SINH-acceleration: efficient evaluation of probability distributions, option pricing, 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 $\xi=\sqrt{-1}\omega_1+b\sinh (\sqrt{-1}\omega+y)$ 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 L\'evy models, the Heston model, the CIR model, and a L\'evy model with the CIR-subordinator. We outline applications to fast and accurate calibration procedures and Monte Carlo simulations in L\'evy models, regime switching L\'evy 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.