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190
Testing ContinuousTime Models of the Spot Interest Rate
 Review of Financial Studies
, 1996
"... Different continuoustime models for interest rates coexist in the literature. We test parametric models by comparing their implied parametric density to the same density estimated nonparametrically. We do not replace the continuoustime model by discrete approximations, even though the data are rec ..."
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Cited by 266 (9 self)
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Different continuoustime models for interest rates coexist in the literature. We test parametric models by comparing their implied parametric density to the same density estimated nonparametrically. We do not replace the continuoustime model by discrete approximations, even though the data are recorded at discrete intervals. The principal source of rejection of existing models is the strong nonlinearity of the drift. Around its mean, where the drift is essentially zero, the spot rate behaves like a random walk. The drift then meanreverts strongly when far away from the mean. The volatility is higher when away from the mean. The continuoustime financial theory has developed extensive tools to price derivative securities when the underlying traded asset(s) or nontraded factor(s) follow stochastic differential equations [see Merton (1990) for examples]. However, as a practical matter, how to specify an appropriate stochastic differential equation is for the most part an unanswered question. For example, many different continuoustime The comments and suggestions of Kerry Back (the editor) and an anonymous referee were very helpful. I am also grateful to George Constantinides,
Likelihood inference for discretely observed nonlinear diffusions. Econometrica 69 959–993. MR1839375
 Ann. Statist
, 2001
"... ..."
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 210 (13 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
A Nonparametric Model of Term Structure Dynamics and the Market Price of Interest Rate Risk
, 1997
"... This article presents a technique for nonparametrically estimating continuoustime di#usion processes which are observed at discrete intervals. We illustrate the methodology by using daily three and six month Treasury Bill data, from January 1965 to July 1995, to estimate the drift and di#usion of t ..."
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Cited by 187 (5 self)
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This article presents a technique for nonparametrically estimating continuoustime di#usion processes which are observed at discrete intervals. We illustrate the methodology by using daily three and six month Treasury Bill data, from January 1965 to July 1995, to estimate the drift and di#usion of the short rate, and the market price of interest rate risk. While the estimated di#usion is similar to that estimated by Chan, Karolyi, Longsta# and Sanders (1992), there is evidence of substantial nonlinearity in the drift. This is close to zero for low and medium interest rates, but mean reversion increases sharply at higher interest rates.
How often to sample a continuoustime process in the presence of market microstructure noise
 Review of Financial Studies
, 2005
"... In theory, the sum of squares of log returns sampled at high frequency estimates their variance. When market microstructure noise is present but unaccounted for, however, we show that the optimal sampling frequency is finite and derives its closedform expression. But even with optimal sampling, usi ..."
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Cited by 144 (13 self)
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In theory, the sum of squares of log returns sampled at high frequency estimates their variance. When market microstructure noise is present but unaccounted for, however, we show that the optimal sampling frequency is finite and derives its closedform expression. But even with optimal sampling, using say 5min returns when transactions are recorded every second, a vast amount of data is discarded, in contradiction to basic statistical principles. We demonstrate that modeling the noise and using all the data is a better solution, even if one misspecifies the noise distribution. So the answer is: sample as often as possible. Over the past few years, price data sampled at very high frequency have become increasingly available in the form of the Olsen dataset of currency exchange rates or the TAQ database of NYSE stocks. If such data were not affected by market microstructure noise, the realized volatility of the process (i.e., the average sum of squares of logreturns sampled at high frequency) would estimate the returns ’ variance, as is well known. In fact, sampling as often as possible would theoretically produce in the limit a perfect estimate of that variance. We start by asking whether it remains optimal to sample the price process at very high frequency in the presence of market microstructure noise, consistently with the basic statistical principle that, ceteris paribus, more data are preferred to less. We first show that, if noise is present but unaccounted for, then the optimal sampling frequency is finite, and we We are grateful for comments and suggestions from the editor, Maureen O’Hara, and two anonymous
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 129 (3 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.
MCMC Analysis of Diffusion Models with Application to Finance
 Journal of Business and Economic Statistics
, 1998
"... This paper proposes a new method for estimation of parameters in diffusion processes from ..."
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Cited by 119 (4 self)
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This paper proposes a new method for estimation of parameters in diffusion processes from
Nonparametric Density Estimation and Tests of Continuous Time Interest Rate Models
 Review of Financial Studies
, 1998
"... A number of recent papers have used nonparametric density estimation or nonparametric regression to study the instantaneous spot interest rate, and to test term structure models. However, little is known about the performance of these methods when applied to persistent timeseries, such as U.S. inte ..."
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Cited by 98 (2 self)
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A number of recent papers have used nonparametric density estimation or nonparametric regression to study the instantaneous spot interest rate, and to test term structure models. However, little is known about the performance of these methods when applied to persistent timeseries, such as U.S. interest rates. This paper uses the Vasicek [1977] model to study the performance of kernel density estimates of the ergodic distribution of the instantaneous spot rate. The model's tractability allows me to analyze the MISE of the kernel estimate as a function of persistence, variance of the ergodic distribution, span of the data, sampling frequency, and kernel bandwidth. Our principle result is that persistence has an important impact on optimal bandwidth selection and on nite sample performance. We also nd that sampling the data more frequently has little e ect on estimator quality. We also examine one of AitSahalia's [1996a] new nonparametric tests of parametric continuoustime Markov models of the instantaneous spot interest rate. The test is based on the distance between parametric and nonparametric (kernel) estimates of the ergodic distribution of the interest rate process. Our principal result is that the test rejects too often when using asymptotic critical values and 22 years of data. The reason for the high rejection rate is probably because the asymptotic distribution of the test does not depend on persistence, but the nite sample performance of the estimator does. After critical values are adjusted for size, the test has low power in distinguishing between the Vasicek and CoxIngersollRoss models when compared with a conditional moment based speci cation test.
Estimating Stochastic Volatility Diffusion Using Conditional Moments of Integrated Volatility
, 2000
"... We exploit the distributional information contained in highfrequency intraday data in constructing a simple conditional moment estimator for stochastic volatility diffusions. The estimator is based on the analytical solutions of the first two conditional moments for the integrated volatility, which ..."
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Cited by 95 (8 self)
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We exploit the distributional information contained in highfrequency intraday data in constructing a simple conditional moment estimator for stochastic volatility diffusions. The estimator is based on the analytical solutions of the first two conditional moments for the integrated volatility, which is effectively approximated by the quadratic variation of the process. We successfully implement the resulting GMM estimator with highfrequency fiveminute foreign exchange and equity index returns. Our simulation evidence and actual empirical results indicate that the method is very reliable and accurate. The computational speed of the procedure compares very favorably to other existing estimation methods in the literature.