Stochastic Volatility: Origins and Overview

Stochastic volatility is the main way time-varying volatility is modelled in financial markets. The development of stochastic volatility is reviewed, placing it in a modeling and historical context. Some recent trends in the literature are highlighted.

[1]  S. Taylor.  Financial Returns Modelled by the Product of Two Stochastic Processes , 1961 .

[2]  B. Mandlebrot.  The Variation of Certain Speculative Prices , 1963 .

[3]  F. Eugene.  FAMA, . The Behavior of Stock-Market Prices, Journal of Business, , . , 1965 .

[4]  E. Fama.  The Behavior of Stock-Market Prices , 1965 .

[5]  Barr Rosenberg..  The Behavior of Random Variables with Nonstationary Variance and the Distribution of Security Prices , 1972 .

[6]  F. Black,et al.  The Valuation of Option Contracts and a Test of Market Efficiency , 1972 .

[7]  P. Clark.  A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices , 1973 .

[8]  S. Afriat,et al.  Measurement of the purchasing power of incomes with linear expansion data , 1974 .

[9]  R. C. Merton,et al.  Option pricing when underlying stock returns are discontinuous , 1976 .

[10]  T. W. Epps,et al.  The Stochastic Dependence of Security Price Changes and Transaction Volumes: Implications for the Mixture-of-Distributions Hypothesis , 1976 .

[11]  L. Rogers,et al.  Diffusions, Markov processes, and martingales , 1979 .

[12]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[13]  George Tauchen,et al.  THE PRICE VARIABILITY-VOLUME RELATIONSHIP ON SPECULATIVE MARKETS , 1983 .

[14]  F. Diebold,et al.  The dynamics of exchange rate volatility: a multivariate latent factor ARCH model , 1986 .

[15]  James B. Wiggins.  Option values under stochastic volatility: Theory and empirical estimates , 1987 .

[16]  D. Shanno,et al.  Option Pricing when the Variance Is Changing , 1987, Journal of Financial and Quantitative Analysis.

[17]  Alan G. White,et al.  The Pricing of Options on Assets with Stochastic Volatilities , 1987 .

[18]  M. Yor.  DIFFUSIONS, MARKOV PROCESSES AND MARTINGALES: Volume 2: Itô Calculus , 1989 .

[19]  S. Turnbull,et al.  Pricing foreign currency options with stochastic volatility , 1990 .

[20]  Daniel B. Nelson.  ARCH models as diffusion approximations , 1990 .

[21]  D. Duffie,et al.  Simulated Moments Estimation of Markov Models of Asset Prices , 1990 .

[22]  Kerry Back,et al.  Asset pricing for general processes , 1991 .

[23]  E. Stein,et al.  Stock Price Distributions with Stochastic Volatility: An Analytic Approach , 1991 .

[24]  S. Heston.  A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options , 1993 .

[25]  N. Gordon,et al.  Novel approach to nonlinear/non-Gaussian Bayesian state estimation , 1993 .

[26]  J. Richard,et al.  Accelerated gaussian importance sampler with application to dynamic latent variable models , 1993 .

[27]  David S. Bates.  Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Thephlx Deutschemark Options , 1993 .

[28]  Neil Shephard,et al.  Fitting Nonlinear Time-Series Models with Applications to Stochastic Variance Models , 1993 .

[29]  Dean P. Foster,et al.  Continuous Record Asymptotics for Rolling Sample Variance Estimators , 1994 .

[30]  Enrique Sentana,et al.  Volatiltiy and Links between National Stock Markets , 1990 .

[31]  Peter E. Rossi,et al.  Bayesian Analysis of Stochastic Volatility Models: Comments: Reply , 1994 .

[32]  N. Shephard,et al.  Multivariate stochastic variance models , 1994 .

[33]  N. Shephard,et al.  Stochastic Volatility: Likelihood Inference And Comparison With Arch Models , 1996 .

[34]  A. Gallant,et al.  Which Moments to Match? , 1995, Econometric Theory.

[35]  A. Pedersen.  A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations , 1995 .

[36]  T. Andersen.  Return Volatility and Trading Volume: An Information Flow Interpretation of Stochastic Volatility , 1996 .

[37]  É. Renault.  Econometric Models of Option Pricing Errors , 1996 .

[38]  Bent E. Sørensen,et al.  GMM Estimation of a Stochastic Volatility Model: A Monte Carlo Study , 1996 .

[39]  R. Engle,et al.  11 – Estimating Diffusion Models of Stochastic Volatility , 1996 .

[40]  A. Harvey,et al.  5 Stochastic volatility , 1996 .

[41]  T. Andersen,et al.  Estimating continuous-time stochastic volatility models of the short-term interest rate , 1997 .

[42]  Pedro Santa-Clara,et al.  Simulated Likeliehood Estimation of Diffusions With an Application to the Short Tem Interest Rate , 1997 .

[43]  K. Singleton,et al.  Specification Analysis of Affine Term Structure Models , 1997 .

[44]  A. Gallant,et al.  Estimation of Stochastic Volatility Models with Diagnostics , 1995 .

[45]  L. Harris,et al.  A maximum likelihood approach for non-Gaussian stochastic volatility models , 1998 .

[46]  T. Bollerslev,et al.  Deutsche Mark–Dollar Volatility: Intraday Activity Patterns, Macroeconomic Announcements, and Longer Run Dependencies , 1998 .

[47]  F. Breidt,et al.  The detection and estimation of long memory in stochastic volatility , 1998 .

[48]  N. Shephard,et al.  Likelihood INference for Discretely Observed Non-linear Diffusions , 2001 .

[49]  David S. Bates.  Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options , 1998 .

[50]  A. Gallant,et al.  Reprojecting Partially Observed Systems with Application to Interest Rate Diffusions , 1998 .

[51]  F. Comte,et al.  Long memory in continuous‐time stochastic volatility models , 1998 .

[52]  Peter E. Rossi,et al.  Stochastic Volatility: Univariate and Multivariate Extensions , 1999 .

[53]  M. Pitt,et al.  Filtering via Simulation: Auxiliary Particle Filters , 1999 .

[54]  D. Duffie,et al.  Transform Analysis and Asset Pricing for Affine Jump-Diffusions , 1999 .

[55]  Simon,et al.  Bayesian Estimation of Continuous-Time Finance Models 1 Introduction , 1999 .

[56]  F. Diebold,et al.  The Distribution of Exchange Rate Volatility , 1999 .

[57]  E. Ghysels,et al.  A study towards a unified approach to the joint estimation of objective and risk neutral measures for the purpose of options valuation , 2000 .

[58]  N. Shephard,et al.  Econometric analysis of realised volatility and its use in estimating stochastic volatility models , 2000 .

[59]  Nicholas G. Polson,et al.  The Impact of Jumps in Volatility and Returns , 2000 .

[60]  Christopher S. Jones,et al.  Bayesian investigation of continuous -time finance models , 2000 .

[61]  Robert F. Dittmar,et al.  Quadratic Term Structure Models: Theory and Evidence , 2000 .

[62]  Michael W. Brandt,et al.  Simulated Likelihood Estimation of Diffusions with an Application to Exchange Rate Dynamics in Incomplete Markets , 2001 .

[63]  R. Gencay,et al.  An Introduc-tion to High-Frequency Finance , 2001 .

[64]  Jun Pan.  The jump-risk premia implicit in options: evidence from an integrated time-series study , 2001 .

[65]  Francis X. Diebold,et al.  Modeling and Forecasting Realized Volatility , 2001 .

[66]  O. Barndorff-Nielsen.  Superposition of Ornstein--Uhlenbeck Type Processes , 2001 .

[67]  N. Shephard,et al.  Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics , 2001 .

[68]  Bjørn Eraker.  MCMC Analysis of Diffusion Models With Application to Finance , 2001 .

[69]  F. Diebold,et al.  The Distribution of Realized Exchange Rate Volatility , 2000 .

[70]  Luca Benzoni,et al.  An Empirical Investigation of Continuous-Time Equity Return Models , 2001 .

[71]  N. Meddahi,et al.  An Eigenfunction Approach for Volatility Modeling , 2001 .

[72]  K. Singleton.  Estimation of affine asset pricing models using the empirical characteristic function , 2001 .

[73]  R. Liesenfeld.  A generalized bivariate mixture model for stock price volatility and trading volume , 2001 .

[74]  G. Roberts,et al.  On inference for partially observed nonlinear diffusion models using the Metropolis–Hastings algorithm , 2001 .

[75]  Hao Zhou,et al.  Term Structure of Interest Rates with Regime Shifts , 2001 .

[76]  B. Werker.  Discussion of "Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics" by Barndorff-Nielsen and Shephard , 2001 .

[77]  Marc Yor,et al.  Stochastic volatility, jumps and hidden time changes , 2002, Finance Stochastics.

[78]  N. Shephard,et al.  Econometric analysis of realized volatility and its use in estimating stochastic volatility models , 2002 .

[79]  A. Gallant,et al.  Numerical Techniques for Maximum Likelihood Estimation of Continuous-Time Diffusion Processes , 2002 .

[80]  Gabriele Fiorentini,et al.  Likelihood-based estimation of latent generalised ARCH structures , 2004 .

[81]  P. Carr,et al.  Time-Changed Levy Processes and Option Pricing ⁄ , 2002 .

[82]  E. Nicolato,et al.  Option Pricing in Stochastic Volatility Models of the Ornstein‐Uhlenbeck type , 2003 .

[83]  Garland B. Durham.  Likelihood-based specification analysis of continuous-time models of the short-term interest rate , 2003 .

[84]  N. Shephard,et al.  Power and bipower variation with stochastic volatility and jumps , 2003 .

[85]  J. Richard,et al.  Univariate and Multivariate Stochastic Volatility Models: Estimation and Diagnostics , 2003 .

[86]  Lan Zhang,et al.  A Tale of Two Time Scales , 2003 .

[87]  N. Shephard,et al.  Impact of jumps on returns and realised variances: econometric analysis of time-deformed Lévy processes , 2006 .

[88]  P. Djurić,et al.  Particle filtering , 2003, IEEE Signal Process. Mag..

[89]  A. Gallant,et al.  Alternative models for stock price dynamics , 2003 .

[90]  Michael S. Johannes,et al.  Earnings announcements and equity options , 2004 .

[91]  Bjørn Eraker.  Do Stock Prices and Volatility Jump? Reconciling Evidence from Spot and Option Prices , 2004 .

[92]  Jeffrey R. Russell,et al.  Separating Microstructure Noise from Volatility , 2004 .

[93]  P. Hansen,et al.  Realized Variance and Market Microstructure Noise , 2005 .

[94]  Michael S. Johannes,et al.  Earnings Announcements and Option Prices , 2005 .

[95]  N. Shephard,et al.  Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise , 2006 .

[96]  Giuseppe Cavaliere.  Stochastic Volatility: Selected Readings , 2006 .

[97]  M. McAleer,et al.  Multivariate Stochastic Volatility: A Review , 2006 .

[98]  Asger Lunde,et al.  Realized Variance and Market Microstructure Noise , 2006 .

[99]  A. Harvey.  Long memory in stochastic volatility , 2007 .

[100]  Francis X. Diebold,et al.  Real-Time Price Discovery in Global Stock, Bond and Foreign Exchange Markets , 2006 .

[101]  Timo Teräsvirta,et al.  Multivariate GARCH models To appear in T. G. Andersen, R. A. Davis, J.-P. Kreiss and T. Mikosch, eds. Handbook of Financial Time Series. New York: Springer. , 2008 .

[102]  Eric Renault,et al.  Moment–Based Estimation of Stochastic Volatility Models , 2009 .

[103]  A. Graja.  Bayesian Analysis of Stochastic Volatility Models , 2009 .

[104]  S. Chib,et al.  Multivariate stochastic volatility , 2009 .

[105]  Richard A. Davis,et al.  Handbook of Financial Time Series , 2009 .