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)

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.

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 (Green's 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 provid...

In this work, the Fourier-cosine series (COS) method has been combined with the Boundary Element Method (BEM) for a fast evaluation of barrier option prices. After a description of its use in the Black and Scholes (BS) model, the focus of the paper is on the application of the proposed methodology to the barrier option evaluation in the Heston model, where its contribution is fundamental to improve computational efficiency and to make BEM appealing among finance practitioners as a valid alternative to Monte Carlo (MC) or other more traditional approaches. An error analysis is provided on the number of terms used in the Fourier-cosine series expansion, where the error bound estimation is based on the characteristic function of the log-asset price process. A Matlab code implementing this technique is attached at the end of the paper.