Multiplicative noise, fast convolution and pricing

Abstract

In this work we detail the application of a fast convolution algorithm computing high dimensional integrals to the context of multiplicative noise stochastic processes. The algorithm provides a numerical solution to the problem of characterizing conditional probability density functions at arbitrary time, and we applied it successfully to quadratic and piecewise linear diffusion processes. The ability in reproducing statistical features of financial return time series, such as thickness of the tails and scaling properties, makes this processes appealing for option pricing. Since exact analytical results are missing, we exploit the fast convolution as a numerical method alternative to the Monte Carlo simulation both in objective and risk neutral settings. In numerical sections we document how fast convolution outperforms Monte Carlo both in velocity and efficiency terms.JEL codes: C63, G13.

Reduction and reconstruction of stochastic differential equations via symmetries

“…Beyond the large number of applications of Ornstein-Uhlenbeck process and of linear SDEs and their spatial transformations, the Kolmogorov-Pearson class (12) has notable applications to finance (see [8,41]), physics (see [17,44]) and biology (see [20]). Moreover, there is a growing interest in the study of statistical inference (see [16]), in the analytical and spectral properties of the Kolmogorov equation associated with (12) (see [4]) and in the development of efficient numerical algorithms for its numerical simulation (see [9]). Finally the Kolmogorov-Pearson diffusions are examples of "polynomial processes" that are becoming quite popular in financial mathematics ( [13]).…”

“…These examples also suggest that the knowledge of closed-form expressions for some mathematical objects related with SDEs can be useful in order to formulate faster or more stable algorithms for numerical simulation (see e.g. [9,23,33]), to propose better estimators for statistical inference (see e.g. [5,6,16]) or to reduce the complexity of the models using asymptotic expansions or perturbation theory techniques (see e.g.…”