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128
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 194 (7 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,
Chaos and Nonlinear Dynamics: Application to Financial Markets
 Journal of Finance
, 1991
"... After the stock market crash of October 19, 1987, interest in nonlinear dynamics, especially deterministic chaotic dynamics, has increased in both the financial press and the academic literature. This has come about because the frequency of large moves in stock markets is greater than would be expec ..."
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Cited by 106 (3 self)
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After the stock market crash of October 19, 1987, interest in nonlinear dynamics, especially deterministic chaotic dynamics, has increased in both the financial press and the academic literature. This has come about because the frequency of large moves in stock markets is greater than would be expected
Continuous Record Asymptotics for Rolling Sample Variance Estimators
 Econometrica
, 1996
"... It is widely known that conditional covariances of asset returns change over time. ..."
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Cited by 89 (0 self)
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It is widely known that conditional covariances of asset returns change over time.
Empirical pricing kernels
, 2001
"... This paper investigates the empirical characteristics of investor risk aversion over equity return states by estimating a timevarying pricing kernel, which we call the empirical pricing kernel (EPK). We estimate the EPK on a monthly basis from 1991 to 1995, using S&P 500 index option data and a sto ..."
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Cited by 70 (1 self)
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This paper investigates the empirical characteristics of investor risk aversion over equity return states by estimating a timevarying pricing kernel, which we call the empirical pricing kernel (EPK). We estimate the EPK on a monthly basis from 1991 to 1995, using S&P 500 index option data and a stochastic volatility model for the S&P 500 return process. We find that the EPK exhibits countercyclical risk aversion over S&P 500 return states. We also find that hedging performance is significantly improved when we use hedge ratios based the EPK rather than a timeinvariant pricing kernel.
Maximum likelihood estimation for stochastic volatility models
 JOURNAL OF FINANCIAL ECONOMICS
, 2007
"... We develop and implement a method for maximum likelihood estimation in closedform of stochastic volatility models. Using Monte Carlo simulations, we compare a full likelihood procedure, where an option price is inverted into the unobservable volatility state, to an approximate likelihood procedure ..."
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Cited by 48 (3 self)
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We develop and implement a method for maximum likelihood estimation in closedform of stochastic volatility models. Using Monte Carlo simulations, we compare a full likelihood procedure, where an option price is inverted into the unobservable volatility state, to an approximate likelihood procedure where the volatility state is replaced by proxies based on the implied volatility of a shortdated atthemoney option. The approximation results in a small loss of accuracy relative to the standard errors due to sampling noise. We apply this method to market prices of index options for several stochastic volatility models, and compare the characteristics of the estimated models. The evidence for a general CEV model, which nests both the affine Heston model and a GARCH model, suggests that the elasticity of variance of volatility lies between that assumed by the two nested models.
Stochastic volatility models as hidden Markov models and statistical applications
 Bernoulli
, 2000
"... This paper deals with the ®xed sampling interval case for stochastic volatility models. We consider a twodimensional diffusion process (Yt, Vt), where only (Yt) is observed at n discrete times with regular sampling interval Ä. The unobserved coordinate (Vt) is ergodic and rules the diffusion coef®c ..."
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Cited by 45 (5 self)
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This paper deals with the ®xed sampling interval case for stochastic volatility models. We consider a twodimensional diffusion process (Yt, Vt), where only (Yt) is observed at n discrete times with regular sampling interval Ä. The unobserved coordinate (Vt) is ergodic and rules the diffusion coef®cient (volatility) of (Yt). We study the ergodicity and mixing properties of the observations (YiÄ). For this purpose, we ®rst present a thorough review of these properties for stationary diffusions. We then prove that our observations can be viewed as a hidden Markov model and inherit the mixing properties of (Vt). When the stochastic differential equation of (Vt) depends on unknown parameters, we derive momenttype estimators of all the parameters, and show almost sure convergence and a central limit theorem at rate n 1=2. Examples of models coming from ®nance are fully treated. We focus on the asymptotic variances of the estimators and establish some links with the small sampling interval case studied in previous papers.
Complete Models with Stochastic Volatility
, 1996
"... The paper proposes an original class of models for the continuous time price process of a financial security with nonconstant volatility. The idea is to define instantaneous volatility in terms of exponentiallyweighted moments of historic logprice. The instantaneous volatility is therefore driven ..."
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Cited by 42 (3 self)
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The paper proposes an original class of models for the continuous time price process of a financial security with nonconstant volatility. The idea is to define instantaneous volatility in terms of exponentiallyweighted moments of historic logprice. The instantaneous volatility is therefore driven by the same stochastic factors as the price process, so that unlike many other models of nonconstant volatility, it is not necessary to introduce additional sources of randomness. Thus the market is complete and there are unique, preferenceindependent options prices. We find a partial differential equation for the price of a European Call Option. Smiles and skews are found in the resulting plots of implied volatility. Keywords: Option pricing, stochastic volatility, complete markets, smiles. Acknowledgement. It is a pleasure to thank the referees of an earlier draft of this paper whose perceptive comments have resulted in many improvements. 1 Research supported in part by Record Treasu...
2003), “Correcting the Errors: Volatility Forecast Evaluation Using HighFrequency Data and Realized Volatilities,” working paper
"... We develop general modelfree adjustment procedures for the calculation of unbiased volatility loss functions based on practically feasible realized volatility benchmarks. The procedures, which exploit the recent nonparametric asymptotic distributional results in BarndorffNielsen and Shephard (200 ..."
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Cited by 41 (11 self)
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We develop general modelfree adjustment procedures for the calculation of unbiased volatility loss functions based on practically feasible realized volatility benchmarks. The procedures, which exploit the recent nonparametric asymptotic distributional results in BarndorffNielsen and Shephard (2002a) along with new results explicitly allowing for leverage effects, are both easytoimplement and highly accurate in empirically realistic situations. On properly accounting for the measurement errors in the volatility forecast evaluations reported in Andersen, Bollerslev, Diebold and Labys (2003), the adjustments result in markedly higher estimates for the true degree of return volatility predictability.
Derivative asset analysis in models with leveldependent and stochastic volatility
 CWI QUARTERLY
, 1996
"... In this survey we discuss models with leveldependent and stochastic volatility from the viewpoint of derivative asset analysis. Both classes of models are generalisations of the classical BlackScholes model; they have been developed in an effort to build models that are flexible enough to cope wit ..."
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Cited by 38 (1 self)
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In this survey we discuss models with leveldependent and stochastic volatility from the viewpoint of derivative asset analysis. Both classes of models are generalisations of the classical BlackScholes model; they have been developed in an effort to build models that are flexible enough to cope with the known deficits of the classical BlackScholes model. We start by briefly recalling the standard theory for pricing and hedging derivatives in complete frictionless markets and the classical BlackScholes model. After a review of the known empirical contradictions to the classical BlackScholes model we consider models with leveldependent volatility. Most of this survey is devoted to derivative asset analysis in stochastic volatility models. We discuss several recent developments in the theory of derivative pricing under incompleteness in the context of stochastic volatility models and review analytical and numerical approaches to the actual computation of option values.