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Stochastic volatility

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  • Torben G. Andersen
  • Luca Benzoni

Abstract

Given the importance of return volatility on a number of practical financial management decisions, the efforts to provide good real- time estimates and forecasts of current and future volatility have been extensive. The main framework used in this context involves stochastic volatility models. In a broad sense, this model class includes GARCH, but we focus on a narrower set of specifications in which volatility follows its own random process, as is common in models originating within financial economics. The distinguishing feature of these specifications is that volatility, being inherently unobservable and subject to independent random shocks, is not measurable with respect to observable information. In what follows, we refer to these models as genuine stochastic volatility models. Much modern asset pricing theory is built on continuous- time models. The natural concept of volatility within this setting is that of genuine stochastic volatility. For example, stochastic-volatility (jump-) diffusions have provided a useful tool for a wide range of applications, including the pricing of options and other derivatives, the modeling of the term structure of risk-free interest rates, and the pricing of foreign currencies and defaultable bonds. The increased use of intraday transaction data for construction of so-called realized volatility measures provides additional impetus for considering genuine stochastic volatility models. As we demonstrate below, the realized volatility approach is closely associated with the continuous-time stochastic volatility framework of financial economics. There are some unique challenges in dealing with genuine stochastic volatility models. For example, volatility is truly latent and this feature complicates estimation and inference. Further, the presence of an additional state variable - volatility - renders the model less tractable from an analytic perspective. We examine how such challenges have been addressed through development of new estimation methods and imposition of model restrictions allowing for closed-form solutions while remaining consistent with the dominant empirical features of the data.

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Bibliographic Info

Paper provided by Federal Reserve Bank of Chicago in its series Working Paper Series with number WP-09-04.

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Date of creation: 2009
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Handle: RePEc:fip:fedhwp:wp-09-04

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Keywords: Stochastic analysis;

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Cited by:
  1. Mc Manus, Des & Watt, David, 1999. "Estimating One-Factor Models of Short-Term Interest Rates," Working Papers 99-18, Bank of Canada.
  2. Asea, Patrick K. & Ncube, Mthuli, 1998. "Heterogeneous information arrival and option pricing," Journal of Econometrics, Elsevier, vol. 83(1-2), pages 291-323.
  3. Théoret, Raymond & Racicot, François-Éric, 2010. "Forecasting stochastic Volatility using the Kalman filter: an application to Canadian Interest Rates and Price-Earnings Ratio," MPRA Paper 35911, University Library of Munich, Germany.
  4. Jacob Boudoukh & Matthew Richardson, 1999. "A Multifactor, Nonlinear, Continuous-Time Model of Interest Rate Volatility," NBER Working Papers 7213, National Bureau of Economic Research, Inc.
  5. Peter F. Christoffersen & Francis X. Diebold, 1998. "How Relevant is Volatility Forecasting for Financial Risk Management?," NBER Working Papers 6844, National Bureau of Economic Research, Inc.
  6. Torben G. Andersen & Tim Bollerslev & Francis X. Diebold, 2002. "Parametric and Nonparametric Volatility Measurement," Center for Financial Institutions Working Papers 02-27, Wharton School Center for Financial Institutions, University of Pennsylvania.
  7. Tsyplakov, Alexander, 2010. "Revealing the arcane: an introduction to the art of stochastic volatility models," MPRA Paper 25511, University Library of Munich, Germany.
  8. Patrick Asea & Mthuli Nube, 1997. "Heterogeneous Information Arrival and Option Pricing," UCLA Economics Working Papers 763, UCLA Department of Economics.
  9. Pierre Collin-Dufresne & Christopher S. Jones & Robert S. Goldstein, 2004. "Can Interest Rate Volatility be Extracted from the Cross Section of Bond Yields? An Investigation of Unspanned Stochastic Volatility," NBER Working Papers 10756, National Bureau of Economic Research, Inc.

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