Simona Sanfelici

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.

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.

An Improved Two-Step Regularization Scheme for Spot Volatility Estimation

We are concerned with the problem of parameter estimation in Finance, namely the estimation of the spot volatility in the presence of the so-called microstructure noise. In [16] a scheme based on the technique of multi-step regularization was presented. It was shown that this scheme can work in a real-time manner. However, the main drawback of this scheme is that it needs a lot of observation data. The aim of the present paper is to introduce an improvement of the scheme such that the modified estimator can work more efficiently and with a data set of smaller size. The technical aspects of implementation of the scheme and its performance on simulated data are analyzed. The proposed scheme is tested against other estimators, namely a realized volatility type estimator, the Fourier estimator and two kernel estimators.

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.

Optimal impulse control on an unbounded domain with nonlinear cost functions

Abstract.In this paper we consider the optimal impulse control of a system which evolves randomly in accordance with a homogeneous diffusion process in ℜ1. Whenever the system is controlled a cost is incurred which has a fixed component and a component which increases with the magnitude of the control applied. In addition to these controlling costs there are holding or carrying costs which are a positive function of the state of the system. Our objective is to minimize the expected discounted value of all costs over an infinite planning horizon. Under general assumptions on the cost functions we show that the value function is a weak solution of a quasi-variational inequality and we deduce from this solution the existence of an optimal impulse policy. The computation of the value function is performed by means of the Finite Element Method on suitable truncated domains, whose convergence is discussed.

Fast Numerical Pricing of Barrier Options under Stochastic Volatility and Jumps

In this paper, we prove the existence of an integral closed-form solution for pricing barrier options in both Heston and Bates frameworks. The option value depends on time, on the price, and on the volatility of the underlying asset, and it can be computed as the solution of a two-dimensional pricing partial integro-differential equation. The integral representation formula of the solution is derived by projection of the differential equation and exploiting the properties of the adjoint operator. We derive the expression of the fundamental solution (Greens function) necessary for the integral representation formula. The computation is based on the interpretation of the fundamental solution as the joint transition probability density function of the underlying asset price and variance and is obtained through Fourier inverse transform of a suitable conditional characteristic function. We propose a numerical scheme to approximate the option price based on the classical boundary element method, and we provide two numerical examples showing the computational efficiency and accuracy of the proposed new method. The algorithm can be modified to compute greeks as well.

A boundary element approach to barrier option pricing in Black–Scholes framework

We treat the way to achieve great computational savings and accuracy in the evaluation of barrier options through boundary element method (BEM). The proposed method applies to quite general pricing models. The only requirement is the knowledge of the characteristic function for the underlying asset distribution, usually available under general asset models. This paper serves as an introductory work to illustrate the implementation of BEM using numerical Fourier inverse transform of the characteristic function and to numerically show its stability and efficiency under simple frameworks such as the Black–Scholes model.

Chapter 24 – Measuring the Leverage Effect in a High-Frequency Trading Framework

Multifactor stochastic volatility models of the financial time series can have important applications in portfolio management and pricing/hedging of financial instruments. Based on the semimartingale paradigm, we focus on the study and the estimation of the leverage effect, defined as the covariance between the price and the volatility process and modeled as a stochastic process. Our estimation procedure is based only on a preestimation of the Fourier coefficients of the volatility process. This approach constitutes a novelty in comparison with the nonparametric leverage estimators proposed in the literature, generally based on a preestimation of spot volatility, and it can be directly applied to estimate the leverage effect in the case of irregular trading observations and in the presence of microstructure noise contaminations, that is, in a high-frequency framework. The finite sample performances of the Fourier estimator of the leverage are tested in numerical simulations and in an empirical application to S&P 500 index futures.

Covariance Estimation and Dynamic Asset Allocation under Microstructure Effects via Fourier Methodology

We analyze the properties of different estimators of multivariate volatilities in the presence of microstructure noise, with particular focus on the Fourier estimator. This estimator is consistent in the case of asynchronous data and robust to microstructure effects; further we prove the positive semi-definiteness of the estimated covariance matrix. The in sample and forecasting properties of Fourier method are analyzed through Monte Carlo simulations. We study the economic benefit of applying the Fourier covariance estimation methodology over other estimators in the presence of market microstructure noise from the perspective of an asset-allocation decision problem. We find that using Fourier methodology yields statistically significant economic gains under strong microstructure effects.

High-frequency volatility of volatility estimation free from spot volatility estimates

We define a new consistent estimator of the integrated volatility of volatility based only on a pre-estimation of the Fourier coefficients of the volatility process. We investigate the finite sample properties of the estimator in the presence of noise contaminations by computing the bias of the estimator due to noise and showing that it vanishes as the number of observations increases, under suitable assumptions. In both simulated and empirical studies, the performance of the Fourier estimator with high-frequency data is investigated and it is shown that the proposed estimator of volatility of volatility is easily implementable, computationally stable and even robust to market microstructure noise.