Citation Context

...n fluid dynamics; see [14] and references therein. The ADI and operator splitting methods have been applied also to the option pricing and, for example, the books [6], [10], and [34] discuss this. In =-=[18]-=- an operator splitting method for pricing American options under stochastic volatility is proposed. It decouples the convection diffusion operator and the early exercise constraint to separate fractio...

Citation Context

...plitting technique to solve the LCP. This technique has been used in other numerical fields and was implemented for financial problems by Ikonen and Toivanen in several articles and reports (see e.g. =-=[12, 13]-=-). This method is built upon a finite difference discretization (though it could be used also for other discretizations) and separates the solution of the linear systems of equations from the enforcin...

Citation Context

...6]. With a well designed multigrid method, the number of iterations does not grow with refined discretizations. For extensive literature on this see, the book [32], for example. For solving LCPs Brandt and Cryer introduced a projected full approximation scheme (PFAS) multigrid method in [33]. Another multigrid method for similar problems was described in [34]. The PFAS method was used to price American options under stochastic volatility by Clarke and Parrott in [9], and Oosterlee in [13]. Some alternative approaches employing multigrid methods for option pricing have been considered in [35], [36], [37]. Reisinger and Wittum described a projected multigrid (PMG) method for LCPs which resembles more closely a classical multigrid method for linear problems in [38]. This method has been used to price American options in [38], [11]. The above mentioned methods are so-called geometrical multigrid methods which means that the spatial operators are discretized on sequence of grids. Furthermore, transfer operators between grids need to be implemented. S. Salmi, J. Toivanen, L. von Sydow / Procedia Computer Science 00 (2013) 000–000 The geometrical multigrid method can be implemented with some ...

Citation Context

... a more sophisticated splitting of the operators, which approximate the continuous LCP by a sequence of simpler problems in every time step, rather than solving the discretised LCP directly, e.g. see =-=[18, 19]-=- and references therein; these methods are typically efficient if the underlying splitting for the European counterpart is accurate. However, with these approaches, there is no sense of solving a disc...

Citation Context

...13, 15, 16, 21, 27, 30, 39, 52, 55, 57, 65, 72, 75, 79, 81, 88, 87, 104, 111, 114, 115, 116], exotic options[11, 14, 22, 33, 36, 41, 43, 50, 82, 86, 93, 94, 95, 102, 105, 118] and multi-asset options =-=[20, 40, 45, 62, 83, 88, 103, 117]-=-. Some of the books that are dealing with various issues (including options) in finanCHAPTER 1. GENERAL INTRODUCTION 35 cial mathematics are [8, 25, 60, 96]. It should be noted that there are many oth...

Citation Context

....5 Operator splitting The operator splitting technique for solving the LCP problem will here be explained in short. The technique was introduced in a series of papers by Ikonen and Toivanen, see e.g. =-=[25, 26]-=-. Let us assume that we have the LCP problem formulated as in (3.7). The operator splitting is then based on a reformulation of the problem using an auxiliary variable λ = λ(x, t). ( ) ∂ − L F (x, t) ...

Citation Context

...sive over-relaxation method (PSOR) – introduced in [20] – the most widely used approach in practice for finite difference matrices (cf. [23, 60, 67]), in spite of its relatively slow convergence (cf. =-=[38]-=-); see also Section 1.2.3, where an outline of PSOR has been given. We aim to demonstrate that the method of policy iteration, developed in [28] for the numerical solution of HJB equations, yields a p...