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# Large Deviations Of The Realized (Co-)Volatility Vector

## Author

Listed:
• Hacène Djellout

(LMBP - Laboratoire de Mathématiques Blaise Pascal - UBP - Université Blaise Pascal - Clermont-Ferrand 2 - CNRS - Centre National de la Recherche Scientifique)

• Arnaud Guillin

(IUF - Institut Universitaire de France - M.E.N.E.S.R. - Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche, LMBP - Laboratoire de Mathématiques Blaise Pascal - UBP - Université Blaise Pascal - Clermont-Ferrand 2 - CNRS - Centre National de la Recherche Scientifique)

• Yacouba Samoura

(LMBP - Laboratoire de Mathématiques Blaise Pascal - UBP - Université Blaise Pascal - Clermont-Ferrand 2 - CNRS - Centre National de la Recherche Scientifique)

## Abstract

Realized statistics based on high frequency returns have become very popular in financial economics. In recent years, different non-parametric estimators of the variation of a log-price process have appeared. These were developed by many authors and were motivated by the existence of complete records of price data. Among them are the realized quadratic (co-)variation which is perhaps the most well known example, providing a consistent estimator of the integrated (co-)volatility when the logarithmic price process is continuous. Limit results such as the weak law of large numbers or the central limit theorem have been proved in different contexts. In this paper, we propose to study the large deviation properties of realized (co-)volatility (i.e., when the number of high frequency observations in a fixed time interval increases to infinity. More specifically, we consider a bivariate model with synchronous observation schemes and correlated Brownian motions of the following form: $dX_{\ell,t} = \sigma_{\ell,t}dB_{\ell,t}+b_{\ell}(t,\omega)dt$ for $\ell=1,2$, where $X_{\ell}$ denotes the log-price, we are concerned with the large deviation estimation of the vector $V_t^n(X)=\left(Q_{1,t}^n(X), Q_{2,t}^n(X), C_{t}^n(X)\right)$ where $Q_{\ell,t}^n(X)$ and $C_{t}^n(X)$ represente the estimator of the quadratic variational processes $Q_{\ell,t}=\int_0^t\sigma_{\ell,s}^2ds$ and the integrated covariance $C_t=\int_0^t\sigma_{1,s}\sigma_{2,s}\rho_sds$ respectively, with $\rho_t=cov(B_{1,t}, B_{2,t})$. Our main motivation is to improve upon the existing limit theorems. Our large deviations results can be used to evaluate and approximate tail probabilities of realized (co-)volatility. As an application we provide the large deviation for the standard dependence measures between the two assets returns such as the realized regression coefficients up to time $t$, or the realized correlation. Our study should contribute to the recent trend of research on the (co-)variance estimation problems, which are quite often discussed in high-frequency financial data analysis.

## Suggested Citation

• Hacène Djellout & Arnaud Guillin & Yacouba Samoura, 2017. "Large Deviations Of The Realized (Co-)Volatility Vector," Post-Print hal-01082903, HAL.
• Handle: RePEc:hal:journl:hal-01082903
Note: View the original document on HAL open archive server: https://hal.archives-ouvertes.fr/hal-01082903
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## References listed on IDEAS

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## More about this item

### Keywords

large deviations; diffusion; discrete-time observation; Realised Volatility and covolatility;
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