SINH-acceleration: efficient evaluation of probability distributions, option pricing, and Monte-Carlo simulations
For quantitative finance practitioners, this method offers a fast and accurate numerical technique for pricing and calibration in complex models, though it is an incremental improvement over existing Fourier-based methods.
The paper introduces a change of variables using sinh and the simplified trapezoid rule to efficiently evaluate probability distributions and option prices via Fourier transforms, achieving fast and accurate results across Lévy, Heston, CIR, and regime-switching models, with applications to calibration and Monte Carlo simulations.
Characteristic functions of several popular classes of distributions and processes admit analytic continuation into unions of strips and open coni around $\mathbb{R}\subset \mathbb{C}$. The Fourier transform techniques reduces calculation of probability distributions and option prices to evaluation of integrals whose integrands are analytic in domains enjoying these properties. In the paper, we suggest to use changes of variables of the form $ξ=\sqrt{-1}ω_1+b\sinh (\sqrt{-1}ω+y)$ and the simplified trapezoid rule to evaluate the integrals accurately and fast. We formulate the general scheme, and apply the scheme for calculation probability distributions and pricing European options in Lévy models, the Heston model, the CIR model, and a Lévy model with the CIR-subordinator. We outline applications to fast and accurate calibration procedures and Monte Carlo simulations in Lévy models, regime switching Lévy models that can account for stochastic drift, volatility and skewness, and the Heston model. For calculation of quantiles in the tails using the Newton or bisection method, it suffices to precalculate several hundred of values of the characteristic exponent at points of an appropriate grid ({\em conformal principal components}) and use these values in formulas for cpdf and pdf.