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27
An empirical investigation of continuoustime equity return models
 Journal of Finance
, 2002
"... This paper extends the class of stochastic volatility diffusions for asset returns to encompass Poisson jumps of timevarying intensity. We find that any reasonably descriptive continuoustime model for equityindex returns must allow for discrete jumps as well as stochastic volatility with a pronou ..."
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Cited by 134 (10 self)
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This paper extends the class of stochastic volatility diffusions for asset returns to encompass Poisson jumps of timevarying intensity. We find that any reasonably descriptive continuoustime model for equityindex returns must allow for discrete jumps as well as stochastic volatility with a pronounced negative relationship between return and volatility innovations. We also find that the dominant empirical characteristics of the return process appear to be priced by the option market. Our analysis indicates a general correspondence between the evidence extracted from daily equityindex returns and the stylized features of the corresponding options market prices. MUCH ASSET AND DERIVATIVE PRICING THEORY is based on diffusion models for primary securities. However, prescriptions for practical applications derived from these models typically produce disappointing results. A possible explanation could be that analytic formulas for pricing and hedging are available for only a limited set of continuoustime representations for asset returns
Numerical Techniques for Maximum Likelihood Estimation of ContinuousTime Diffusion Processes
 JOURNAL OF BUSINESS AND ECONOMIC STATISTICS
, 2001
"... Stochastic differential equations often provide a convenient way to describe the dynamics of economic and financial data, and a great deal of effort has been expended searching for efficient ways to estimate models based on them. Maximum likelihood is typically the estimator of choice; however, sinc ..."
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Cited by 87 (0 self)
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Stochastic differential equations often provide a convenient way to describe the dynamics of economic and financial data, and a great deal of effort has been expended searching for efficient ways to estimate models based on them. Maximum likelihood is typically the estimator of choice; however, since the transition density is generally unknown, one is forced to approximate it. The simulationbased approach suggested by Pedersen (1995) has great theoretical appeal, but previously available implementations have been computationally costly. We examine a variety of numerical techniques designed to improve the performance of this approach. Synthetic data generated by a CIR model with parameters calibrated to match monthly observations of the U.S. shortterm interest rate are used as a test case. Since the likelihood function of this process is known, the quality of the approximations can be easily evaluated. On data sets with 1000 observations, we are able to approximate the maximum likelihood estimator with negligible error in well under one minute. This represents something on the order of a 10,000fold reduction in computational effort as compared to implementations without these enhancements. With other parameter settings designed to stress the methodology, performance remains strong. These ideas are easily generalized to multivariate settings and (with some additional work) to latent variable models. To illustrate, we estimate a simple stochastic volatility model of the U.S. shortterm interest rate.
The Dynamics of Stochastic Volatility: Evidence from Underlying and Option Markets
, 2000
"... This paper proposes and estimates a more general parametric stochastic variance model of equity index returns than has been previously considered using data from both underlying and options markets. The parameters of the model under both the objective and riskneutral measures are estimated simultane ..."
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Cited by 72 (1 self)
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This paper proposes and estimates a more general parametric stochastic variance model of equity index returns than has been previously considered using data from both underlying and options markets. The parameters of the model under both the objective and riskneutral measures are estimated simultaneously. I conclude that the square root stochastic variance model of Heston (1993) and others is incapable of generating realistic returns behavior and find that the data are more accurately represented by a stochastic variance model in the CEV class or a model that allows the price and variance processes to have a timevarying correlation. Specifically, I find that as the level of market variance increases, the volatility of market variance increases rapidly and the correlation between the price and variance processes becomes substantially more negative. The heightened heteroskedasticity in market variance that results generates realistic crash probabilities and dynamics and causes returns to display values of skewness and kurtosis much more consistent with their sample values. While the model dramatically improves the fit of options prices relative to the square root process, it falls short of explaining the implied volatility smile for shortdated options.
Simulated likelihood estimation of diffusions with an application to exchange rate dynamics in incomplete markets
, 2002
"... ..."
Continuoustime methods in finance: A review and an assessment
 Journal of Finance
, 2000
"... I survey and assess the development of continuoustime methods in finance during the last 30 years. The subperiod 1969 to 1980 saw a dizzying pace of development with seminal ideas in derivatives securities pricing, term structure theory, asset pricing, and optimal consumption and portfolio choices. ..."
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Cited by 32 (0 self)
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I survey and assess the development of continuoustime methods in finance during the last 30 years. The subperiod 1969 to 1980 saw a dizzying pace of development with seminal ideas in derivatives securities pricing, term structure theory, asset pricing, and optimal consumption and portfolio choices. During the period 1981 to 1999 the theory has been extended and modified to better explain empirical regularities in various subfields of finance. This latter subperiod has seen significant progress in econometric theory, computational and estimation methods to test and implement continuoustime models. Capital market frictions and bargaining issues are being increasingly incorporated in continuoustime theory. THE ROOTS OF MODERN CONTINUOUSTIME METHODS in finance can be traced back to the seminal contributions of Merton ~1969, 1971, 1973b! in the late 1960s and early 1970s. Merton ~1969! pioneered the use of continuoustime modeling in financial economics by formulating the intertemporal consumption and portfolio choice problem of an investor in a stochastic dynamic programming setting.
Optimal filtering of jump diffusions: extracting latent states from asset prices”, Working Paper, http://wwwstat.wharton.upenn.edu/ stroud/pubs.html
, 2006
"... This paper provides a methodology for computing optimal filtering distributions in discretely observed continuoustime jumpdiffusion models. Although it has received little attention, the filtering distribution is useful for estimating latent states, forecasting volatility and returns, computing mo ..."
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Cited by 20 (5 self)
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This paper provides a methodology for computing optimal filtering distributions in discretely observed continuoustime jumpdiffusion models. Although it has received little attention, the filtering distribution is useful for estimating latent states, forecasting volatility and returns, computing model diagnostics such as likelihood ratios, and parameter estimation. Our approach combines timediscretization schemes with Monte Carlo methods to compute the optimal filtering distribution. Our approach is very general, applying in multivariate jumpdiffusion models with nonlinear characteristics and even nonanalytic observation equations, such as those that arise when option prices are available. We provide a detailed analysis of the performance of the filter, and analyze four applications: disentangling jumps from stochastic volatility, forecasting realized volatility, likelihood based model comparison, and filtering using both option prices and underlying returns. 2 1
Nonlinear Mean Reversion in the ShortTerm Interest Rate
, 2003
"... Using a new Bayesian method for the analysis of diffusion processes, this article finds that the nonlinear drift in interest rates found in a number of previous studies can be confirmed only under prior distributions that are best described as informative. The assumption of stationarity, which is co ..."
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Cited by 20 (2 self)
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Using a new Bayesian method for the analysis of diffusion processes, this article finds that the nonlinear drift in interest rates found in a number of previous studies can be confirmed only under prior distributions that are best described as informative. The assumption of stationarity, which is common in the literature, represents a nontrivial prior belief about the shape of the drift function. This belief and the use of ``flat'' priors contribute strongly to the finding of nonlinear mean reversion. Implementation of an approximate Jeffreys prior results in virtually no evidence for mean reversion in interest rates unless stationarity is assumed. Finally, the article documents that nonlinear drift is primarily a feature of daily rather than monthly data, and that these data contain a transitory element that is not reflected in the volatility of longermaturity yields.
LikelihoodBased Specification Analysis of ContinuousTime Models of the ShortTerm Interest Rate
 JOURNAL OF FINANCIAL ECONOMICS
, 2003
"... An extensive collection of continuoustime models of the shortterm interest rate are evaluated over data sets that have appeared previously in the literature. The analysis, which uses the simulated maximum likelihood procedure proposed by Durham and Gallant (1999), provides new insights regardin ..."
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Cited by 18 (0 self)
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An extensive collection of continuoustime models of the shortterm interest rate are evaluated over data sets that have appeared previously in the literature. The analysis, which uses the simulated maximum likelihood procedure proposed by Durham and Gallant (1999), provides new insights regarding several previously unresolved questions. For single factor models, I find that the volatility rather than the drift is the critical component in model specification. Allowing for additional flexibility beyond a constant term in the drift provides negligible benefit. While constant drift would appear to imply that the short rate is nonstationary, in fact stationarity is volatilityinduced. The simple constant elasticity of volatility model fits weekly observations of the threemonth Treasury bill rate remarkably well but is easily rejected when compared to more flexible volatility specifications over daily data. The methodology of Durham and Gallant can also be used to estimate stochastic volatility models. While adding the latent volatility component provides a large improvement in the likelihood for the physical process, it does little to improve bondpricing performance.
A Bayesian Analysis of Return Dynamics with Lévy Jumps, forthcoming, Review of Financial Studies
, 2007
"... and the 2004 Institute of Mathematical Statistics Annual Meeting/6th Bernoulli World Congress for helpful comments. We are responsible for any remaining errors. A Bayesian Analysis of Return Dynamics with Stochastic Volatility and Lévy Jumps We develop Bayesian Markov chain Monte Carlo methods for i ..."
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Cited by 16 (1 self)
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and the 2004 Institute of Mathematical Statistics Annual Meeting/6th Bernoulli World Congress for helpful comments. We are responsible for any remaining errors. A Bayesian Analysis of Return Dynamics with Stochastic Volatility and Lévy Jumps We develop Bayesian Markov chain Monte Carlo methods for inferences of continuoustime models with stochastic volatility and infiniteactivity Lévy jumps using discretely sampled data. Simulation studies show that (i) our methods provide accurate joint identification of diffusion, stochastic volatility, and Lévy jumps, and (ii) affine jumpdiffusion models fail to adequately approximate the behavior of infiniteactivity jumps. In particular, the affine jumpdiffusion models fail to capture the “infinitely many ” small LévyjumpswhicharetoobigforBrownianmotiontomodelandtoosmall for compound Poisson process to capture. Empirical studies show that infiniteactivity Lévy jumps are essential for modeling the S&P 500 index returns. The continuoustime finance literature in the past few decades has mainly relied on Brownian motion and compound Poisson process as basic model building blocks. Sophisticated models based solely on Brownian motion and compound Poisson process have been developed to capture important
Stochastic volatility: origins and overview
 Handbook of Financial Time Series
, 2008
"... Stochastic volatility (SV) models are used heavily within the fields of financial economics and mathematical finance to capture the impact of timevarying volatility on financial markets and decision making. The development of the subject has been highly multidisciplinary, with results drawn from fi ..."
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Cited by 7 (0 self)
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Stochastic volatility (SV) models are used heavily within the fields of financial economics and mathematical finance to capture the impact of timevarying volatility on financial markets and decision making. The development of the subject has been highly multidisciplinary, with results drawn from financial economics, probability theory and econometrics blending to produce methods that