The price of a fixed-term option is the expected value of the payoff at the time of maturity. When not analytically available, the option price is computed using stochastic or deterministic numerical methods. The most common approach when using deterministic methods is to solve a backward partial differential equation (PDE) such as the Black-Scholes equation for the option value. The problem can alternatively be formulated based on a forward PDE for the probability of the asset value at the time of maturity. This enables simultaneous pricing of several contracts with different payoffs written on the same underlying asset. The main drawback is that the initial condition is a (non-smooth) Dirac function. We show that by using an analytical expansion of the solution for the first part of the time interval, and applying a high-order accurate radial basis function (RBF) approximation in space, we can derive a competitive forward pricing method. We evaluate the proposed method on European call options and barrier options, and show that even for just one payoff it is more efficient than solving the corresponding backward PDE.
Self-consistent modelling of ICRH requires calculations of the wave field consistent with the distribution function of the resonant species. Because of the difference in time scales for wave propagation and the evolution of distribution functions this is commonly done by iterations. A robust code SELFO-light, suitable for routine calculations was recently developed, based on coupling a 1D time dependent Fokker-Planck code with the global wave solver LION using a FEM. Here the structure of an upgraded version of the SELFO-light code is presented calculating the distribution function with a 2D Fokker-Planck code. This requires new interfaces calculating the quasi-linear diffusion coefficient from the wave field and the susceptibility tensor from distribution functions.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.