## Thesis etd-09192013-130357 |

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Thesis type

Tesi di dottorato di ricerca

Author

CURATO, IMMA VALENTINA

URN

etd-09192013-130357

Thesis title

Non parametric estimations of volatility of volatility and leverage using integral transforms

Academic discipline

SECS-S/06

Course of study

SCIENZE AZIENDALI, ECONOMICHE E MATEMATICO-STATISTICHE APPLICATE ALL'ECONOMIA

Supervisors

**tutor**Mancino, Maria Elvira

Keywords

- Fourier transform
- high-frequency data
- Laplace transform
- leverage
- non-parametric estimation
- semi-martingale
- Spot volatility
- volatility of volatility

Graduation session start date

04/10/2013

Availability

Full

Summary

In this dissertation, we focus on the study of new estimators of the volatility, of the leverage and the volatility of volatility based on the Fourier methodology proposed by Malliavin and Mancino, applied for the first time to the computation of the spot volatility in Malliavin and Mancino, 2002 and developed in Malliavin and Mancino, 2009 and in Mancino and Sanfelici, 2012 for the computation of covariance and quarticity. Such methodology is based on the integration of the time series of returns rather than on its differentiation. On an important note, the spot estimator is obtained without performing any numerical derivative, which causes numerical instabilities. Moreover, as shown in the papers Mancino and Sanfelici, 2008,2011, 2012, respectively the integrated estimators of

volatility, covariance and quarticity are robust under microstructure noise.

We define estimators incorporating the above good features, together with some additional properties that aim to solve some bottlenecks of the

classical Fourier estimations and of the non parametric procedures designed to estimate

the leverage and the volatility of volatility in the presence of microstructure contamination effects.

The present dissertation has the following structure.

In Chapter 1, we define an estimator of the spot volatility based on the Laplace transform.

We aim to avoid the artificial periodization subjacent to Fourier series methodology,

responsible for the low precision of the spot Fourier estimator at the boundary of

the time window. The Laplace transform allows to build a spot estimator based on a long time series of prices by smooting past data and retaining

recent price observations.

From this procedure, we obtain a spot estimator that represents a bridge between two different methods of computation of volatility: the method based on quadratic variation and the classical

Fourier spot estimation based on a convolution product formula.

Doing so, we prove that the Laplace transform spot estimator is consistent and by means of simulation results we show the higher precision of the provided estimation at the boundary of the time horizon.

In Chapter 2, we define a new estimator of the leverage stochastic process based only on a pre-estimation of the Fourier coefficients of the volatility process.

This feature constitutes a novelty in comparison with the leverage estimators proposed in the literature generally based on a

pre-estimation of the spot volatility. Our estimator is proved to be consistent and in virtue of its definition it can be directly applied to estimate the leverage effect in case of irregular trading observations of the price path.

We also perform a simulation study in which the robustness of the Fourier estimator

is shown in the presence of microstructure noise effects.

Finally, in Chapter 3, we deal with an estimator of the volatility of volatility obtained only by means of a pre-estimation of the Fourier coefficients of the volatility.

We investigate its finite sample properties in presence of i.i.d Gaussian noise contamination by computing the bias of the estimator due to noise,

showing that it does not diverge when the number of observations increases.

Recently, some papers have proposed methods for the estimation of stochastic volatility of volatility based on a pre-estimation of the spot volatility.

However, these estimators does not take into account the microstructure contamination

effects, which would generate some difficulties because of the estimator's sensitivity to noise in the absence of manipulations on the initial dataset.

In a simulation study, the performances of the Fourier estimator are compared

with the pre-estimated spot variance based realized variance estimator proposed in Barndorff-Nielsen and Varaart, 2009, in the absence and in the presence of microstructure noise contaminations.

In virtue of its definition, the Fourier estimator performs considerably better than the aforementioned estimator in relation to the

bias and the mean squared error of the volatility of volatility.

volatility, covariance and quarticity are robust under microstructure noise.

We define estimators incorporating the above good features, together with some additional properties that aim to solve some bottlenecks of the

classical Fourier estimations and of the non parametric procedures designed to estimate

the leverage and the volatility of volatility in the presence of microstructure contamination effects.

The present dissertation has the following structure.

In Chapter 1, we define an estimator of the spot volatility based on the Laplace transform.

We aim to avoid the artificial periodization subjacent to Fourier series methodology,

responsible for the low precision of the spot Fourier estimator at the boundary of

the time window. The Laplace transform allows to build a spot estimator based on a long time series of prices by smooting past data and retaining

recent price observations.

From this procedure, we obtain a spot estimator that represents a bridge between two different methods of computation of volatility: the method based on quadratic variation and the classical

Fourier spot estimation based on a convolution product formula.

Doing so, we prove that the Laplace transform spot estimator is consistent and by means of simulation results we show the higher precision of the provided estimation at the boundary of the time horizon.

In Chapter 2, we define a new estimator of the leverage stochastic process based only on a pre-estimation of the Fourier coefficients of the volatility process.

This feature constitutes a novelty in comparison with the leverage estimators proposed in the literature generally based on a

pre-estimation of the spot volatility. Our estimator is proved to be consistent and in virtue of its definition it can be directly applied to estimate the leverage effect in case of irregular trading observations of the price path.

We also perform a simulation study in which the robustness of the Fourier estimator

is shown in the presence of microstructure noise effects.

Finally, in Chapter 3, we deal with an estimator of the volatility of volatility obtained only by means of a pre-estimation of the Fourier coefficients of the volatility.

We investigate its finite sample properties in presence of i.i.d Gaussian noise contamination by computing the bias of the estimator due to noise,

showing that it does not diverge when the number of observations increases.

Recently, some papers have proposed methods for the estimation of stochastic volatility of volatility based on a pre-estimation of the spot volatility.

However, these estimators does not take into account the microstructure contamination

effects, which would generate some difficulties because of the estimator's sensitivity to noise in the absence of manipulations on the initial dataset.

In a simulation study, the performances of the Fourier estimator are compared

with the pre-estimated spot variance based realized variance estimator proposed in Barndorff-Nielsen and Varaart, 2009, in the absence and in the presence of microstructure noise contaminations.

In virtue of its definition, the Fourier estimator performs considerably better than the aforementioned estimator in relation to the

bias and the mean squared error of the volatility of volatility.

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