On the Acceleration of Explicit Finite Difference Methods for Option Pricing

Implicit finite difference methods are conventionally preferred over their explicit counterparts for the numerical valuation of options. In large part the reason for this is a severe stability constraint known as the Courant–Friedrichs–Lewy (CFL) condition which limits the latter class’s efficiency. Implicit methods, however, are difficult to implement for all but the most simple of pricing models, whereas explicit techniques are easily adapted to complex problems. For the first time in a financial context, we present an acceleration technique, applicable to explicit finite difference schemes describing diffusive processes with symmetric evolution operators, called Super-Time-Stepping. We show that this method can be implemented as part of a more general approach for non-symmetric operators. Formal stability is thereby deduced for the exemplar cases of European and American put options priced under the Black–Scholes equation. Furthermore, we introduce a novel approach to describing the efficiencies of finite difference schemes as semi-empirical power laws relating the minimal real time required to carry out the numerical integration to a solution with a specified accuracy. Tests are described in which the method is shown to significantly ameliorate the severity of the CFL constraint whilst retaining the simplicity of the underlying explicit method. Degrees of acceleration are achieved yielding comparable, or superior, efficiencies to a set of benchmark implicit schemes. We infer that the described method is a powerful tool, the explicit nature of which makes it ideally suited to the treatment of symmetric and non-symmetric diffusion operators describing complex financial instruments including multi-dimensional systems requiring representation on decomposed and/or adaptive meshes

Particle Transport and Stochastic Acceleration in the Giant Lobes of Centaurus A

The conditions within the giant lobes of Centaurus A are reviewed in light of recent radio and γ‐ray observations. Data from WMAP and ground‐based telescopes in conjunction with measurements from Fermi‐LAT constrain the characteristic field strength and the maximum electron energy. The implications for the transport of energetic particles are discussed in terms of residence times and cooling times within the lobes. Acceleration of electrons and UHECR via the second order Fermi mechanism is discussed

An Explicit Scheme for Multifluid Magnetohydrodynamics

When modelling astrophysical fluid flows, it is often appropriate to discard the canonical magnetohydrodynamic approximation, thereby freeing the magnetic field to diffuse with respect to the bulk velocity field. As a consequence, however, the induction equation can become problematic to solve via standard explicit techniques. In particular, the Hall diffusion term admits fast-moving whistler waves which can impose a vanishing time-step limit.Within an explicit differencing framework, a multifluid scheme for weakly ionized plasmas is presented which relies upon a new approach to integrating the induction equation efficiently. The first component of this approach is a relatively unknown method of accelerating the integration of parabolic systems by enforcing stability over large compound time-steps rather than over each of the constituent substeps. This method, Super Time-Stepping, proves to be very effective in applying a part of the Hall term up to a known critical value. The excess of the Hall term above this critical value is then included via a new scheme for pure Hall diffusion

An Explicit Scheme for Multifluid Magnetohydrodynamics

When modelling astrophysical fluid flows, it is often appropriate to discard the canonical magnetohydrodynamic approximation, thereby freeing the magnetic field to diffuse with respect to the bulk velocity field. As a consequence, however, the induction equation can become problematic to solve via standard explicit techniques. In particular, the Hall diffusion term admits fast-moving whistler waves which can impose a vanishing time-step limit.Within an explicit differencing framework, a multifluid scheme for weakly ionized plasmas is presented which relies upon a new approach to integrating the induction equation efficiently. The first component of this approach is a relatively unknown method of accelerating the integration of parabolic systems by enforcing stability over large compound time-steps rather than over each of the constituent substeps. This method, Super Time-Stepping, proves to be very effective in applying a part of the Hall term up to a known critical value. The excess of the Hall term above this critical value is then included via a new scheme for pure Hall diffusion

A Three‐Dimensional Numerical Method for Modelling Weakly Ionized Plasmas

Astrophysical fluids under the influence of magnetic fields are often subjected to single- or two-fluid approximations. In the case of weakly ionized plasmas, however, this can be inappropriate due to distinct responses from the multiple constituent species to both collisional and non-collisional forces. As a result, in dense molecular clouds and protostellar accretion discs, for instance, the conductivity of the plasma may be highly anisotropic leading to phenomena such as Hall and ambipolar diffusion strongly influencing the dynamics

Nonideal Magnetohydrodynamic Turbulent Decay in Molecular Clouds

It is well known that non-ideal magnetohydrodynamic effects are important in the dynamics of molecular clouds: both ambipolar diffusion and possibly the Hall effect have been identified as significant. We present the results of a suite of simulations with a resolution of 512-cubed of turbulent decay in molecular clouds incorporating a simplified form of both ambipolar diffusion and the Hall effect simultaneously. The initial velocity field in the turbulence is varied from being super-Alfvénic and hypersonic, through to trans-Alfvénic but still supersonic

The European Central Bank and the Federal Reserve

In the 4 years of its existence, the European Central Bank (ECB) has made significant contributions to the macroeconomic stability of the euro area. This paper takes a critical look at the ECB and compares its institutional structure, policy framework, and operational procedures with those of the longer-established US central bank. We discuss the implications of various differences between the ECB and the Federal Reserve with a view toward identifying successful elements of the practices of both these institutions. The paper recommends that the ECB abandon the first pillar of its monetary policy strategy that affords a special role to monetary aggregates in the evaluation of financial market conditions. It also suggests that the Federal Reserve should follow the ECB\u27s lead and provide an explicit definition of price stability

Particle Acceleration at the Discontinuous Flow Boundary of Collimated Cylindrical Jets

We revisit the issue of particle acceleration at the interface between a collimated laminar jet and a static ambient medium. The contrast between standard diffusive scattering treatments and anomalous transport in synthetic field constructions is explored. A particular emphasis is placed on the necessity for physically consistent particle transport considerations. The temporal, spatial and spectral features of the process are discussed, in the context of potential UHECR production, as well as further observational consequences

Asset Prices in the Measurement of Inflation

The debate over including asset prices in the construction of an inflation statistic has attracted renewed attention in recent years. Virtually all of this (and earlier) work on incorporating asset prices into an aggregate price statistic has been motivated by a presumed, but unidentified transmission mechanism through which asset prices are leading indicators of inflation at the retail level. This paper takes an alternative, longer-term perspective on the issue and argues that the exclusion of asset prices introduces an excluded goods bias in the computation of the inflation statistic that is of interest to the monetary authority. This idea is implemented using a relatively modern statistical technique, a dynamic factor index. This statistical algorithm allows researchers to see through the excessively noisy asset price data that have frustrated earlier researchers who have attempted to integrate these prices into an aggregate measure