Role of scaling in the statistical modeling of finance

Details Role of scaling in the statistical modeling of finance Role of scaling in the statistical modeling of finance

2008-04-02

arxiv.org/abs/0804.0331v1

Pramana - Journal of Physics 71, 341 (2008)

Economy

Quantitative Finance

Statistical Finance

Based on the Key Note lecture by A.L. Stella at the Conference on ``Statistical Physics Approaches to Multi-Disciplinary Probl

Scientific paper

10.1007/s12043-008-0167-0

Modeling the evolution of a financial index as a stochastic process is a problem awaiting a full, satisfactory solution since it was first formulated by Bachelier in 1900. Here it is shown that the scaling with time of the return probability density function sampled from the historical series suggests a successful model. The resulting stochastic process is a heteroskedastic, non-Markovian martingale, which can be used to simulate index evolution on the basis of an auto-regressive strategy. Results are fully consistent with volatility clustering and with the multi-scaling properties of the return distribution. The idea of basing the process construction on scaling, and the construction itself, are closely inspired by the probabilistic renormalization group approach of statistical mechanics and by a recent formulation of the central limit theorem for sums of strongly correlated random variables.

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