I. David Abrahams

An exact analytical solution for discrete barrier options

Abstract.In the present paper we provide an analytical solution for pricing discrete barrier options in the Black-Scholes framework. We reduce the valuation problem to a Wiener-Hopf equation that can be solved analytically. We are able to give explicit expressions for the Greeks of the contract. The results from our formulae are compared with those from other numerical methods available in the literature. Very good agreement is obtained, although evaluation using the present method is substantially quicker than the alternative methods presented.

On the solution of Wiener-Hopf problems involving noncommutative matrix kernel decompositions

Many problems in physics and engineering with semi-infinite boundaries or interfaces are exactly solvable by the Wiener--Hopf technique. It has been used successfully in a multitude of different disciplines when the Wiener--Hopf functional equation contains a single scalar kernel. For complex boundary value problems, however, the procedure often leads to coupled equations which therefore have a kernel of matrix form. The key step in the technique is to decompose the kernel into a product of two functions, one analytic in an upper region of a complex (transform) plane and the other analytic in an overlapping lower half-plane. This is straightforward for scalar kernels but no method has yet been proposed for general matrices.In this article a new procedure is introduced whereby Pade approximants are employed to obtain an approximate but explicit noncommutative factorization of a matrix kernel. As well as being simple to apply, the use of approximants allows the accuracy of the factorization to be increased ...

On nonlinear viscoelastic deformations: a reappraisal of Fung's quasi-linear viscoelastic model

This paper offers a reappraisal of Fungs model for quasi-linear viscoelasticity. It is shown that a number of negative features exhibited in other works, commonly attributed to the Fung approach, are merely a consequence of the way it has been applied. The approach outlined herein is shown to yield improved behaviour and offers a straightforward scheme for solving a wide range of models. Results from the new model are contrasted with those in the literature for the case of uniaxial elongation of a bar: for an imposed stretch of an incompressible bar and for an imposed load. In the latter case, a numerical solution to a Volterra integral equation is required to obtain the results. This is achieved by a high-order discretization scheme. Finally, the stretch of a compressible viscoelastic bar is determined for two distinct materials: Horgan–Murphy and Gent.

On the existence of flexural edge waves on thin orthotropic plates

This paper is concerned with an investigation into the existence of waves propagating along a free edge of an orthotropic plate, where the edge is inclined at arbitrary angle to a principal direction of the material. After deriving the governing equation and edge conditions, an edge wave ansatz is substituted into this system to reduce it to a set of algebraic equations for the edge wave wave number and wave vector. These are solved numerically for several typical composite materials although analytic expressions can be obtained in the case of special values of the material parameters and inclination angle. It is found that a unique edge wave solution, which generally exhibits oscillation as well as decay away from the free edge, exists in all cases, and its wave speed is independent of its direction of propagation along the plate.

Mean residence time for drugs subject to enterohepatic cycling.

A physiologically realistic model of enterohepatic cycling (EHC) which includes separate liver and gallbladder compartments, discontinuous gallbladder emptying and first-order absorption from both an oral formulation and secreted bile (kapo and kab, respectively) has been developed. The effect of EHC on area under the first-moment curve (AUMC) of drug concentration in plasma and on parameters derived from the AUMC was investigated. Unlike AUC, AUMC is dependent on the time and time-course of gallbladder emptying, increasing as the interval between gallbladder emptying increases. Consequently, mean residence time (MRT) is also a time-dependent parameter. Analytical solutions for MRTiv and MRTpo were derived. Mean absorption time (MAT = MRTpo — MRTivj is also time-dependent, contrary to findings previously published for a model of EHC with a continuous time lag. MAT is also dependent on kapo, kab and the hepatic extraction ratio. The difference between MRTpos two formulations with unequal kapo values may deviate from the difference in the inverse of their absorption rate constants. Implications for design and interpretation of pharmacokinetic studies include (i) MAT values may be dominated by the time-course of recycling rather than the time-course of the initial absorption, depending on the extent of EHC and (ii) the unpredictable nature of the time of gallbladder emptying will contribute to intrasubject variability in derived parameters during crossover studies. Knowledge of the extent of EHC is invaluable in deciding whether modification of the in vitro release characteristics of an oral formulation will have any effect on the overall time-course of absorption in vivo. Techniques to monitor or control gallbladder emptying may be helpful for reducing variability in pharmaco-kinetic studies for compounds which are extensively cycled in bile.

On transient oscillations of plates in moving fluids

In recent years, various groups of researchers have looked at the two-dimensional motions of an undamped infinite thin elastic plate lying under a uniformly moving incompressible inviscid fluid. The plate is driven, usually by a single frequency time-harmonic line-source switched on at a finite time. The system’s behaviour is interesting as it can be shown to be absolutely unstable for flow velocities above a critical value, and below this the long-time solution is convectively unstable (downstream of the source) for a sufficiently low forcing frequency. These results do not appear particularly plausible from a physical point of view, and there is some question regarding the realisation of long-time steady behaviour, and so this article attempts to examine ways in which the model problem can be improved. In particular, the effects of introducing plate thickness and fluid compressibility to the model are studied. This is carried out by comparing the morphology of the original and modified solutions in the complex wavenumber space. It is found that, in the limit of small fluid-to-plate density ratio, the two problems exhibit qualitatively identical behaviour. However, the addition of structural damping is shown herein to lead to a very different solution – the initial boundary value problem is absolutely unstable at all flow velocities. Various other modifications to the original model, including finiteness of the plate, three-dimensional effects and nonlinearity, are discussed and their impact on the long-time response of the system is assessed.

On the application of the Wiener-Hopf technique to problems in dynamic elasticity

Many problems in linear elastodynamics, or dynamic fracture mechanics, can be reduced to Wiener–Hopf functional equations defined in a strip in a complex transform plane. Apart from a few special cases, the inherent coupling between shear and compressional body motions gives rise to coupled systems of equations, and so the resulting Wiener–Hopf kernels are of matrix form. The key step in the solution of a Wiener–Hopf equation, which is to decompose the kernel into a product of two factors with particular analyticity properties, can be accomplished explicitly for scalar kernels. However, apart from special matrices which yield commutative factorizations, no procedure has yet been devised to factorize exactly general matrix kernels. This paper shall demonstrate, by way of example, that the Wiener–Hopf approximant matrix (WHAM) procedure for obtaining approximate factors of matrix kernels (recently introduced by the author in [SIAM J. Appl. Math. 57 (2) (1997) 541]) is applicable to the class of matrix kernels found in elasticity, and in particular to problems in QNDE. First, as a motivating example, the kernel arising in the model of diffraction of skew incident elastic waves on a semi-infinite crack in an isotropic elastic space is studied. This was first examined in a seminal work by Achenbach and Gautesen [J. Acoust. Soc. Am. 61 (2) (1977) 413] and here three methods are offered for deriving distinct non-commutative factorizations of the kernel. Second, the WHAM method is employed to factorize the matrix kernel arising in the problem of radiation into an elastic half-space with mixed boundary conditions on its face. Third, brief mention is made of kernel factorization related to the problems of flexural wave diffraction by a crack in a thin (Mindlin) plate, and body wave scattering by an interfacial crack.

On the existence of flexural edge waves on submerged elastic plates

The existence of flexural waves confined to the free edge of a fluid-loaded plate is established theoretically. Whereas analogous in vacuo edge waves exist for all parameter values, submerged plates are shown herein to support such waves only under very light fluid–loading conditions. For example, thin plates of aluminium, brass or Plexiglas will not support edge waves in water, although edge waves are permissible for each of these materials in air. The analysis is based on classical thinplate theory and employs the Wiener–Hopf technique to derive the dispersion relation for the edge–wave wavenumber as a function of frequency. In the limit of zero fluid loading the dispersion relation predicts the well–known result of Konenkov for edge waves on thin plates in vacuo.

The Wiener–Hopf Technique and Discretely Monitored Path-Dependent Option Pricing

Fusai, Abrahams, and Sgarra (2006) employed the Wiener–Hopf technique to obtain an exact analytic expression for discretely monitored barrier option prices as the solution to the Black–Scholes partial differential equation. The present work reformulates this in the language of random walks and extends it to price a variety of other discretely monitored path-dependent options. Analytic arguments familiar in the applied mathematics literature are used to obtain fluctuation identities. This includes casting the famous identities of Baxter and Spitzer in a form convenient to price barrier, first-touch, and hindsight options. Analyzing random walks killed by two absorbing barriers with a modified Wiener–Hopf technique yields a novel formula for double-barrier option prices. Continuum limits and continuity correction approximations are considered. Numerically, efficient results are obtained by implementing Pade approximation. A Gaussian Black–Scholes framework is used as a simple model to exemplify the techniques, but the analysis applies to Levy processes generally.

Multiple point scattering to determine the effective wavenumber and effective material properties of an inhomogeneous slab

In this article we attempt to clarify various notions regarding multiple point scattering. We consider several predictions for the effective material properties of an inhomogeneous slab region which can be derived from classical multiple scattering theories. In particular we are interested in the point scattering limit when wavelengths λ0 ≫ l ∼ a where l is the characteristic length-scale of the distance between inclusions and a is the characteristic length-scale of inclusions. In this limit we are able to derive effective properties which are physically valid for any volume fraction φ, except in the sound-soft scatterer case where there is a condition on the size of φ. We shall confine attention to random distributions of inclusions and employ the Quasi-Crystalline Approximation to yield results. In particular we discuss the different scenarios of acoustics and antiplane elasticity and stress the reciprocity between these two problems which means that they can be solved simultaneously. We make various statements regarding the efficacy of the various multiple scattering theories in the prediction of effective material properties in the quasi-static limit.