Luca Vincenzo Ballestra

The evaluation of American options in a stochastic volatility model with jumps: An efficient finite element approach

We consider the problem of pricing American options in the framework of a well-known stochastic volatility model with jumps, the Bates model. According to this model the asset price is described by a jump-diffusion stochastic differential equation in which the jump term consists of a Levy process of compound Poisson type, while the volatility is modeled as a CIR-type process correlated with the asset price. Pricing American options under the Bates model requires us to solve a partial integro-differential equation with the final condition and boundary conditions prescribed on a free boundary. In this paper a numerical method for solving such a problem is proposed. In particular, first of all, using a Richardson extrapolation technique, the problem is reduced to a problem with fixed boundary. Then the problem obtained is solved using an ad hoc finite element method which efficiently combines an implicit/explicit time stepping, an operator splitting technique, and a non-uniform mesh of right-angled triangles. Numerical experiments are presented showing that the option pricing algorithm developed in this paper is extremely accurate and fast. In particular it is significantly more efficient than other numerical methods that have recently been proposed for pricing American options under the Bates model.

Pricing European and American options by radial basis point interpolation

A RBPI method for pricing European and American options is proposed.RBPI has not yet been used for option pricing.The overall efficiency is improved by coupling RBPI with other approaches.Three different algorithms for American options are tested and compared.Numerical results demonstrate the effectiveness of the techniques employed. We propose the use of the meshfree radial basis point interpolation (RBPI) to solve the Black-Scholes model for European and American options. The RBPI meshfree method offers several advantages over the more conventional radial basis function approximation, nevertheless it has never been applied to option pricing, at least to the very best of our knowledge. In this paper the RBPI is combined with several numerical techniques, namely: an exponential change of variables, which allows us to approximate the option prices on their whole spatial domain, a mesh refinement algorithm, which turns out to be very suitable for dealing with the non-smooth options payoff, and an implicit Euler Richardson extrapolated scheme, which provides a satisfactory level of time accuracy. Finally, in order to solve the free boundary problem that arises in the case of American options three different approaches are used and compared: the projected successive overrelaxation method (PSOR), the Bermudan approximation, and the penalty approach. Numerical experiments are presented which demonstrate the computational efficiency of the RBPI and the effectiveness of the various techniques employed.

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 fast numerical method to price American options under the Bates model

We consider the problem of pricing American options in the framework of a well-known stochastic volatility model with jumps, the Bates model. According to this model, the price of an American option can be obtained as the solution of a linear complementarity problem governed by a partial integro-differential equation. In this paper, a numerical method for solving such a problem is proposed. In particular, first of all, using a Bermudan approximation and a Richardson extrapolation technique, the linear complementarity problem is reduced to a set of standard linear partial differential problems (see, for example, Ballestra and Sgarra, 2010; Chang etźal. 2007, 2012). Then, these problems are solved using an ad hoc pseudospectral method which efficiently combines the Chebyshev polynomial approximation, an implicit/explicit time stepping and an operator splitting technique. Numerical experiments are presented showing that the novel algorithm is very accurate and fast and significantly outperforms other methods that have recently been proposed for pricing American options under the Bates model.

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.

Repeated spatial extrapolation: An extraordinarily efficient approach for option pricing

Various finite difference methods for option pricing have been proposed. In this paper we demonstrate how a very simple approach, namely the repeated spatial extrapolation, can perform extremely better than the finite difference schemes that have been developed so far. In particular, we consider the problem of pricing vanilla and digital options under the Black-Scholes model, and show that, if the payoff functions are dealt with properly, then errors close to the machine precision are obtained in only some hundredths of a second.

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.

Semiconductor device simulation using a viscous-hydrodynamic model

Abstract In this article we deal with a hydrodynamic model of Navier–Stokes (NS) type for semiconductors including a physical viscosity in the momentum and energy equations. A stabilized finite difference scheme with upwinding based on the characteristic variables is used for the discretization of the NS equations, while a mixed finite element scheme is employed for the approximation of the Poisson equation. A consistency result for the method is established showing that the scheme is first-order accurate in both space and time. We also perform a stability analysis of the numerical method applied to a linearized incompletely parabolic system in two space dimensions with vanishing viscosity. A thorough numerical parametric study as a function of the heat conductivity and of the momentum viscosity is carried out in order to investigate their effect on the development of shocks in both one and two space dimensional devices.

A numerical method to price discrete double Barrier options under a constant elasticity of variance model with jump diffusion

We develop a numerical method to price discrete barrier options on an underlying described by the constant elasticity of variance model with jump-diffusion (CEVJD). In particular, the partial integro differential equation associated to this model is discretized in time using an operator splitting scheme whose accuracy is enhanced by repeated Richardson extrapolation. Such an approach allows us to approximate the differential terms and the jump integral by means of two different numerical techniques. Precisely, the spatial derivatives, which exist only in the weak sense, are discretized using a finite element method based on piecewise quadratic polynomials, whereas the jump integral is directly collocated at the mesh points, so that it can be easily evaluated by Simpson numerical quadrature. As shown by extensive numerical simulation, the proposed approach is very efficient from the computational standpoint, and performs significantly better than the finite difference scheme developed in Wade et al. [On smoothing of the Crank–Nicolson scheme and higher order schemes for pricing barrier options, J. Comput. Appl. Math. 204 (2007), pp. 144–158].

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.