Numerical Methods

Notes on the SWIFT method based on Shannon Wavelets for Option Pricing – Revisited ArXiv ID: 2401.01758 “View on arXiv” Authors: Unknown Abstract This note revisits the SWIFT method based on Shannon wavelets to price European options under models with a known characteristic function in 2023. In particular, it discusses some possible improvements and exposes some concrete drawbacks of the method. Keywords: Shannon Wavelets, Option Pricing, Characteristic Function, Spectral Methods, Numerical Methods, Derivatives ...

Root-finding: from Newton to Halley and beyond ArXiv ID: 2312.12305 “View on arXiv” Authors: Unknown Abstract We give a new improvement over Newton’s method for root-finding, when the function in question is doubly differentiable. It generally exhibits faster and more reliable convergence. It can be also be thought of as a correction to Halley’s method, as this can exhibit undesirable behaviour. Keywords: Root-finding, Newton’s method, Optimization algorithms, Numerical methods, General / Quantitative Methods ...

From characteristic functions to multivariate distribution functions and European option prices by the damped COS method ArXiv ID: 2307.12843 “View on arXiv” Authors: Unknown Abstract We provide a unified framework to obtain numerically certain quantities, such as the distribution function, absolute moments and prices of financial options, from the characteristic function of some (unknown) probability density function using the Fourier-cosine expansion (COS) method. The classical COS method is numerically very efficient in one-dimension, but it cannot deal very well with certain integrands in general dimensions. Therefore, we introduce the damped COS method, which can handle a large class of integrands very efficiently. We prove the convergence of the (damped) COS method and study its order of convergence. The method converges exponentially if the characteristic function decays exponentially. To apply the (damped) COS method, one has to specify two parameters: a truncation range for the multivariate density and the number of terms to approximate the truncated density by a cosine series. We provide an explicit formula for the truncation range and an implicit formula for the number of terms. Numerical experiments up to five dimensions confirm the theoretical results. ...

Efficient inverse $Z$-transform: sufficient conditions ArXiv ID: 2305.10725 “View on arXiv” Authors: Unknown Abstract We derive several sets of sufficient conditions for applicability of the new efficient numerical realization of the inverse $Z$-transform. For large $n$, the complexity of the new scheme is dozens of times smaller than the complexity of the trapezoid rule. As applications, pricing of European options and single barrier options with discrete monitoring are considered; applications to more general options with barrier-lookback features are outlined. In the case of sectorial transition operators, hence, for symmetric Lévy models, the proof is straightforward. In the case of non-symmetric Lévy models, we construct a non-linear deformation of the dual space, which makes the transition operator sectorial, with an arbitrary small opening angle, and justify the new realization. We impose mild conditions which are satisfied for wide classes of non-symmetric Stieltjes-Lévy processes. ...