**Study of Financial Systems Volatility Using Suboptimal Estimation Algorithms**

**Felipe A. Tobar**

*Electrical and Electronic Engineering Department, Imperial College London*

London SW7 2AZ, U.K.

**Marcos E. Orchard**

*Electrical Engineering Department, Universidad de Chile*

Santiago 8370451, Chile

**Abstract**:

This paper presents and implements a novel stochastic volatility (SV) model, based on the structure of the GARCH model, to describe the relationship between an observed financial return series and its standard deviation, namely volatility. The proposed approach has been compared to the standard GARCH as the underlying modeling structure within a particle-filtering-based scheme for state estimation. The proposed structure has been implemented to estimate the volatility of both a simulated return series and the NASDAQ Composite index during the period July 21, 2008 – July 17, 2009. The results of this procedure show that the parameters of the GARCH model can be used in the uGARCH structure, allowing the latter representing the hidden volatility as accurate as the standard GARCH, but also providing an estimate of the whole probability density.

**Keywords**:

Parameter identification, state estimation, time-series analysis, financial systems, particle filters, GARCH model.

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**CITE THIS PAPER AS**:

Felipe A. TOBAR, Marcos E. ORCHARD, **Study of Financial Systems Volatility Using Suboptimal Estimation Algorithms**, *Studies in Informatics and Control*, ISSN 1220-1766, vol. 21 (1), pp. 59-66, 2012.

**1. Introduction**

In the field of Mathematical Finance -i.e., the branch of Applied Mathematics that aims to model the behavior of variables in a financial system- it is of great interest to quantify the stability of an asset given its realized returns. A measure of this stability is the so-called volatility, which refers to the standard deviation of the continuously compounded returns of a financial instrument.

From a Bayesian standpoint, it is possible to relate the volatility to the observed returns by a state-space model, and in order to account for the returns stylized facts, such as higher order moments and volatility clustering, this model should include non-linearities, non‑Gaussian innovations, and unobservable states. A structure that is commonly used to model volatility of financial instruments is the Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) model proposed by [1], which assumes that the volatility evolves in a deterministic fashion given the observations.

On the other hand, the stochastic volatility model considers that the volatility is a hidden state driven by an innovation process. Although the models that consider stochastic variables allow representing the uncertainty of some observed phenomena, the identification of its parameters is not straightforward due to a likelihood function defined as an intractable integral. Moreover, even if a suitable set of parameters is available, when themodel is non-linear/non-Gaussian the estimation of the hidden states turns outinto a difficult issue, since for this kind of structures there is no algorithm that guarantees optimal estimation as the Kalman filter does for the linear/Gaussian case. Consequently, sub-optimal filtering and identification approaches are needed to overcome the estimation of the volatility when a hidden-state, non-linear/non-Gaussian model is used.

The implementation of numerical techniques to estimate unobserved components has received the attention of several scientific disciplines due to the high computational power, and increasing storage capacity, developed throughout the last decades. Among the suboptimal techniques for state estimation, particle filters (PF) have recently caught the attention of the scientific community [2]-[5].

These methods are capable to approximate expectations w.r.t. a sequence of time-varying, growing dimension, probability density functions (pdf) through a finite set of weighted samples. In the case of the filtering problem, this pdf is the posterior density of the state. PF have been widely used -particularly in Financial Mathematics- due to their flexibility, and capability to be implemented along dynamical models characterized, for instance, by non-linearities, jump-diffusions, and non-Gaussian or multiplicative noise [6]-[9].

This paper is organized as follows. Section 2 introduces the concept of volatility, and presents two important classes of volatility models. Section 3 proposes a novel stochastic volatility model based on the structure of a widely used deterministic model, while Section 4 presents the results of both the previous and the introduced approach. Finally, Section 5 states the concluding remarks and suggests areas of further study according to the results obtained.

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