Volatility Library

Article by: David Ruppert
From: Statistics and Data Analysis for Financial Engineering
Published by: Springer New York
Date: 2010

“…financial markets data often exhibit volatility clustering, where time series show periods of high volatility and periods of low volatility;…. In fact, with economic and financial data, time-varying volatility is more common than constant volatility, and accurate modeling of time-varying volatility is of great importance in financial engineering.

“…ARMA models are used to model the conditional expectation of a process given the past, but in an ARMA model the conditional variance given the past is constant. What does this mean for, say, modeling stock returns? Suppose we have noticed that recent daily returns have been unusually volatile. We might expect that tomorrow’s return is also more variable than usual. However, an ARMA model cannot capture this type of behavior because its conditional variance is constant. So we need better time series models if we want to model the nonconstant volatility. In this chapter we look at GARCH time series models that are becoming widely used in econometrics and finance because they have randomly varying volatility.”

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Article by: Lyndon Lyons and Prof. Dr. Notger Carl
Published by: Würzburg-Schweinfurt University of Applied Sciences
Date: April 2005

“The volatility index, sometimes called by financial professionals and academics as
“the investor gauge of fear” has developed overtime to become one of the highlights
of modern day financial markets. Due to the many financial mishaps during the last
two decades such as LTCM (Long Term Capital Management), the Asian Crisis just
to name a few and also the discovery of the volatility skew, many financial experts
are seeing volatility risk as one of the prime and hidden risk factors on capital
markets. This paper will mainly emphasize on the developments in measuring and
estimating volatility with a concluding analysis of the historical time series of the new
volatility indices at the Deutsche Boerse.”

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Article by: Ahsan Amin
Published by: Infiniti Derivatives Technologies
Date: 13 Nov 2012

“In this paper, we solve the problem of solution of stochastic volatility models in which the volatility diffusion can be solved by a one dimensional Fokker-planck equation. We use one dimensional transition probabilities for the evolution of PDE of variance. We also find dynamics of evolution of expected value of any path dependent function of stochastic volatility variable along the PDE grid. Using this technique, we find the conditional expected values of moments of log of terminal asset price along every node of one dimensional forward Kolmogorov PDE. We use the conditional distribution of moments of above path integrals along the variance grid and use Edgeworth expansions to calculate the density of log of asset price. Main result of the paper gives dynamics of evolution of conditional expected value of a path dependent function of volatility (or any other SDE) at any node on the PDE grid using just one dimensional PDE if we can describe its one step conditional evolution between different nodes of the PDE.”

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