The valuation of self-funding instalment warrants

© 2017 World Scientific Publishing Company. We present two models for the fair value of a self-funding instalment warrant. In both models we assume the underlying stock process follows a geometric Brownian motion. In the first model, we assume that the underlying stock pays a continuous dividend yield and in the second we assume that it pays a series of discrete dividend yields. We show that both models admit similarity reductions and use these to obtain simple finite-difference and Monte Carlo solutions. We use the method of multiple scales to connect these two models and establish the first-order correction term to be applied to the first model in order to obtain the second, thereby establishing that the former model is justified when many dividends are paid during the life of the warrant. Further, we show that the functional form of this correction may be expressed in terms of the hedging parameters for the first model and is, from this point of view, independent of the particular payoff in the first model. In two appendices we present approximate solutions for the first model which are valid in the small volatility and the short time-To-expiry limits, respectively, by using singular perturbation techniques. The small volatility solutions are used to check our finite-difference solutions and the small time-To-expiry solutions are used as a means of systematically smoothing the payoffs so we may use pathwise sensitivities for our Monte Carlo methods

Differential equations and asymptotic solutions for arithmetic Asian options: 'Black-Scholes formulae' for Asian rate calls

In this article, we present a simplified means of pricing Asian options using partial differential equations (PDEs). We first provide a concise derivation of the well-known similarity reduction and exact Laplace transform solution. We then analyse the problem afresh as a power series in the volatility-scaled contract duration, with a view to obtaining an asymptotic solution for the low-volatility limit, a limit which presents difficulties in the context of the general Laplace transform solution. The problem is approached anew from the point of view of asymptotic expansions and the results are compared with direct, high precision, inversion of the Laplace transform and with numerical results obtained by V. Linetsky and J. Vecer. Our asymptotic formulae are little more complicated than the standard Black-Scholes formulae and, working to third order in the volatility-scaled expiry, are accurate to at least four significant figures for standard test problems. In the case of zero risk-neutral drift, we have the solution to fifth order and, for practical purposes, the results are effectively exact. We also provide comparisons with the hybrid analytic and finite-difference method of Zhang. © 2008 Cambridge University Press

General string theory for dynamic curved viscida with surface tension

This work deals with the asymptotic derivation and numerical investigation of a model for the dynamics of curved inertial viscous fibres under surface tension, as they occur in rotational spinning processes. The resulting string model accounts for the inner viscous transport and places no restriction on either motion or shape of the fibre centre-line. The boundary conditions for the free end of the fibre yield a description for its temporal evolution, depending on the ratio of viscous and surface tension (capillary number). The behaviour of the fibre is studied numerically as function of the effects of viscosity, gravity, rotation and surface tension

Can we fabricate that fibre?

This paper reviews the development of an efficient mathematical model for the drawing of optical fibres using extensional flow theory which is applicable for fibres of arbitrary geometry. The model is comprised of a 1D axial stretching problem describing the change in area of the cross-section from preform to fibre coupled with a 2D cross-plane problem describing the evolution of a cross-section. The solution of the axial stretching problem may be written in an exact form while the cross-plane problem must, in general, be solved numerically. The model may be used to solve forward and inverse problems and gives results that compare well with experiments.Yvonne M. Stokes, Darren G. Crowdy, Heike Ebendorff-Heidepriem, Peter Buchak and Michael J. Che

Interface conditions for fast-reaction fronts in wet porous mineral materials : the case of concrete carbonation

Reaction–diffusion processes, where slow diffusion balances fast reaction, usually exhibit internal loci where the reactions are concentrated. Some modeling and simulation aspects of using kinetic free-boundary conditions to drive fast carbonation reaction fronts into unsaturated porous cement-based materials are discussed. Providing full control on the velocity of the reaction front, such conditions offer a rich description of the coupling between transport, reaction, and change in the shape of the a priori unknown time-dependent regions. New models are formulated and validated by means of numerical simulations and experimental data.cited By 6</p