LAPLACE TRANSFORMS AND INSTALLMENT OPTIONS

2004 ◽  
Vol 14 (08) ◽  
pp. 1167-1189 ◽  
Author(s):  
GHADA ALOBAIDI ◽  
ROLAND MALLIER ◽  
A. STANLEY DEAKIN
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Integral Equation ◽  
Free Boundary ◽  
Asymptotic Analysis ◽  
Laplace Transform ◽  
Financial Security ◽  
Present Value ◽  
Lump Sum ◽  
A Free Boundary Problem

An installment option is a derivative financial security where the price is paid in installments instead of as a lump sum at the time of purchase. The valuation of these options involves a free boundary problem in that at each installment date, the holder of the derivative has the option of continuing to pay the premiums or allowing the contract to lapse, and the decision will depend upon whether the present value of the expected pay-off is greater or less than the present value of the remaining premiums. Using a model installment option where the premiums are paid continuously rather than on discrete dates, an integral equation is derived for the position of this free boundary by applying a partial Laplace transform to the underlying partial differential equation for the value of the security. Asymptotic analysis of this integral equation allows us to deduce the behavior of the free boundary close to expiry.

scholarly journals Laplace Transform Methods for a Free Boundary Problem of Time-Fractional Partial Differential Equation System

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Zhiqiang Zhou ◽  
Xuemei Gao
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Free Boundary ◽  
Laplace Transform ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Fractional Partial Differential Equation ◽  
Partial Differential ◽  
Problem Of Time ◽  
A Free Boundary Problem

We study the pricing of the American options with fractal transmission system under two-state regime switching models. This pricing problem can be formulated as a free boundary problem of time-fractional partial differential equation (FPDE) system. Firstly, applying Laplace transform to the governing FPDEs with respect to the time variable results in second-order ordinary differential equations (ODEs) with two free boundaries. Then, the solutions of ODEs are expressed in an explicit form. Consequently the early exercise boundaries and the values for the American option are recovered using the Gaver-Stehfest formula. Numerical comparisons of the methods with the finite difference methods are carried out to verify the efficiency of the methods.

The integral of geometric Brownian motion

2001 ◽  
Vol 33 (1) ◽  
pp. 223-241 ◽  
Author(s):  
Daniel Dufresne
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Brownian Motion ◽  
Laplace Transform ◽  
Finite Time ◽  
Geometric Brownian Motion ◽  
Time Interval ◽  
Finite Time Interval ◽  
Special Cases ◽  
The Laplace Transform

This paper is about the probability law of the integral of geometric Brownian motion over a finite time interval. A partial differential equation is derived for the Laplace transform of the law of the reciprocal integral, and is shown to yield an expression for the density of the distribution. This expression has some advantages over the ones obtained previously, at least when the normalized drift of the Brownian motion is a non-negative integer. Bougerol's identity and a relationship between Brownian motions with opposite drifts may also be seen to be special cases of these results.

scholarly journals A boundary integral equation method for the numerical solution of a second order elliptic equation with variable coefficients

1980 ◽  
Vol 22 (2) ◽  
pp. 218-228 ◽  
Author(s):  
D. L. Clements
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Integral Equation ◽  
Elliptic Partial Differential Equation ◽  
Integral Equation Method ◽  
Variable Coefficients ◽  
Boundary Integral ◽  
Second Order ◽  
Order Elliptic Equation ◽  
Second Order Elliptic Equation

AbstractA method is derived for the solution of boundary value problems governed by a second-order elliptic partial differential equation with variable coefficients. The method is obtained by expressing the solution to a particular problem in terms of an integral taken round the boundary of the region under consideration.

scholarly journals Integral Equation-Wavelet Collocation Method for Geometric Transformation and Application to Image Processing

2014 ◽  
Vol 2014 ◽  
pp. 1-17
Author(s):  
Lina Yang ◽  
Yuan Yan Tang ◽  
Xiang Chu Feng ◽  
Lu Sun
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Integral Equation ◽  
Boundary Integral Equation ◽  
Boundary Integral ◽  
Transformation Model ◽  
Geometric Transformation ◽  
Mathematical Form ◽  
Partial Differential ◽  
Distorted Image

Geometric (or shape) distortion may occur in the data acquisition phase in information systems, and it can be characterized by geometric transformation model. Once the distorted image is approximated by a certain geometric transformation model, we can apply its inverse transformation to remove the distortion for the geometric restoration. Consequently, finding a mathematical form to approximate the distorted image plays a key role in the restoration. A harmonic transformation cannot be described by any fixed functions in mathematics. In fact, it is represented by partial differential equation (PDE) with boundary conditions. Therefore, to develop an efficient method to solve such a PDE is extremely significant in the geometric restoration. In this paper, a novel wavelet-based method is presented, which consists of three phases. In phase 1, the partial differential equation is converted into boundary integral equation and representation by an indirect method. In phase 2, the boundary integral equation and representation are changed to plane integral equation and representation by boundary measure formula. In phase 3, the plane integral equation and representation are then solved by a method we call wavelet collocation. The performance of our method is evaluated by numerical experiments.

scholarly journals Semi-Hyers–Ulam–Rassias Stability of the Convection Partial Differential Equation via Laplace Transform

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2980
Author(s):  
Daniela Marian
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Partial Differential ◽  
Rassias Stability

In this paper, we study the semi-Hyers–Ulam–Rassias stability and the generalized semi-Hyers–Ulam–Rassias stability of some partial differential equations using Laplace transform. One of them is the convection partial differential equation.

scholarly journals About an inverse problem for a free boundary compressible problem in hydrodynamic lubrication

2016 ◽  
Vol 24 (5) ◽  
Author(s):  
Khalid Ait Hadi ◽  
Guy Bayada ◽  
Mohamed El Alaoui Talibi
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Inverse Problem ◽  
Free Boundary ◽  
Hydrodynamic Lubrication ◽  
Optimal Solution ◽  
Mass Conservation ◽  
Cavitation Model ◽  
Slightly Compressible ◽  
Cavitation Phenomenon

AbstractIn this paper an inverse problem is considered for a non-coercive partial differential equation, issued from a mass conservation cavitation model for a slightly compressible fluid. The cavitation phenomenon and compressibility take place and are described by the Elrod model. The existence of an optimal solution is proven. Optimality conditions are derived and some numerical results are given.

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