Time Series with Stochastic Volatility

Consider a multivariate time series where each component series is assumed to be a linear mixture of latent mutually independent stationary time series. Classical independent component analysis (ICA) tools, such as fastICA, are often used to extract latent series, but they don't utilize any information on temporal dependence. Also financial time series often have periods of low and high volatility. In such settings second order source separation methods, such as SOBI, fail. We review here some classical methods used for time series with stochastic volatility, and suggest modifications of them by proposing a family of vSOBI estimators. These estimators use different nonlinearity functions to capture nonlinear autocorrelation of the time series and extract the independent components. Simulation study shows that the proposed method outperforms the existing methods when latent components follow GARCH and SV models. This paper is an invited extended version of the paper presented at the CDAM 2016 conference.

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Financial Models

10.26686/wgtn.16946047 ◽

2021 ◽

Author(s):

◽

Konstantin Kvatch

Keyword(s):

Time Series ◽

Stochastic Volatility ◽

Real World ◽

Financial Time Series ◽

Closed Form Solution ◽

Financial Industry ◽

The Real ◽

Financial Time ◽

Black Scholes ◽

Standard Models

<p>The thesis will have two main parts. First, let us start with an example. In finance, the standard version of the Black-Scholes formula is a beautiful closed form solution used to price European options. This famous formula is ingenious, but has a flaw that relegates it to something that should be admired, and perhaps not be used in the real world. It relies on the assumption that prices of shares evolve according to geometric Brownian motion. This means that we are willing to accept that extreme shocks to prices are almost impossible. Is this a realistic assumption? Of course not. The stock market crashes of 1929, 1987 are great examples to show that extreme events do happen. More recently, the 1997 Asian crisis and 2000 crash of the NASDAQ show that in addition, such events are not so rare. These jumps occur even more frequently and are larger in magnitude for share prices of individual companies. This problem is by no means new, and a plethora of models and pricing techniques have been developed. The standard Black-Scholes formula is just one example, but this is simply illustration of the matter at hand. The process that we use to model a financial time series is of paramount importance, whether we do it for forecasting purposes or for pricing financial derivatives. If we choose to use a model that does not capture the key empirical aspects of the data, then any subsequent inference may be very unfavourably biased. It is because of this problem that we should investigate the more standard modeling that assumes continuity and normal or log-normal distribution of financial time series. We will begin from the very basics and we will see that this is a wonderful piece of theory, deserving of the reputation it has in being simple, groundbreaking and extremely useful. This work should bring us to a position where we can evaluate a second goal. Stochastic processes with jumps and "heavy-tails" have existed for some time, but have begun to filter through to the financial industry only recently. This lag is due to the perceived added conceptual difficulty in the introduction of such models, although we will see that this should not be the case. There is plenty of real evidence that nancial time series exhibit discontinuous behaviour and that these series are far from normally or log-normally distributed. Rather than looking at standard models as correct, and jump or stochastic volatility models as complicated, we should look upon standard models as educational but not sufficient for the real world. Stochastic volatility or jump models should instead be viewed as natural. The theme of the thesis is the importance of choosing a correct model for the underlying process. Although we may speak of the implications of some models to hedging, we will not actually look at specific hedging techniques. The particular aspect of pricing is also not considered in full scope although we will see the Black-Scholes pricing formula. We will consider that the main problem is to specify the model correctly where the method of pricing is a subsequent technicality. In examples we may take pricing tools like Monte-Carlo simulation as a given. We will not strive for full generality or formality, but rather take a physical approach and aim for clarity and understanding. Let us now move on to the beginning, with the introduction of our primary source of randomness.</p>

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Linear and nonlinear linkage of conditional stochastic volatility of interbank interest rates: Empirical evidence of the BRICS countries

BRICS Journal of Economics ◽

10.38050/2712-7508-2021-2-1 ◽

2021 ◽

Vol 2 (2) ◽

pp. 4-16

Author(s):

Zouhaier Dhifaoui ◽

Faicel Gasmi

Keyword(s):

Time Series ◽

Interest Rates ◽

Stochastic Volatility ◽

Causal Relationship ◽

Granger Causality ◽

Granger Causality Test ◽

Causality Test ◽

Brics Countries ◽

Volatility Model ◽

Linear And Nonlinear

The purpose of this article is to detect a possible linear and nonlinear causal relationship between the conditional stochastic volatility of log return of interbank interest rates for the BRICS countries in the period from January 2015 to October 2018. To extract the volatility of the analyzed time series, we use a stochastic volatility model with moving average innovations. To test a causal relationship between the estimated stochastic volatilities, two steps are applied. First, we used the Granger causality test and a vector autoregressive model (VAR). Secondly, we applied the nonlinear Granger causality test to the raw data to explore a new nonlinear causal relationship between stochastic volatility time series, and also applied it to the residual of the VAR model to confirm the causality detected in the first step. This study demonstrates the existence of some unidirectional/bidirectional linear/nonlinear causal relationships between the studied stochastic volatility time series.

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A note on stochastic volatility model estimation

Brazilian Review of Finance ◽

10.12660/rbfin.v17n4.2019.79892 ◽

2019 ◽

Vol 17 (4) ◽

pp. 22

Author(s):

Omar Abbara ◽

Mauricio Zevallos

Keyword(s):

Time Series ◽

Monte Carlo ◽

Stochastic Volatility ◽

Stochastic Volatility Model ◽

Parameter Estimates ◽

Model Estimation ◽

Stochastic Volatility Models ◽

Volatility Models ◽

Volatility Model ◽

Monte Carlo Evaluation

<p>The paper assesses the method proposed by Shumway and Stoffer (2006, Chapter 6, Section 10) to estimate the parameters and volatility of stochastic volatility models. First, the paper presents a Monte Carlo evaluation of the parameter estimates considering several distributions for the perturbations in the observation equation. Second, the method is assessed empirically, through backtesting evaluation of VaR forecasts of the S&P 500 time series returns. In both analyses, the paper also evaluates the convenience of using the Fuller transformation.</p>

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