# On the Modeling of Financial Time Series

Aleksey Kutergin and Vladimir Filimonov

Abstract This paper discusses issues related to modeling of financial time series. We discuss so-called empirical “stylized facts” of real price time-series and the evolution of financial models from trivial random walk introduced by Louis Bachelier in 1900 to modern multifractal models, that nowadays are the most parsimonious and flexible models of stochastic volatility. We focus on a particular model of Multifractal Random Walk (MRW), which is the only continuous stochastic stationary causal process with exact multifractal properties and Gaussian infinitesimal increments. The paper presents a method of numerical simulation of realizations of MRW using the Circulant Embedding Method and discuss methods of its calibration.

Keywords Circulant Embedding Method • Estimation of parameters • Financial time series • Multifractal Random Walk • Numerical simulations • Stylized facts

Financial modeling is being one of the most actively evolved topics of quantitative finance for many decades. Having a numerous practical applications especially in the fields of derivative pricing or risk management, it is of an extreme interest of academic research as well. Financial markets are a global social system in which many agents make decisions, every minute being exposed to risk and uncertainty. Participants interact with each other trying to make profit, taking into account not only recent news and internal market events but also action of other participants as well. However the result of such complex behavior is reduced to a small set of entities, by far the most important of which is the price of some given asset.

A. Kutergin (H)

Prognoz Risk Lab, Perm, Russian Federation e-mail: kutergin@prognoz. ru

V. Filimonov

Department of Management, Technology and Economics ETH Zurich, Zurich, Switzerland e-mail: vfilimonov@ethz. ch

© Springer International Publishing Switzerland 2015

A. K. Bera et al. (eds.), Financial Econometrics and Empirical Market

Microstructure, DOI 10.1007/978-3-319-09946-0_____ 10

The first attempt to describe observable assets price dynamics was made by Bachelier in 1900 with his seminal work “Theorie de la speculation” (Bachelier 1900). Bachelier suggested that asset prices follows random walk. In other words, increments (returns) of the price are independent identically distributed (iid) random variables which he suggested to have Gaussian probability distribution. Having many merits, such simple model is not able to fully account for complex interaction of many random factors underling the price formation process. However, analytical tractability due to simple construction and underlying Gaussian distribution, allowed to construct on top of the random walk process many financial theories, such as Black and Scholes option pricing theory or Markowitz portfolio theory.

Rebounding from the naive random walk, the evolution of the financial modeling has brought a variety of models that are aimed to describe the complex statistical properties of real price time series, summarized in the so-called “stylized facts”. One of the most widely used is the “conditional volatility” models such as ARCH/- GARCH family, that model volatility as an autoregressive process on the past values of volatility and returns as well. Another class of models represent volatility as some stochastic process. The most interesting subclass of these “stochastic volatility” models are so-called multifractal models. Despite having simple construction, multifractal models are able to represent most of the empirically observed “stylized facts”. In this paper we focus on particularly one multifractal model, namely on the Multifractal Random Walk (MRW) that was proposed by Bacry, Delour and Muzy in 2000 (Bacry et al. 2001), which is the only continuous stochastic stationary causal process with exact multifractal properties and Gaussian infinitesimal increments. We describe the procedure of numerical simulation of the realization of MRW process and discuss issues related to estimation of parameters.

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