It is widely known that conditional covariances of asset returns change over time.

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We propose a functional estimation procedure for homogeneous stochastic differential equations based on a discrete sample of observations and with minimal requirements on the data generating process. We show how to identify the drift and diffusion function in situations where one or the other function is considered a nuisance parameter. The asymptotic behavior of the estimators is examined as the observation frequency increases and as the time span lengthens. We prove almost sure consistency and weak convergence to mixtures of normal laws, where the mixing variates depend on the chronological local time of the underlying diffusion process, that is the random time spent by the process in the vicinity of a generic spatial point. The estimation method and asymptotic results apply to both stationary and nonstationary recurrent processes.

Abstract. This paper gives a brief overview of the nonparametric techniques that are useful for financial econometric problems. The problems include estimation and inference for instantaneous returns and volatility functions of time-homogeneous and time-dependent diffusion processes, and estimation of transition densities and state price densities. We first briefly describe the problems and then outline the main techniques and main results. Some useful probabilistic aspects of diffusion processes are also briefly summarized to facilitate our presentation and applications.

This paper is a survey of existing estimation techniques for stationary and ergodic diffusion processes observed at discrete points in time. The reader is introduced to the following techniques: (i) estimating functions with special emphasis on martingale estimating functions and so-called simple estimating functions; (ii) analytical and numerical approximations of the likelihood which can in principle be made arbitrarily accurate; (iii) Bayesian analysis and MCMC methods; and (iv) indirect inference and EMM which both introduce auxiliary (but wrong) models and correct for the implied bias by simulation

Ito processes are the most common form of continuous semimartingales, and include diffusion processes. The paper is concerned with the nonparametric regression relationship between two such Ito processes. We are interested in the quadratic variation (integrated volatility) of the residual in this regression, over a unit of time (such as a day). A main conceptual finding is that this quadratic variation can be estimated almost as if the residual process were observed, the difference being that there is also a bias which is of the same asymptotic order as the mixed normal error term. The proposed methodology, “ANOVA for diffusions and Ito processes”, can be used to measure the statistical quality of a parametric model, and, nonparametrically, the appropriateness of a one-regressor model in general. On the other hand, it also helps quantify and characterize the trading (hedging) error in the case of financial applications.

The paper shows how to convert statistical prediction sets into worst case hedging strategies for derivative securities. The prediction sets can, in particular, be ones for volatilities and correlations of the underlying securities, and for interest rates. This permits a transfer of statistical conclusions into prices for options and similar financial instruments. A prime feature of our results is that one can construct the trading strategy as if the prediction set had a 100 % probability. If, in fact, the set has probability 1−α, the hedging strategy will work with at least the same probability. Different types of prediction regions are considered. The starting value A0 for the trading strategy corresponding to the 1 − α prediction region is a form of long term value at risk. At the same time, A0 is coherent.

In an effort to capture the time variation on the instantaneous return and volatility functions, a family of time-dependent diffusion processes is introduced to model the term structure dynamics. This allows one to examine how the instantaneous return and price volatility change over time and price level. Nonparametric techniques, based on kernel regression, are used to estimate the time-varying coefficient functions in the drift and diffusion. The newly proposed semiparametric model includes most of the well-known short-term interest rate models, such as those proposed by Cox, Ingersoll and Ross (1985) and Chan, Karolyi, Longstaff and Sanders (1992). It can be used to test the goodness-of-fit of these famous time-homogeneous short rate models. The newly proposed method complements the time-homogeneous nonparametric estimation techniques of Stanton (1997) and Fan and Yao (1998), and is shown through simulations to truly capture the heteroscedasticity and time-inhomogeneous structure in volatility. A family of new statistics is introduced to test whether the time-homogeneous models adequately fit interest rates for certain periods of the economy. We illustrate the new methods by using weekly three-month treasury bill data.

This paper is concerned with the estimation of stochastic di#erential equations when only discrete observations are available. It primarily focuses on deriving a closed form solution for the one-step ahead conditional transition density using the Milstein scheme. This higher order Taylor approximation enables us to obtain an order of improvement in accuracy in estimating the parameters in a non-linear di#usion, as compared to use of the Euler-Maruyama discretization scheme. Examples using simulated data are presented. The method can easily be extended to the situation where auxiliary points are introduced between the observed values. The Milstein scheme can be used to obtain the approximate transition density as in a Pedersen (1995) type of simulated likelihood method or within an MCMC method as proposed in Elerian, Chib, and Shephard (1998). Keywords: Bayes estimation, nonlinear di#usion, Euler-Maruyama approximation, Maximum Likelihood, Markov chain Monte Carlo, Metropolis Hastings algorithm, Milstein scheme, Simulation, Stochastic Di#erential Equation. 1 1

In this paper, a Monte Carlo simulation is performed to investigate the finite sample properties of various estimators, based on discretely sampled observations, of the continuous-time Itô diffusion process. The simulation study aims to compare the performance of the nonparametric estimators proposed in Jiang and Knight (1996) with common parametric estimators based on those diffusion processes which have explicit transition density functions. The simulation results show that, with a large sample over a short sampling period, although all the parametric diffusion estimators perform very well, the parametric drift estimators perform very poorly. However, both the nonparametric diffusion and drift estimators perform reasonably well.