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In this paper we study stock loans of finite maturity with different dividend distributions semi-analytically using the analytical approximation method in Zhu (2006). Stock loan partial differential equations (PDEs) are established under Black–Scholes framework. Laplace transform method is used to solve the PDEs. Optimal exit price and stock loan value are obtained in Laplace space. Values in the original time space are recovered by numerical Laplace inversion. To demonstrate the efficiency and accuracy of our semi-analytic method several examples are presented, the results are compared with those calculated using existing methods. We also present a calculation of fair service fee charged by the lender for different loan parameters. [ABSTRACT FROM AUTHOR]

Abstract: An efficient integral equation formulation for two-dimensional crack problems is proposed with the displacement equation being used on the outer boundary and the traction equation being used on one of the crack faces. Discontinuous quarter point elements are used to correctly model the displacement in the vicinity of crack tips. Using this formulation a general crack problem can be solved in a single-region formulation, and only one of the crack faces needs to be discretised. Once the relative displacements of the cracks are solved numerically, physical quantities of interest, such as crack tip stress intensity factors can be easily obtained. Numerical examples are provided to demonstrate the accuracy and efficiency of the present formulation. [Copyright &y& Elsevier]

• European options prices with transaction costs under Heston stochastic volatility. • Non-linear pricing partial differential equation is solved by the finite difference method. • Bid and ask prices for the buyer and the writer in the presence of transaction costs. • Model can be extended to price American options with appropriate boundary conditions. In this work, we formulate a pricing model for European options with transaction costs under Heston-type stochastic volatility. The resulting pricing partial differential equations (PDEs) are a pair of nonlinear convection-diffusion-reaction equations with mixed derivative terms, for the writing and holding prices, respectively. The equations are solved numerically by the explicit Euler method. Numerical experiments are presented to illustrate the order of convergence and the effect of the transaction costs on option prices. [ABSTRACT FROM AUTHOR]