Closed-Form Likelihood Expansions for Multivariate Diffusions

This paper provides closed-form expansions for the transition density and likelihood function of arbitrary multivariate diffusions. The expansions are based on a Hermite series, whose coefficients are calculated explicitly by exploiting the special structure afforded by the diffusion hypothesis. Because the transition function for most diffusion models is not known explicitly, the expansions of this paper can help make maximum-likelihood a practical estimation method for discretely sampled multivariate diffusions. Examples of interest in financial econometrics are included.

Fisher's Information for Discretely Sampled Levy Processes

This paper studies the asymptotic behavior of the Fisher information for a Levy process discretely sampled at an increasing frequency. We show that it is possible to distinguish not only the continuous part of the process from its jumps part, but also different types of jumps, and derive the rates of convergence of efficient estimators.Comment: 17 novembre 200

Maximum Likelihood Estimation of Stochastic Volatility Models

We develop and implement a new method for maximum likelihood estimation in closed-form 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 the implied volatility of a short dated at-the-money option. We find that the approximation results in a negligible loss of accuracy. 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 model of Heston (1993) and a GARCH model, suggests that the elasticity of variance of volatility lies between that assumed by the two nested models.

Nonparametric Option Pricing under Shape Restrictions

Frequently, economic theory places shape restrictions on functional relationships between economic variables. This paper develops a method to constrain the values of the first and second derivatives of nonparametric locally polynomial estimators. We apply this technique to estimate the state price density (SPD), or risk-neutral density, implicit in the market prices of options. The option pricing function must be monotonic and convex. Simulations demonstrate that nonparametric estimates can be quite feasible in the small samples relevant for day-to-day option pricing, once appropriate theory-motivated shape restrictions are imposed. Using S&P500 option prices, we show that unconstrained nonparametric estimators violate the constraints during more than half the trading days in 1999, unlike the constrained estimator we propose.

How Often to Sample a Continuous-Time Process in the Presence of Market Microstructure Noise

Classical statistics suggest that for inference purposes one should always use as much data as is available. We study how the presence of market microstructure noise in high-frequency financial data can change that result. We show that the optimal sampling frequency at which to estimate the parameters of a discretely sampled continuous-time model can be finite when the observations are contaminated by market microstructure effects. We then address the question of what to do about the presence of the noise. We show that modelling the noise term explicitly restores the first order statistical effect that sampling as often as possible is optimal. But, more surprisingly, we also demonstrate that this is true even if one misspecifies the assumed distribution of the noise term. Not only is it still optimal to sample as often as possible, but the estimator has the same variance as if the noise distribution had been correctly specified, implying that attempts to incorporate the noise into the analysis cannot do more harm than good. Finally, we study the same questions when the observations are sampled at random time intervals, which are an essential feature of transaction-level data.

The Leverage Effect Puzzle: Disentangling Sources of Bias at High Frequency

The leverage effect refers to the generally negative correlation between an asset return and its changes of volatility. A natural estimate consists in using the empirical correlation between the daily returns and the changes of daily volatility estimated from high-frequency data. The puzzle lies in the fact that such an intuitively natural estimate yields nearly zero correlation for most assets tested, despite the many economic reasons for expecting the estimated correlation to be negative. To better understand the sources of the puzzle, we analyze the different asymptotic biases that are involved in high frequency estimation of the leverage effect, including biases due to discretization errors, to smoothing errors in estimating spot volatilities, to estimation error, and to market microstructure noise. This decomposition enables us to propose novel bias correction methods for estimating the leverage effect.

Telling from Discrete Data Whether the Underlying Continuous-Time Model is a Diffusion

Asset returns have traditionally been modeled in the literature as following continuous-time Markov processes, and in many cases diffusions. Can discretely sampled financial rate data help us decide which continuous-time models are sensible? Diffusion processes are characterized by the continuity of their sample paths. This cannot be verified from the discrete sample path: by nature, even if the underlying sample path were continuous, the discretely sampled data will always appear as a sequence of discrete jumps. Instead, this paper relies on a characterization of the transition density of the discrete data to determine whether the discontinuities observed in the discrete data are the result of the discreteness of sampling, or rather evidence of genuine jump dynamics for the underlying continuous-time process. I then focus on the implications of this approach for option pricing models.

Disentangling Volatility from Jumps

Realistic models for financial asset prices used in portfolio choice, option pricing or risk management include both a continuous Brownian and a jump components. This paper studies our ability to distinguish one from the other. I find that, surprisingly, it is possible to perfectly disentangle Brownian noise from jumps. This is true even if, unlike the usual Poisson jumps, the jump process exhibits an infinite number of small jumps in any finite time interval, which ought to be harder to distinguish from Brownian noise, itself made up of many small moves.

Ultra High Frequency Volatility Estimation with Dependent Microstructure Noise

We analyze the impact of time series dependence in market microstructure noise on the properties of estimators of the integrated volatility of an asset price based on data sampled at frequencies high enough for that noise to be a dominant consideration. We show that combining two time scales for that purpose will work even when the noise exhibits time series dependence, analyze in that context a refinement of this approach based on multiple time scales, and compare empirically our different estimators to the standard realized volatility.

Ultra high frequency volatility estimation with dependent microstructure noise

We analyze the impact of time series dependence in market microstructure noise on the properties of estimators of the integrated volatility of an asset price based on data sampled at frequencies high enough for that noise to be a dominant consideration. We show that combining two time scales for that purpose will work even when the noise exhibits time series dependence, analyze in that context a refinement of this approach based on multiple time scales, and compare empirically our different estimators to the standard realized volatility. --Market microstructure,Serial dependence,High frequency data,Realized volatility,Subsampling,Two Scales Realized Volatility