**Volatility Estimation of Financial Returns Using Risk-Sensitive Particle Filters**

**Karel MUNDNICH, Marcos E. ORCHARD, Jorge F. SILVA, Patricio PARADA **

Electrical Engineering Department, Universidad de Chile

Av. Tupper 2007, Santiago, 8370451, CHILE

**Abstract**: This article presents and analyzes the implementation of risk-sensitive particle filtering algorithm for volatility estimation of continuously compounded returns of financial assets. The proposed approach uses a stochastic state-space representation for the evolution of the dynamic system -the unobserved generalized autoregressive conditional heteroskedasticity (uGARCH)model- and an Inverse Gamma distribution as risk functional (and importance density distribution) to ensure the allocation of particles in regions of the state-space that are associated to sudden changes in the volatility of the system. A set of ad-hoc performance and entropy-based measures is used to compare the performance of this scheme with respect to a classic implementation of sequential Monte Carlo methods, both in terms of accuracy and precision of the resulting volatility estimates; considering for this purpose data sets generated in a blind-test format with GARCH structures and time-varying parameters.

**Keywords**: Bayesian estimation, stochastic volatility, particle filters, entropy.

**>>Full Text**

**CITE THIS PAPER AS**:

Karel MUNDNICH, Marcos E. ORCHARD, Jorge F. SILVA, Patricio PARADA**, Volatility Estimation of Financial Returns Using Risk-Sensitive Particle Filters,**

*Studies in Informatics and Control*, ISSN 1220-1766, vol. 22 (3), pp. 297-306, 2013. https://doi.org/10.24846/v22i3y201306

**Introduction**

In finance, the term volatility refers to the standard deviation of the continuously compounded returns of a financial asset. Calling *p* the price or level of a financial asset, the returns of the financial index are defined according to

To define the concept of volatility in a proper manner, it is first necessary to introduce the concept of variance conditional to the returns:

where is the conditional expectation of the returns. Considering that is the -algebra defined by the observations up to time , the financial returns can be modeled as a stochastic Gaussian process

(3)

where and thus, . Consequently, one can interpret volatility as a measure of the intensity of the variations undergone by a given financial asset or, equivalently, a measure of the risk associated to that asset. It is important to mention that different definitions may be associated to the concept of financial volatility; including “historic volatility”, which refers to the volatility experimented by an asset within a specific time period, and “current volatility”, which refers to the volatility of current prices. This work is focused on the estimation of the latter, which is assumed to be time-variant.

From an information technology perspective, volatility estimation plays an important role in the development of trading algorithms for today’s stock market. Although several strategies that are currently in use include arbitrage and decision-making processes based on trend (moving average) evaluation, just few of those strategies are able to quantify (and manage) the risk that is associated to high volatility periods from the perspective of fault detection (and control) systems.

Considering Eq.(1)-(3), volatility estimation can be seen as a Bayesian filtering problem. In this regard, it is important to note that particle filters (also known as sequential Monte Carlo methods) have gained notable attention in the past few years, due to simple implementation and the excellent results they confer. Contrary to the Kalman filter, Particle Filters (PF) can be used to track the state trajectory in non-linear or non-Gaussian systems by approximating the state probability density function through a set of samples (called particles) and their correspondent weights. These characteristics make PF and their variants an interesting choice to solve the stochastic volatility estimation problem.

In literature, few examples and applications of Bayesian filtering applied in economics and finance may be found. (Harvey and Koopman, 2009) introduces a survey on the use of state-space models in economics applied to CPI and GDP modeling. They also describe some of the most common volatility models, including the GARCH model (Bollerslev, 1986) and stochastic volatility (SV) models. He briefly describes the Kalman filter and PF, but does not show any results on stochastic volatility estimation.

(Tsay, 2010) describes the use of Gibbs sampling in volatility estimation of SV models, but there is no mention about the use of PF for volatility estimation. (Tobar and Orchard, 2012) introduce the uGARCH model, stochastic variation of the GARCH(1,1) model, and use Kalman filters (classic and extended) and PF to compare the results of estimating stochastic volatility in the uGARCH and log-VE models. On the other hand, several examples can be found in literature that incorporate information-theoretic measures to analyze the output of particle filtering algorithms (Ajgl and Šimandl, 2011; Lanz, 2007; Boers *et al.*, 2010; Skoglar*et al.*, 2009). Most of these are related to uncertainty characterization, optimality testing, and evaluation of control strategies. In (Orchard *et al.*, 2012), a detection scheme using entropy as a particle filter output processing tool is proposed.

This article proposes a risk-sensitive particle filter approach for volatility estimation. This approach also explores the use of differential entropy to implement a risk management scheme that detects high-volatility clusters based on estimates of the state probability density function. Validation of the proposed scheme is performed using ground truth data that is generated via a GARCH model. The structure of this document is as follows: In Section 2, the uGARCH model is introduced, and sub-optimal Bayesian filtering routines are referenced. In the same section, the concept of entropy is included, and insight is given towards its use in a detection scheme. In Section 3, the held experiments are described, and results of these experiments are displayed. Finally, in Section 4, conclusions are presented.

**REFERENCES:**

- AJGL J., M. ŠIMANDL,
**Particle Based Probability Density Fusion with Differential Shannon Entropy Criterion**Proceeding of the 14th International Conference on Information Fusion ISIF, 2011, pp. 803-810, Chicago, Illinois, USA. - ARULAMPALAM, M. S., S. MASKELL, N. GORDON, T. CLAPP,
**A Tutorial on Particle Filters for On-line Nonlinear / Non-Gaussian Bayesian Tracking**, IEEE Transactions on Signal Processing, vol. 50, no. 2, 2002, pp. 174-188. - BOERS, Y., J. N. DRIESSEN, A. BAGCHI, P. K. MANDAL,
**Particle Filter Based Entropy**, Proceedings of FUSION 2010, Edinburgh, UK. - BOLLERSLEV, T.,
**Generalized Auto-regressive Conditional Heterokedasticity**, Journal of Econometrics, vol. 31, no. 3, 1986, pp. 307-327. - COVER, T., J. THOMAS,
**Elements of Information Theory**, Wiley Interscience, New York, 1991. - DOUCET, A.,
**On Sequential Monte Carlo Methods for Bayesian Filtering**, Technical Report, Engineering Department, Univ. Cambridge, 1998, UK. - DOUCET, A., N. DE FREITAS, N. GORDON,
**An Introduction to Sequential Monte Carlo Methods**, in Sequential Monte Carlo Methods in Practice, A. Doucet, N. de Freitas, and N. Gordon, Eds. NY: Springer-Verlag. - HARTLEY, R. V. L.,
**Transmission of Information**, International Congress of Telegraphy and Telephony, Lake Como, Italy, 1927. - HARVEY, A., S. M. KOOPMAN,
**Unobserved Components Models in Economics and Finance**, Control Systems, IEEE, vol. 29, no. 6, 2009, pp. 71-81. - HAYKIN, S.,
**Neural Networks and Learning Machines**, Prentice Hall, vol. 10, 3^{rd}edition, 2009. - LANZ, O.,
**An Information Theoretic Rule for Sample Size Adaptation in Particle Filtering**, Proceedings of the 14th International Conference on Image Analysis and Processing (ICIAP), 2007, pp. 317–322. - MANDELBROT, B.,
**The Variation of Certain Speculative Prices**, The journal of Business, vol. 36, no. 4, 1963, pp. 394-419. - MASKELL, S., GORDON, N.,
**A Tutorial on Particle Filters for On-line Nonlinear/Non-Gaussian Bayesian Tracking**, Technical Report, QinetiQ Ltd., and Cambridge University, 2001, UK. - ORCHARD, M., B. OLIVARES, M. CERDA, J. F. SILVA,
**Anomaly Detection based on Information-Theoretic Measures and Particle Filtering Algorithms**, Annual Conference of the PHM Society 2012, Minneapolis, USA. - ORGUNER, U.,
**Entropy Calculation in Particle Filters**, IEEE Signal Processing and Communications Applications Conference, 2009. - SKOGLAR, P., U. ORGUNER, F. GUSTAFSSON,
**On Information Measures based on Particle Mixture for Optimal Bearings-only Tracking**, Proceedings of IEEE Aerospace Conference 2009, Big Sky, Montana, USA - THRUN, S., J. LANGFORD, V. VERMA,
**Risk Sensitive Particle Filters**, In Advances in Neural Information Processing Systems, vol. 14, 2002. - TOBAR, F., M. ORCHARD,
**Study of Financial Systems Volatility Using Suboptimal Estimation Algorithms**, Studies in Informatics and Control, vol. 21, Issue 1, March 2012, pp. 59-66. - TSAY, R.,
**Analysis of Financial Time Series**, John Wiley & Sons, 3^{rd}edition, 2010.