(s):  
Karl Kunisch ◽  
Philip Trautmann
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
Eikonal Equation ◽  
State Equation ◽  
High Accuracy ◽  
Well Posedness ◽  
Numerical Examples ◽  
Marquardt Method ◽  
Math Biol ◽  
Levenberg Marquardt

AbstractIn this work we discuss the reconstruction of cardiac activation instants based on a viscous Eikonal equation from boundary observations. The problem is formulated as a least squares problem and solved by a projected version of the Levenberg–Marquardt method. Moreover, we analyze the well-posedness of the state equation and derive the gradient of the least squares functional with respect to the activation instants. In the numerical examples we also conduct an experiment in which the location of the activation sites and the activation instants are reconstructed jointly based on an adapted version of the shape gradient method from (J. Math. Biol. 79, 2033–2068, 2019). We are able to reconstruct the activation instants as well as the locations of the activations with high accuracy relative to the noise level.

THE HEAT-KERNEL MOST-LIKELY-PATH APPROXIMATION

2012 ◽  
Vol 15 (01) ◽  
pp. 1250001 ◽  
Author(s):  
JIM GATHERAL ◽  
TAI-HO WANG
Keyword(s):  
Heat Kernel ◽  
Implied Volatility ◽  
Order Term ◽  
Time Integration ◽  
Natural Extension ◽  
Approximation Formula ◽  
Local Volatility ◽  
Improved Performance ◽  
Volatility Function ◽  
Definition Of

In this article, we derive a new most-likely-path (MLP) approximation for implied volatility in terms of local volatility, based on time-integration of the lowest order term in the heat-kernel expansion. This new approximation formula turns out to be a natural extension of the well-known formula of Berestycki, Busca and Florent. Various other MLP approximations have been suggested in the literature involving different choices of most-likely-path; our work fixes a natural definition of the most-likely-path. We confirm the improved performance of our new approximation relative to existing approximations in an explicit computation using a realistic S&P500 local volatility function.

scholarly journals Numerical method of identification of an unknown source term in a heat equation

2002 ◽  
Vol 8 (2) ◽  
pp. 161-168 ◽  
Author(s):  
Afet Golayoğlu Fatullayev
Keyword(s):  
Inverse Problem ◽  
Heat Equation ◽  
Numerical Method ◽  
Minimization Problem ◽  
Unknown Function ◽  
Source Term ◽  
Numerical Procedure ◽  
Numerical Examples ◽  
Unknown Source

A numerical procedure for an inverse problem of identification of an unknown source in a heat equation is presented. Approach of proposed method is to approximate unknown function by polygons linear pieces which are determined consecutively from the solution of minimization problem based on the overspecified data. Numerical examples are presented.

AN IMPLIED VOLATILITY MODEL DETERMINED BY CREDIT DEFAULT SWAPS

2012 ◽  
Vol 15 (07) ◽  
pp. 1250049
Author(s):  
PASCAL HEIDER
Keyword(s):  
Capital Structure ◽  
Stock Price ◽  
Implied Volatility ◽  
Contingent Claims ◽  
Stock Volatility ◽  
Numerical Examples ◽  
Volatility Model ◽  
Credit Default ◽  
Spread Dynamics ◽  
A Company

In this paper we propose a diffusion model relating the stock price dynamics to the CDS spread dynamics of a company by assuming a linear relationship between instantaneous stock volatility and CDS spread. To value contingent claims under this model we apply a finite elements discretization to the associated pricing partial differential equation. A robust calibration strategy is presented and numerical examples are studied to validate the model assumptions. Besides option pricing, we discuss further applications which are e.g. the identification of market situations allowing volatility and capital structure arbitrage.

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