Arun Verma - Academia.edu (original) (raw)
Papers by Arun Verma
Journal of Computational Physics, Jan 1, 2000
Wave propagational inverse problems arise in several applications including medical imaging and g... more Wave propagational inverse problems arise in several applications including medical imaging and geophysical exploration. In these problems, one is interested in obtaining the parameters describing the medium from its response to excitations. The problems are characterized by their large size, and by the hyperbolic equation which models the physical phenomena. The inverse problems are often posed as a nonlinear datafitting where the unknown parameters are found by minimizing the misfit between the predicted data and the actual data. In order to solve the problem numerically using a gradient-type approach, one must calculate the action of the Jacobian and its adjoint on a given vector. In this paper, we explore the use of automatic differentiation (AD) to develop codes that perform these calculations.
… of the 1996 International Conference on …, Jan 1, 1996
Differentiation is one of the fundamental problems in numerical mathemetics. The solution of many... more Differentiation is one of the fundamental problems in numerical mathemetics. The solution of many optimization problems and other applications require knowledge of the gradient, the Jacobian matrix, or the Hessian matrix of a given function.
We compare the dynamic hedging performance of the deterministic local volatility function approac... more We compare the dynamic hedging performance of the deterministic local volatility function approach with the implied/constant volatility method. Using an example in which the underlying price follows an absolute diffusion process, we illustrate that hedge parameters computed from the implied/constant volatility method can have significant error even though the implied volatility method is able to calibrate the current option prices of different strikes and maturities. In particular the delta hedge parameter produced by the implied/constant volatility method is consistently significantly larger than that of the exact delta when the underlying price follows an absolute diffusion.
SIAM Interdiscplinary Workshop on Object Oriented …, Jan 1, 2002
Differentiation is one of the fundamental problems in numerical mathematics. The solution of many... more Differentiation is one of the fundamental problems in numerical mathematics. The solution of many optimization problems and other applications require knowledge of the gradient, the Jacobian matrix, or the Hessian matrix of a given function.
Proceedings of the SIAM Workshop on …, Jan 1, 1998
Journal of Computational Finance, Jan 1, 2002
The variational inequality formulation provides a mechanism to determine both the option value an... more The variational inequality formulation provides a mechanism to determine both the option value and the early exercise curve implicitly . Standard finite difference approximation typically leads to linear complementarity problems with tridiagonal coefficient matrices. The second order upwind finite difference formulation gives rise to finite dimensional linear complementarity problems with nontridiagonal matrices, whereas the upstream weighting finite difference approach with the van Leer flux limiter for the convection term yields nonlinear complementarity problems.
SIAM Journal on Scientific Computing, Jan 1, 2000
Shape sensitivity analysis is a tool that provides quantitative information about the influence o... more Shape sensitivity analysis is a tool that provides quantitative information about the influence of shape parameter changes on the solution of a partial differential equation (PDE). These shape sensitivities are described by a continuous sensitivity equation (CSE). Automatic ...
ACM Transactions on Mathematical …, Jan 1, 2000
ADMIT-1 enables the computation of sparse Jacobian and Hessian matrices, using automatic differen... more ADMIT-1 enables the computation of sparse Jacobian and Hessian matrices, using automatic differentiation technology, from a MATLAB environment. Given a function to be differentiated, ADMIT-1 will exploit sparsity if present to yield sparse derivative matrices (in sparse MATLAB form). A generic automatic differentiation tool, subject to some functionality requirements, can be plugged into ADMIT-1; examples include ADOL-C (C/Cϩϩ target functions) and ADMAT (MATLAB target functions). ADMIT-1 also allows for the calculation of gradients and has several other related functions. This article provides an introduction to the design and usage of ADMIT-1.
Computational Optimization and Applications, Jan 1, 2001
We propose a new framework for the application of preconditioned conjugate gradients in the solut... more We propose a new framework for the application of preconditioned conjugate gradients in the solution of large-scale linear equality constrained minimization problems. This framework allows for the exploitation of structure and sparsity in the context of solving the reduced Newton system (despite the fact that the reduced system may be dense).
… : collected papers of the New York …, Jan 1, 2001
Using market European option prices, a method for computing a smooth local volatility function in... more Using market European option prices, a method for computing a smooth local volatility function in a 1-factor continuous diffusion model is proposed. Smoothness is introduced to facilitate accurate approximation of the true local volatility function from a finite set of observation data. It is emphasized that accurately approximating the true local volatility function is crucial in hedging even simple European options, and pricing exotic options. A spline functional approach is used: the local volatility function is represented by a spline whose values at chosen knots are determined by solving a constrained nonlinear optimization problem. The optimization formulation is amenable to various option evaluation methods; a partial differential equation implementation is discussed. Using a synthetic European call option example, we illustrate the capability of the proposed method in reconstructing the unknown local volatility function. Accuracy of pricing and hedging is also illustrated. Moreover, it is demonstrated that, using a different constant implied volatility for an option with different strike/maturity can produce erroneous hedge factors. In addition, real market European call option data on the S&P 500 stock index is used to compute the local volatility function; stability of the approach is demonstrated.
Journal of Computational Physics, Jan 1, 2000
Wave propagational inverse problems arise in several applications including medical imaging and g... more Wave propagational inverse problems arise in several applications including medical imaging and geophysical exploration. In these problems, one is interested in obtaining the parameters describing the medium from its response to excitations. The problems are characterized by their large size, and by the hyperbolic equation which models the physical phenomena. The inverse problems are often posed as a nonlinear datafitting where the unknown parameters are found by minimizing the misfit between the predicted data and the actual data. In order to solve the problem numerically using a gradient-type approach, one must calculate the action of the Jacobian and its adjoint on a given vector. In this paper, we explore the use of automatic differentiation (AD) to develop codes that perform these calculations.
… of the 1996 International Conference on …, Jan 1, 1996
Differentiation is one of the fundamental problems in numerical mathemetics. The solution of many... more Differentiation is one of the fundamental problems in numerical mathemetics. The solution of many optimization problems and other applications require knowledge of the gradient, the Jacobian matrix, or the Hessian matrix of a given function.
We compare the dynamic hedging performance of the deterministic local volatility function approac... more We compare the dynamic hedging performance of the deterministic local volatility function approach with the implied/constant volatility method. Using an example in which the underlying price follows an absolute diffusion process, we illustrate that hedge parameters computed from the implied/constant volatility method can have significant error even though the implied volatility method is able to calibrate the current option prices of different strikes and maturities. In particular the delta hedge parameter produced by the implied/constant volatility method is consistently significantly larger than that of the exact delta when the underlying price follows an absolute diffusion.
SIAM Interdiscplinary Workshop on Object Oriented …, Jan 1, 2002
Differentiation is one of the fundamental problems in numerical mathematics. The solution of many... more Differentiation is one of the fundamental problems in numerical mathematics. The solution of many optimization problems and other applications require knowledge of the gradient, the Jacobian matrix, or the Hessian matrix of a given function.
Proceedings of the SIAM Workshop on …, Jan 1, 1998
Journal of Computational Finance, Jan 1, 2002
The variational inequality formulation provides a mechanism to determine both the option value an... more The variational inequality formulation provides a mechanism to determine both the option value and the early exercise curve implicitly . Standard finite difference approximation typically leads to linear complementarity problems with tridiagonal coefficient matrices. The second order upwind finite difference formulation gives rise to finite dimensional linear complementarity problems with nontridiagonal matrices, whereas the upstream weighting finite difference approach with the van Leer flux limiter for the convection term yields nonlinear complementarity problems.
SIAM Journal on Scientific Computing, Jan 1, 2000
Shape sensitivity analysis is a tool that provides quantitative information about the influence o... more Shape sensitivity analysis is a tool that provides quantitative information about the influence of shape parameter changes on the solution of a partial differential equation (PDE). These shape sensitivities are described by a continuous sensitivity equation (CSE). Automatic ...
ACM Transactions on Mathematical …, Jan 1, 2000
ADMIT-1 enables the computation of sparse Jacobian and Hessian matrices, using automatic differen... more ADMIT-1 enables the computation of sparse Jacobian and Hessian matrices, using automatic differentiation technology, from a MATLAB environment. Given a function to be differentiated, ADMIT-1 will exploit sparsity if present to yield sparse derivative matrices (in sparse MATLAB form). A generic automatic differentiation tool, subject to some functionality requirements, can be plugged into ADMIT-1; examples include ADOL-C (C/Cϩϩ target functions) and ADMAT (MATLAB target functions). ADMIT-1 also allows for the calculation of gradients and has several other related functions. This article provides an introduction to the design and usage of ADMIT-1.
Computational Optimization and Applications, Jan 1, 2001
We propose a new framework for the application of preconditioned conjugate gradients in the solut... more We propose a new framework for the application of preconditioned conjugate gradients in the solution of large-scale linear equality constrained minimization problems. This framework allows for the exploitation of structure and sparsity in the context of solving the reduced Newton system (despite the fact that the reduced system may be dense).
… : collected papers of the New York …, Jan 1, 2001
Using market European option prices, a method for computing a smooth local volatility function in... more Using market European option prices, a method for computing a smooth local volatility function in a 1-factor continuous diffusion model is proposed. Smoothness is introduced to facilitate accurate approximation of the true local volatility function from a finite set of observation data. It is emphasized that accurately approximating the true local volatility function is crucial in hedging even simple European options, and pricing exotic options. A spline functional approach is used: the local volatility function is represented by a spline whose values at chosen knots are determined by solving a constrained nonlinear optimization problem. The optimization formulation is amenable to various option evaluation methods; a partial differential equation implementation is discussed. Using a synthetic European call option example, we illustrate the capability of the proposed method in reconstructing the unknown local volatility function. Accuracy of pricing and hedging is also illustrated. Moreover, it is demonstrated that, using a different constant implied volatility for an option with different strike/maturity can produce erroneous hedge factors. In addition, real market European call option data on the S&P 500 stock index is used to compute the local volatility function; stability of the approach is demonstrated.