The evaluation of option premium is a very delicate issue arising from the assumptions made under a financial market model, and pricing of a wide range of options is generally feasible only when numerical methods are involved. This paper is based on our recent research on numerical pricing of path-dependent multi-asset options and extends these results also to the case of Asian options with fixed strike. First, we recall the three-dimensional backward parabolic PDE describing the evolution of European-style...
Option pricing models are an important part of financial markets worldwide. The PDE formulation of these models leads to analytical solutions only under very strong simplifications. For more general models the option price needs to be evaluated by numerical techniques. First, based on an ideal pure diffusion process for two risky asset prices with an additional path-dependent variable for continuous arithmetic average, we present a general form of PDE for pricing of Asian option contracts on two...
The paper presents a discontinuous Galerkin method for solving partial integro-differential equations arising from the European as well as American option pricing when the underlying asset follows an exponential variance gamma process. For practical purposes of numerical solving we introduce the modified option pricing problem resulting from a localization to a bounded domain and an approximation of small jumps, and we discuss the related error estimates. Then we employ a robust numerical procedure...
Under real market conditions, there exist many cases when it is inevitable to adopt numerical approximations of option prices due to non-existence of analytical formulae. Obviously, any numerical technique should be tested for the cases when the analytical solution is well known. The paper is devoted to the discontinuous Galerkin method applied to European option pricing under the Merton jump-diffusion model, when the evolution of the asset prices is driven by a Lévy process with finite activity....
This paper is devoted to barrier options and the main objective is to develop a sufficiently robust, accurate and efficient method for computation of their values driven according to the well-known Black-Scholes equation. The main idea is based on the discontinuous Galerkin method together with a spatial adaptive approach. This combination seems to be a promising technique for the solving of such problems with discontinuous solutions as well as for consequent optimization of the number of degrees...
We deal with a scalar nonstationary convection-diffusion equation with nonlinear convective as well as diffusive terms which represents a model problem for the solution of the system of the compressible Navier-Stokes equations describing a motion of viscous compressible fluids. We present a discretization of this model equation by the discontinuous Galerkin finite element method. Moreover, under some assumptions on the nonlinear terms, domain partitions and the regularity of the exact solution,...
The real options approach interprets a flexibility value, embedded in a project, as an option premium. The object of interest is to valuate real options to change operating scale, typical for natural resources industry. The evolution of the project as well as option prices is decribed by partial differential equations of the Black-Scholes type, linked through a payoff function given by a type of the flexibility provided. The governing equations are discretized by the discontinuous Galerkin method...
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