American Options under Proportional Transaction Costs: Pricing, Hedging and Stopping Algorithms for Long and Short Positions

American options are studied in a general discrete market in the presence of proportional transaction costs, modelled as bid-ask spreads. Pricing algorithms and constructions of hedging strategies, stopping times and martingale representations are presented for short (seller’s) and long (buyer’s) positions in an American option with an arbitrary payoff. This general approach extends the special cases considered in the literature concerned primarily with computing the prices of American puts under transaction costs by relaxing any restrictions on the form of the payoff, the magnitude of the transaction costs or the discrete market model itself. The largely unexplored case of pricing, hedging and stopping for the American option buyer under transaction costs is also covered. The pricing algorithms are computationally efficient, growing only polynomially with the number of time steps in a recombinant tree model. The stopping times realising the ask (seller’s) and bid (buyer’s) option prices can differ from one another. The former is generally a so-called mixed (randomised) stopping time, whereas the latter is always a pure (ordinary) stopping time.     

Pointwise Estimates for Laplace Equation. Applications to the Free Boundary of the Obstacle Problem with Dini Coefficients

In this paper we are interested in pointwise regularity of solutions to elliptic equations. In a first result, we prove that if the modulus of mean oscillation of Δu at the origin is Dini (in L ^p average), then the origin is a Lebesgue point of continuity (still in L ^p average) for the second derivatives D ² u. We extend this pointwise regularity result to the obstacle problem for the Laplace equation with Dini right hand side at the origin. Under these assumptions, we prove that the solution to the obstacle problem has a Taylor expansion up to the order 2 (in the L ^p average). Moreover we get a quantitative estimate of the error in this Taylor expansion for regular points of the free boundary. In the case where the right hand side is moreover double Dini at the origin, we also get a quantitative estimate of the error for singular points of the free boundary. Our method of proof is based on some decay estimates obtained by contradiction, using blow-up arguments and Liouville Theorems. In the case of singular points, our method uses moreover a refined monotonicity formula.      

On the Numerical Solution to a Nonlinear Wave Equation Associated with the First Painlev′e Equation: an Operator-Splitting Approach

The main goal of this article is to discuss the numerical solution to a nonlinear wave equation associated with the first of the celebrated Painlevé transcendent ordinary differential equations. In order to solve numerically the above equation, whose solutions blow up in finite time, the authors advocate a numerical methodology based on the Strang’s symmetrized operator-splitting scheme. With this approach, one can decouple nonlinearity and differential operators, leading to the alternate solution at every time step of the equation as follows: (i) The first Painlevé ordinary differential equation, (ii) a linear wave equation with a constant coefficient. Assuming that the space dimension is two, the authors consider a fully discrete variant of the above scheme, where the space-time discretization of the linear wave equation sub-steps is achieved via a Galerkin/finite element space approximation combined with a second order accurate centered time discretization scheme. To handle the nonlinear sub-steps, a second order accurate centered explicit time discretization scheme with adaptively variable time step is used, in order to follow accurately the fast dynamic of the solution before it blows up. The results of numerical experiments are presented for different coefficients and boundary conditions. They show that the above methodology is robust and describes fairly accurately the evolution of a rather “violent” phenomenon.     

Weak Solutions to the Cahn-Hilliard Equation with Degenerate Diffusion Mobility in ℝN

This paper is concerned with a popular form of Cahn-Hilliard equation which plays an important role in understanding the evolution of phase separation. We get the existence and regularity of a weak solution to nonlinear parabolic, fourth order Cahn-Hilliard equation with degenerate mobility M(u)=u^m(1-u)^m which is allowed to vanish at 0 and 1. The existence and regularity of weak solutions to the degenerate Cahn-Hilliard equation are obtained by getting the limits of Cahn-Hilliard equation with non-degenerate mobility. We explore the initial value problem with compact support and obtain the local non-negative result. Further, the above derivation process is also suitable for the viscous Cahn-Hilliard equation with degenerate mobility.     

Optimal Control and “Strange” Term Arising from Homogenization of the Poisson Equation in the Perforated Domain with the Robin-type Boundary Condition in the Critical Case

Doklady Mathematics - The present paper is devoted to the study of the asymptotic behavior of the optimal control for the boundary value problem in an ε-periodically perforated domain with...     

On Universality of Critical Behavior in the Focusing Nonlinear Schrödinger Equation, Elliptic Umbilic Catastrophe and the Tritronquée Solution to the Painlevé-I Equation

We argue that the critical behavior near the point of “gradient catastrophe” of the solution to the Cauchy problem for the focusing nonlinear Schrödinger equation $i\epsilon \varPsi _{t}+\frac{\epsilon^{2}}{2}\varPsi _{xx}+|\varPsi |^{2}\varPsi =0We argue that the critical behavior near the point of “gradient catastrophe” of the solution to the Cauchy problem for the focusing nonlinear Schr?dinger equation , ε ≪1, with analytic initial data of the form is approximately described by a particular solution to the Painlevé-I equation.      

Dispersive Estimates of Solutions to the Wave Equation with a Potential in Dimensions n ≥ 4

We prove dispersive estimates for solutions to the wave equation with a real-valued potential V ∈ L ^∞(R ⁿ ), n ≥ 4, satisfying V(x) = O(?x?^?(n+1)/2?ε), ε > 0.     

A DEA methodology to evaluate the impact of information asymmetry on the efficiency of not-for-profit organizations with an application to higher education in Brazil

This paper presents a conceptual framework and an analytical DEA model for evaluating the impact of information asymmetry on organizational efficiency. The framework uses concepts from agency theory to estimate the extent of moral hazard by comparing the objectives of the principal to those of the agent. The framework and model are useful in the analysis of both for-profit and not-for-profit organizations because DEA is applicable whether or not inputs and/or outputs are subject to pricing mechanisms. An illustration focusing on the Brazilian not-for-profit federal university system finds that the agency problem indeed exists for a subset of those institutions, indicating the desirability of improved incentive and control mechanisms on the part of the principal.     

Wavelet Approximation in Weighted Sobolev Spaces of Mixed Order with Applications to the Electronic Schr?dinger Equation

We study the approximation of functions in weighted Sobolev spaces of mixed order by anisotropic tensor products of biorthogonal, compactly supported wavelets. As a main result, we characterize these spaces in terms of wavelet coefficients, which also enables us to explicitly construct approximations. In particular, we derive approximation rates for functions in exponentially weighted Sobolev spaces discretized on optimized general sparse grids. Under certain regularity assumptions, the rate of convergence is independent of the number of dimensions. We apply these results to the electronic Schr?dinger equation and obtain a convergence rate which is independent of the number of electrons; numerical results for the helium atom are presented.