Estimating quadratic variation of prices and spreads from
the Swedish limit order book
Suad Elezović∗
Department of Statistics,
Umeå University, Sweden
Abstract
This paper is concerned with some recent theoretical and empirical studies about variation of asset prices in high-frequency financial data with applications to the Swedish
limit order book (LOB) data. The main objective here is to study volatility aspect
of the stock prices and the corresponding returns through a genuine measure called
quadratic variation (QV). The most obvious estimator of the unknown QV process is
the realized qudratic variation (RQV), defined as the sum of squared returns of asset
prices (e.g. Barndorff-Nielsen and Shephard, 2005). We are also interested in studying
liquidity which may be thought as the difference between the ask and the bid prices
(spread).
In our approach we extend the estimation of volatility of efficient prices to estimation of volatility of the bid (ask) price impact curves. These curves are the average
prices of specified quantities of shares on the bid (ask) side of the book. Then, the
functional mid-quotes and spreads are created by using the price impact curves of
each quantity. Finally, the daily sums of squared returns are calculated for each such
collection of mid-quotes and spread curves, obtaining the daily RQV estimates.
This approach is motivated by an assumption that the bid and the ask prices impact curves, since they are functions of quantities, might be more informative about
liquidity than the ordinary bid and ask prices. By applying this modified RQV estimator to the Swedish (Ericsson B stock) LOB data, we obtain empirical results that do
not indicate any remarkable violation of the basic theoretical assumptions. Particularly, we focus on finding the optimal time interval for calculations of squared returns.
Our results confirm the main empirical results from the comparing studies about the
microstructure effects: The bias of the estimator gets increasing as the interpolation
time interval approaches to zero.
Keywords: Quadratic variation, realized volatility, limit order book, liquidity, price
impact curves.
References
Barndorff-Nielsen, O. E. and Shephard, N. (2005) Variation, jumps, market frictions and
high frequency data in financial econometrics. Prepared for the invited symposium in
Financial Econometrics, 9th Congress of the Econometric Society, London, 20th August
2005.
∗
Tel.[+46] 090–786–5635, e-mail:[email protected]
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