Pricing Parisian down-and-in options

An efficient numerical method for pricing American put options under the CEV model

Closed Form Optimal Exercise Boundary of the American Put Option

This paper proposes an innovative algorithm that significantly improves on the approximation of the optimal early exercise boundary obtained with simulation based methods for American option pricing. The method works by exploiting and leveraging the information in multiple cross sectional regressions to the fullest by averaging the individually obtained estimates at each early exercise step, starting from just before maturity, in the backwards induction algorithm. With this method less errors are accumulated, and as a result of this the price estimate is essentially unbiased even for long maturity options. Numerical results demonstrate the improvements from our method and show that these are robust to the choice of simulation setup, the characteristics of the option, and the dimensionality of the problem. Finally, because our method naturally disassociates the estimation of the optimal early exercise boundary from the pricing of the option, significant efficiency gains can be obtained by using less simulated paths and repetitions to estimate the optimal early exercise boundary than with the regular method.

We consider the problem of pricing American options using the generalized Black–Scholes model. The generalized Black–Scholes model is a modified form of the standard Black–Scholes model with the effect of interest and consumption rates. In general, because the American option problem does not have an exact closed-form solution, some type of approximation is required. A simple numerical method for pricing American put options under the generalized Black–Scholes model is presented. The proposed method corresponds to a free boundary (also called an optimal exercise boundary) problem for a partial differential equation. We use a transformed function that has Lipschitz character near the optimal exercise boundary to determine the optimal exercise boundary. Numerical results indicating the performance of the proposed method are examined. Several numerical results are also presented that illustrate a comparison between our proposed method and others.

Classical theory for systems of the first order partial differential equations for a scalar function can be rephrased as the submanifold theory of contact manifolds (geometric first order jet spaces). In the same spirit, we will develop the geometric theory of systems of partial differential equations of second order for a scalar function as the Contact Geometry of Second Order, following E. Cartan.We will formulate the submanifold theory of second order jet spaces as the geometry of PD manifolds (R;D 1 ,D 2 ) of second order. Moreover we will establish the First Reduction Theorem for (R;D 1,D 2) admitting non-trivial Cauchy characteristic systems. By utilizing Parabolic Geometry, we will give, directly or combined with reduction theorems, several classes of systems of partial differential equations of second order, for which the model equation of each class admits the Lie algebra of infinitesimal contact transformations, which is finite dimensional and simple.

Exact solutions of some nonlinear systems of partial differential equations by using the first integral method

In many democratic parliamentary systems, election timing is an important decision availed to governments according to sovereign political systems. Prudent governments can take advantage of this constitutional option in order to maximize their expected remaining life in power. The problem of establishing the optimal time to call an election based on observed poll data has been well studied with several solution methods and various degrees of modeling complexity. The derivation of the optimal exercise boundary holds strong similarities with the American option valuation problem from mathematical finance. A seminal technique refined by Longstaff and Schwartz in 2001 provided a method to estimate the exercise boundary of the American options using a Monte Carlo method and a least squares objective. In this paper, we modify the basic technique to establish the optimal exercise boundary for calling a political election. Several innovative adaptations are required to make the method work with the additional complexity in the electoral problem. The transfer of Monte Carlo methods from finance to determine the optimal exercise of real-options appears to be a new approach.

A method for reducing systems of partial differential equations to corresponding systems of ordinary differential equations is proposed. A system of equations describing two-dimensional, cylindrical, and spherical flows of a polytropic gas; a system of dimensionless Stokes equations for the dynamics of a viscous incompressible fluid; a system of Maxwell’s equations for vacuum; and a system of gas dynamics equations in cylindrical coordinates are studied. It is shown how this approach can be used for solving certain problems (shockless compression, turbulence, etc.).

A Moving Boundary Approach to American Option Pricing

A Closed-Form Solution to American Options under General Diffusions

An integral equation approach for the valuation of American-style down-and-out calls with rebates

Using the Lie symmetry approach, we have examined herein the system of partial differential equations corresponding to the Einstein–Maxwell equations for a static axially symmetric spacetime. The method used reduces the system of partial differential equations to a system of ordinary differential equations according to the Lie symmetry admitted. In particular, we found the relevant system of ordinary differential equations is all optimal subgroups. The system of ordinary differential equations is further solved in general to obtain exact solutions. Several new physically important families of exact solutions are derived.

In this paper, we present a fast and accurate numerical approach applied to specific American-style derivatives, namely American power call and put options, whose main feature is that the underlying asset is raised to a power. The study is set in the Black–Scholes framework, and we consider continuously paying dividends assets and arbitrary positive values for the power. It is important to note that although a log-normal process raised to a power is again log-normal, the resulting change in variables may lead to a negative dividend rate, and this case remains largely understudied in the literature. We derive closed-form formulas for the perpetual options’ optimal boundaries and for the fair prices. For finite maturities, we approximate the optimal boundary using some first-hitting properties of the Brownian motion. As a consequence, we obtain the option price quickly and with relatively high accuracy—the error is at the third decimal position. We further provide a comprehensive analysis of the impact of the parameters on the options’ value, and discuss ordinary European and American capped options. Various numerical examples are provided.

In the theory of systems of quasilinear partial differential equations of the first order, the main questions are the solvability of initial values problem and justification of the approximate methods. This is due to problems in gas dynamics and hydromechanics. In the second half of the previous century attempts were made to construct a correct theory of solvability of problems or the systems of quasilinear partial differential equations of the first order. The necessity of the correct way of introductions the nothions of a generalized solution of initial values problems is connected with this. In this paper a class of systems of quasilinear partial differential equations of the first order is singled out for which the concept of a generalized solution is introduced. A method for constructing approximate methods for solving initial values problem is proposed. We obtained estimates of the convergence speed in approximate methods and proved the existence and uniqueness of the solution of initial values problem for systems of quasilinear partial differential equations of the first order of a certain form.

For a class of nonlinear systems of partial differential equations of high order in a cylindrical domain the following boundary value problem is considered: Cauchy-type conditions are prescribed on the upper and lower bases of the cylinder, and a Robin-type condition is prescribed on the lateral part of the boundary. This boundary value problem is reduced in an equivalent way to a nonlinear functional equation in a certain subspace of the Sobolev space. Under certain assumptions about the nonlinear terms an a priori estimate is obtained for the solution of the problem in question and the existence of the solution is established, while in the case when these conditions fail, the absence of a solution is established. The question of uniqueness is also discussed for the solution. Bibliography: 18 titles.