Maria Elvira Mancino

Fourier series method for measurement of multivariate volatilities

Abstract. We present a methodology based on Fourier series analysis to compute time series volatility when the data are observations of a semimartingale. The procedure is not based on the Wiener theorem for the quadratic variation, but on the computation of the Fourier coefficients of the process and therefore it relies on the integration of the time series rather than on its differentiation. The method is fully model free and nonparametric. These features make the method well suited for financial market applications, and in particular for the analysis of high frequency time series and for the computation of cross volatilities.

Robustness of Fourier estimator of integrated volatility in the presence of microstructure noise

The finite sample properties of the Fourier estimator of integrated volatility under market microstructure noise are studied. Analytic expressions for the bias and the mean squared error (MSE) of the contaminated estimator are derived. These formulae can be practically used to design optimal MSE-based estimators, which are very robust and efficient in the presence of noise. Moreover an empirical analysis based on a simulation study and on high-frequency logarithmic prices of the Italian stock index futures (FIB30) validates the theoretical results.

Asset pricing with a forward-backward stochastic differential utility

Abstract In an intertemporal setting we model the anticipation–disappointment effect through a habit formation process which is a function of past consumption and of past expected utility. We show that in equilibrium the anticipation effect reduces the risk premium, whereas the disappointment effect induces a higher risk premium.

Estimating Covariance Via Fourier Method in the Presence of Asynchronous Trading and Microstructure Noise

We analyze the effects of market microstructure noise on the Fourier estimator of multivariate volatilities. We prove that the estimator is consistent in the case of asynchronous data and asymptotically unbiased in the presence of various types of microstructure noise. This result is obtained through an analytical computation of the bias and the mean squared error of the Fourier estimator and confirmed by Monte Carlo experiments. A comparison with several covariance estimators is performed. (JEL: C14, C32, G1) Copyright The Author 2010. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org, Oxford University Press.

Estimation of quarticity with high-frequency data

We propose a new methodology based on Fourier analysis to estimate the fourth power of the volatility function (spot quarticity) and, as a byproduct, the integrated function. We prove the consistency of the proposed estimator of the integrated quarticity. Further, we analyse its efficiency in the presence of microstructure noise, from both a theoretical and empirical viewpoint. Extensions to higher powers of volatility and to the multivariate case are also discussed.

Computation of Volatility in Stochastic Volatility Models with High Frequency Data

We consider general stochastic volatility models driven by continuous Brownian semimartingales, we show that the volatility of the variance and the leverage component (covariance between the asset price and the variance) can be reconstructed pathwise by exploiting Fourier analysis from the observation of the asset price. Specifying parametrically the asset price model we show that the method allows us to compute the parameters of the model. We provide a Monte Carlo experiment to recover the volatility and correlation parameters of the Heston model.

A comparison result for FBSDE with applications to decisions theory

In general, a comparison Lemma for the solutions of Forward-Backward Stochastic Differential Equations (FBSDE) does not hold. Here we prove one for the backward component at the initial time, relying on certain monotonicity conditions on the coefficients of both components. Such a result is useful in applications. Indeed, one can use FBSDEs to define a utility functional able to capture the disappointment-anticipation effect for an agent in an intertemporal setting under risk. Exploiting our comparison result, we prove some “desirable” properties for the utility functional, such as continuity, concavity, monotonicity and risk aversion. Finally, for completeness, in a Markovian setting, we characterize the utility process by means of a degenerate parabolic partial differential equation.

Fourier volatility forecasting with high frequency data and microstructure noise

We study the forecasting performance of the Fourier volatility estimator in the presence of microstructure noise. Analytical comparison and simulation studies indicate that the Fourier estimator significantly outperforms realized volatility-type estimators, particularly for high-frequency data and when the noise component is relevant. We show that the Fourier estimator generally exhibits better performance, even compared with methods specifically designed to handle market microstructure contamination.

Optimal strategies in a risky debt context

This paper analyses structural models for the evaluation of risky debt following Leland (J. Finance 49 (1994), pp. 1213–1252) with an approach of optimal stopping problem. Moreover, we introduce an investment control parameter and we optimize with respect to the failure threshold and coupon rate. We show that the value of the optimal coupon policy decreases if the strict priority rule is removed.

Fourier Spot Volatility Estimator: Asymptotic Normality and Efficiency with Liquid and Illiquid High-Frequency Data

The recent availability of high frequency data has permitted more efficient ways of computing volatility. However, estimation of volatility from asset price observations is challenging because observed high frequency data are generally affected by noise-microstructure effects. We address this issue by using the Fourier estimator of instantaneous volatility introduced in Malliavin and Mancino 2002. We prove a central limit theorem for this estimator with optimal rate and asymptotic variance. An extensive simulation study shows the accuracy of the spot volatility estimates obtained using the Fourier estimator and its robustness even in the presence of different microstructure noise specifications. An empirical analysis on high frequency data (U.S. S&P500 and FIB 30 indices) illustrates how the Fourier spot volatility estimates can be successfully used to study intraday variations of volatility and to predict intraday Value at Risk.