Graziella Pacelli

A boundary element method to price time-dependent double barrier options

Abstract In this paper we propose a new method for pricing double-barrier options with moving barriers under the Black–Scholes and the CEV models. First of all, by applying a variational technique typical of the boundary element method, we derive an integral representation of the double-barrier option price in which two of the integrand functions are not given explicitly but must be obtained solving a system of Volterra integral equations of the first kind. Second, we develop an ad hoc numerical method to regularize and solve the system of integral equations obtained. Several numerical experiments are carried out showing that the overall algorithm is extraordinarily fast and accurate, even if the barriers are not differentiable functions. Moreover the numerical method presented in this paper performs significantly better than the finite difference approach.

A new formalism for wave scattering from a bounded obstacle

Let Ω⊂R3 be an obstacle that is a simply connected bounded domain. The exterior Dirichlet problem for the Helmholtz equation in R3\Ω with the Sommerfeld radiation condition at infinity is considered. Based on an integral representation formula, a new method to compute the solution of the exterior boundary value problem mentioned above is proposed. This method generalizes the formalism introduced for an unbounded obstacle by Milder [J. Acoust. Soc. Am. 89, 529–541 (1991)] and consists in computing a perturbation series whose coefficients are integrals. These integrals are independent one from the other so that the computation of the series is fully parallelizable. Finally, some numerical results obtained on test problems are shown. In particular, numerical experiments for obstacles with nonsmooth boundaries such as polyhedra and obstacles with multiscale corrugations are shown.

An interior point algorithm for global optimal solutions and KKT points

In this paper some theorems which characterize the global optimal solutions of nonlinear programming problems are proved. Two algorithms are derived using these results. The first is a path following algorithm to approximate the Karush Kuhn Tucker points of linearly constrained optimization problems and the second is an algorithm to solve linearly constrained global optimization problems. The convergence of these algorithms is proved under suitable assumptions. Numerical results obtained on several test problems are shown.

The constant elasticity of variance model: calibration, test and evidence from the Italian equity market

We present a robust and reliable methodology to calibrate and test the Constant Elasticity of Variance (CEV) model. Precisely, the parameters of the model are estimated by maximum likelihood, and an efficient numerical method to maximize the likelihood function is developed. Furthermore, a consistent and effective goodness-of-fit test of the CEV model is obtained using the Rosenblatt probability transformation and the χ 2 analysis. The novel procedure is employed to investigate the performances of the model on the Italian market. This analysis reveals that the CEV model does not offer a correct description of equity prices.

Stability Switches and Hopf Bifurcation in a Kaleckian Model of Business Cycle

This paper considers a Kaleckian type model of business cycle based on a nonlinear delay differential equation, whose associated characteristic equation is a transcendental equation with delay dependent coefficients. Using the conventional analysis introduced by Beretta and Kuang (2002), we show that the unique equilibrium can be destabilized through a Hopf bifurcation and stability switches may occur. Then some properties of Hopf bifurcation such as direction, stability, and period are determined by the normal form theory and the center manifold theorem.

Characterization of Convex Premium Principles

In actuarial literature the properties of risk measures or insurance premium principles have been extensively studied . We propose a characterization of a particular class of coherent risk measures defined in [1]. The considered premium principles are obtained by expansion of TVar measures, consequently they look like very interesting in insurance pricing where TVar measures is frequently used to value tail risks.

Inverse problem for a class of two-dimensional diffusion equations with piecewise constant coefficients

In this paper, we consider an inverse problem for a class of two-dimensional diffusion equations with piecewise constant coefficients. This problem is studied using an explicit formula for the relevant spectral measures and an asymptotic expansion of the solution of the diffusion equations. A numerical method that reduces the inverse problem to a sequence of nonlinear least-square problems is proposed and tested on synthetic data.

Stability Switches and Bifurcation Analysis of a Time Delay Model for the Diffusion of a New Technology

We deal with the time delay model for the diffusion of innovation technologies proposed by Fanelli and Maddalena [2012]. For this model, the stability switches and the occurrence of Hopf bifurcations are still largely undetermined, and in the present paper we perform some analysis on these topics. In particular, by applying the theory of delay differential equations and the analytical-geometrical approach developed by Beretta and Kuang [2002], we show that the equilibrium may lose stability and Hopf bifurcations may occur. Moreover, using the normal form theory and the center manifold theorem, we derive closed-form expressions that allow us to determine the direction of the Hopf bifurcations and the stability of the periodic solutions. Numerical results are presented which confirm and illustrate the theoretical predictions obtained.

Computing survival probabilities based on stochastic differential models

We develop a new numerical method to compute survival probabilities based on stochastic differential models, a matter of great importance in several areas of science, such as finance, biology, medicine and geophysics. This novel approach is based on polynomial differential quadrature, which is combined with a high-order time discretization scheme. Numerical experiments are presented showing that the proposed method performs extremely well and is more efficient than the approaches recently developed in Costabile et?al. (2013) and Guarin et?al. (2011). We propose a very efficient method to compute survival probabilities.We combine polynomial differential quadrature with high-order time-stepping.We consider a reduced-form model and a structural model that arise from finance and insurance.The method is model independent and could also be extended to other stochastic processes.Numerical comparison with other recent approaches is provided.

Pricing Double-Barrier Options Using the Boundary Element Method

A numerical method to price double-barrier options with moving barriers is proposed. Using the so-called Boundary Element Method, an integral representation of the double-barrier option price is derived in which two of the integrand functions are not given explicitly but must be obtained solving a system of Volterra integral equations of the first kind. This system of equations is affected by several kinds of singularities, therefore it is first regularized and then solved using a low-order finite element method based on product integration. Several numerical experiments are carried out showing that the method proposed is extraordinarily fast and accurate, also when the barriers are not differentiable functions. Moreover the numerical algorithm presented in this paper performs significantly better than the finite difference approach.