Multiplicative noise, fast convolution and pricing

Abstract

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

Keywords:

• Computational finance
• Stochastic processes
• Non-Gaussian option pricing
• Numerical methods for option pricing

Keywords:

• JEL Classification
• C63
• G13

Acknowledgements

The authors acknowledge the support of the Scuola Normale Superiore Grant ‘Giovani Ricercatori (2011/2012)’; S.C. also acknowledges the financial support of the Swiss Finance Institute and of FinRisk.

Last but not least, they warmly thank Melissa Lehman and Erik Lundberg for their help provided in editing the final version of the manuscript.

Notes

†By virtue of the properties of g, τ is a monotonically increasing function of t, implying the well-definiteness of the inverse function t(τ).

†From now on we will drop the explicit dependence on the time variable τ.

†Random number generators and FFT algorithms are provided by GNU Scientific Library.

‡The sign function is defined according to the convention sign(0) = 0.

†Gabaix (2009) comprehensively reviews power law regularity in economics and finance, while Gabaix et al. (2003) propose a model providing theoretical explanation.

‡The relation between Borland's parameter q and α is given by q = α + 1.