Song-Ping Zhu

An exact and explicit solution for the valuation of American put options

In this paper, an exact and explicit solution of the well-known Black–Scholes equation for the valuation of American put options is presented for the first time. To the best of the authors knowledge, a closed-form analytical formula has never been found for the valuation of American options of finite maturity, although there have been quite a few approximate solutions and numerical approaches proposed. The closed-form exact solution presented here is written in the form of a Taylors series expansion, which contains infinitely many terms. However, only about 30 terms are actually needed to generate a convergent numerical solution if the solution of the corresponding European option is taken as the initial guess of the solution series. The optimal exercise boundary, which is the main difficulty of the problem, is found as an explicit function of the risk-free interest rate, the volatility and the time to expiration. A key feature of our solution procedure, which is based on the homotopy-analysis method, is the optimal exercise boundary being elegantly and temporarily removed in the solution process of each order, and, consequently, the solution of a linear problem can be analytically worked out at each order, resulting in a completely analytical and exact series-expansion solution for the optimal exercise boundary and the option price of American put options.

Solving linear diffusion equations with the dual reciprocity method in Laplace space

Abstract The dual reciprocity method (DRM) is applied in the Laplace space to solve efficiently time-dependent diffusion problems. Since there is no discretation in time and there are no domain integrals involved in a calculation, the proposed approach seems to have provided considerable savings on computer operating costs and in data preparation, and thus leads to certain advantages over existing methods. Three numerical examples are presented, which demonstrate well the efficiency and accuracy of the new approach.

NEW SOLUTIONS FOR THE PROPAGATION OF LONG WATER WAVES OVER VARIABLE DEPTH

Based on the linearized long-wave equation, two new analytical solutions are obtained respectively for the propagation of long surface gravity waves around a conical island and over a paraboloidal shoal. Having been intensively studied during the last two decades, these two problems have practical significance and are physically revealing for wave propagation over variable water depth. The newly derived analytical solutions are compared with several previously obtained numerical solutions and the accuracy of those numerical solutions is discussed. The analytical method has the potential to be used to find solutions for wave propagation over more natural bottom topographies.

On the choice of interpolation functions used in the dual-reciprocity boundary-element method

Abstract The dual-reciprocity boundary-element method is a very powerful technique for solving general elliptic equations of the type ∇ 2 u = b . In this method, a series of interpolation functions is used to approximate b in order to convert the associated domain integral, which it is necessary to evaluate in a traditional boundary-element analysis, into boundary integrals only. Hence the choice of interpolation functions has direct effects on the numerical results. According to Partridge and Brebbia, the adoption of a comparatively simple form of interpolation function gives the best results. Unfortunately, when b contains partial derivatives of the unknown function u ( x , y ), the adoption of such a type of interpolation function inevitably leads to the creation of singularities on all boundary and internal nodes used in a dual-reciprocity boundary-element analysis, as was pointed out by Zhu and Zhang in 1992. To avoid this problem, a functional transformation, which applies only to linear governing equations, can be employed to eliminate these derivative terms and thus to obtain better numerical results. In this paper, two new interpolation functions are proposed and examined; they are proven to be generally applicable and satisfactory.

A New Analytical Approximation Formula For The Optimal Exercise Boundary Of American Put Options

In this paper, a new analytical formula as an approximation to the value of American put options and their optimal exercise boundary is presented. A transform is first introduced to better deal with the terminal condition and, most importantly, the optimal exercise price which is an unknown moving boundary and the key reason that valuing American options is much harder than valuing its European counterparts. The pseudo-steady-state approximation is then used in the performance of the Laplace transform, to convert the systems of partial differential equations to systems of ordinary differential equations in the Laplace space. A simple and elegant formula is found for the optimal exercise boundary as well as the option price of the American put with constant interest rate and volatility. Other hedge parameters as the derivatives of this solution are also presented.

Diffraction of short-crested waves around a circular cylinder

Abstract In this paper, an exact solution for the diffraction of short-crested waves incident on a circular cylinder is presented. The pressure distribution and water run-up on the cylinder was found to be quite different from those of plane incident waves. The total force exerted on the cylinder in the direction of the wave propagation was found to be smaller compared to that induced by plane waves with the same wave number in the direction of the wave propagation. The total wave load increases as the wave number in the direction perpendicular to the direction of the wave propagation increases, or as the incident waves become shorter. These results show that if the wave loading is calculated, as a design criterion, according to plane incident waves, it will be over-estimated when the incident waves are short-crested. However, from the safety point of view, the wave loading formula derived from a plane incident wave may still serve as a good engineering design criterion.

SCATTERING OF LONG WAVES AROUND A CIRCULAR ISLAND MOUNTED ON A CONICAL SHOAL

Abstract Scattering of simple harmonic long waves by a cylindrical island mounted on a conical shoal in an otherwise open sea of constant depth is solved analytically based on the shallow-water wave (long-wave) theory. The new analytical solution not only confirms some conclusions and conjectures previously drawn from purely numerical studies, such as those showing how the slope of the shoal affects the amplification of the ocean waves around the coastline of islands, but also provides another useful check for numerical model developers.

A predictor–corrector scheme based on the ADI method for pricing American puts with stochastic volatility

Abstract In this paper, we introduce a new numerical scheme, based on the ADI (alternating direction implicit) method, to price American put options with a stochastic volatility model. Upon applying a front-fixing transformation to transform the unknown free boundary into a known and fixed boundary in the transformed space, a predictor–corrector finite difference scheme is then developed to solve for the optimal exercise price and the option values simultaneously. Based on the local von Neumann stability analysis, a stability requirement is theoretically obtained first and then tested numerically. It is shown that the instability introduced by the predictor can be damped, to some extent, by the ADI method that is used in the corrector. The results of various numerical experiments show that this new approach is fast and accurate, and can be easily extended to other types of financial derivatives with an American-style exercise. Another key contribution of this paper is the proposition of a set of appropriate boundary conditions, particularly in the volatility direction, upon realizing that appropriate boundary conditions in the volatility direction for stochastic volatility models appear to be controversial in the literature. A sound justification is also provided for the proposed boundary conditions mathematically as well as financially.

A general DRBEM model for wave refraction and diffraction

Abstract A numerical model based on the dual reciprocity boundary element method (DRBEM) is presented here for the study of combined wave diffraction and refraction. The model is more general than that presented by Zhu [Zhu S-P. Engng Anal Boundary Elements 1993;12:261–274] in the sense that areas or coastlines where water depth is zero can be dealt with as well. Our comparative study show that the new model is very accurate for long waves (tsunami waves). It is numerically very efficient in comparison with models based on finite elements too. Using the new model, the interaction between the diffraction and refraction effects is examined. It is shown that the diffraction effect is significantly enhanced when there is a combined diffraction and refraction than when there is just diffraction alone.

Numerical calculation of forces induced by short-crested waves on a vertical cylinder of arbitrary cross-section

Abstract Most off-shore oil platforms are supported by vertical cylinders extending to the ocean floor. An important problem in off-shore engineering is the calculation of the wave loading exerted on these vertical cylinders. Analytical solutions have been found for the case of plane incident waves incident on a circular cylinder by MacCamy and Fuchs [(1954), Wave forces on piles: a diffraction theory. U.S. Army Corps of Engineering, Beach Erosion Board, Technical Memorandum No. 69] and also for short-crested waves incident on a circular cylinder by Zhu [(1993), Diffraction of short-crested waves around a circular cylinder. Ocean Engng 20 , 389–407]. However, for a cylinder of arbitrary cross-section, no analytic solutions currently exist. Au and Brebbia [(1983), Diffraction of water waves for vertical cylinders using boundary elements. Appl. Math. Modelling 7 , 106–114] proposed an efficient numerical approach to calculate the wave loads induced by plane waves on vertical cylinders by using the boundary element method. However, wind-generated waves are better modelled by short-crested waves. Whether or not these short-crested waves can induce larger wave forces on a structure is of great concern to ocean engineers. In this paper wave loads, induced by short-crested incident waves, on a vertical cylinder of arbitrary cross-section are discussed. For a cylinder of certain cross-section, the wave loads induced by short-crested waves can be larger than those induced by plane waves with the same total wave number.