Abstract
The goal of this note is to describe a model for ultra high frequency prices and durations, the model with uncertainty zones developed in [27]. We also give some results from [28] and [29] which show how it can be used in practice for statistical estimation or in order to hedge derivatives. Before introducing this model, we briefly recall the classical approaches of price modelling in the so-called microstructure noise literature.
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References
Andersen, T.G., Bollerslev, T., Diebold, F.X., Labys, P. (2000) Great Realizations. Risk, 13, 105–108
Ball, C. (1988) Estimation bias induced by discrete security prices. Journal of Finance, 43, 841–865
Bandi, F.M., Russel, J.R. (2005) Microstructure noise, realized variance and optimal sampling. To appear in Review of Economic Studies
Barndorff-Nielsen O.E., Hansen P., Lunde A., Shephard, N. (2008) Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise. Econometrica, 76, 1481–1536
Bertsimas, D., Kogan, L., Lo, A.W. (2000) When is time continuous? Journal of Financial Economics, 55, 173–204
Black, F., Scholes, M. (1973) The pricing of options and corporate liabilities. Journal of Political Economy, 81, 637–654
Cetin, U., Soner, M., Touzi, N. (2009) Option hedging under liquidity costs. To appear in Finance and Stochastics
Cho, D., Frees, E. (1988) Estimating the volatility of discrete stock prices. Journal of Finance, 43, 451–466
Delbaen, F., Schachermayer, W. (1994) A general version of the fundamental theorem of asset pricing. Mathematische Annalen, 300, 463–520
Delattre, S., Jacod, J. (1997) A central limit theorem for normalized functions of the increments of a diffusion process in the presence of round-off errors. Bernoulli, 3, 1–28
El Karoui, N., Jeanblanc-Picqué, M., Shreve, S. (1998) Robustness of the Black and Scholes formula. Mathematical Finance, 8, 93–126
Engle, R.F. (2000) The econometrics of ultra high frequency data. Econometrica, 68, 1–22
Gloter, A., Jacod, J. (1997) Diffusions with measurement errors, I. Local Asymptotic Normality, II. Optimal estimators. ESAIM PS, 5, 225–260
Gottlieb, G., Kalay, A. (1985) Implications of the Discreteness of Observed Stock Prices. Journal of Finance, 40, 135–154
Gourieroux, C., Jasiak, J. (2001) Financial Econometrics: Problems, Models, and Methods. Princeton University Press
Hansen, P.R., Lunde, A. (2006) Realized variance and market microstructure noise. Journal of Business and Economics Statistics, 24, 127–161
Hayashi, T., Mykland, P.A. (2005) Hedging errors: an asymptotic approach. Mathematical Finance, 15, 309–343
Hayashi, T., Yoshida, N. (2005) On covariance estimation of non-synchronously observed diffusion processes. Bernoulli, 11, 359–379
Jacod, J., Shiryaev, A.N. (2002) Limit Theorems for Stochastic Processes. Springer-Verlag, Second Edition
Jacod, J., Li, Y., Mykland, P.A., Podolskij, M., Vetter, M. (2009) Microstructure noise in the continuous case: the pre-averaging Approach. Stochastic Processes and their Applications, 119, 2249–2276
Jouini, E., Kallal, H. (1995) Arbitrage in securities markets with short sales constraints. Mathematical Finance, 5, 197–232
Kabanov, Y.M., Last, G. (2002) Hedging under transaction costs in currency markets: a continuous-time model. Mathematical Finance, 12, 63–70
Kabanov, Y.M., Safarian, M.M. (1997) On Leland’s strategy of option pricing with transactions costs. Finance and Stochastics, 1, 239–250
Kalnina, I., Linton, O.B. (2008) Estimating quadratic variation consistently in the presence of correlated measurement error. Journal of Econometrics, 147, 47–59
Koehl, P.F., Pham, H., Touzi, N. (1999) Hedging in discrete time under transaction costs and continuous-time limit. Journal of Applied Probability, 36, 163–178
Li, Y., Mykland, P.A. (2007) Are volatility estimators robust with respect to modelling assumptions? Bernoulli, 13, 601–622
Robert, C.Y., Rosenbaum, M. (2009) A new approach for the dynamics of ultra high frequency data: the model with uncertainty zones. To appear in Journal of Financial Econometrics
Robert, C.Y., Rosenbaum, M. (2009) Volatility and covariation estimation when microstructure noise and trading times are endogenous. To appear in Mathematical Finance
Robert, C.Y., Rosenbaum, M. (2010) On the microstructural hedging error. SIAM Journal on Financial Mathematics, 1, 427–453
Rosenbaum, M. (2009) Integrated volatility and round-off error. Bernoulli, 15, 687–720
Tankov, P., Voltchkova, E. (2008) Asymptotic analysis of hedging errors in models with jumps. Stochastic Processes and their Applications, 119, 2004–2027
Voev, V., Lunde, A. (2007) Integrated covariance estimation using high-frequency data in the presence of noise. Journal of Financial Econometrics, 5, 68–104
Zhang, L. (2006a) Efficient estimation of stochastic volatility using noisy observations: a multi-scale approach. Bernoulli, 12, 1019–1043
Zhang, L. (2006b) Estimating covariation: Epps effect and microstructure noise. A paraître dans Journal of Econometrics
Zhang, L., Mykland, P.A., Aït-Sahalia, Y. (2005) A Tale of two time scales: determining integrated volatility with noisy high-frequency data. Journal of the American Statistical Association, 100, 1394–1411
Zhou, B. (1996) High-frequency data and volatility in foreign-exchange rates. Journal of Business and Economic Statistics, 14, 45–52
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Robert, C.Y., Rosenbaum, M. (2011). The Model with Uncertainty Zones for Ultra High Frequency Prices and Durations: Applications to Statistical Estimation and Mathematical Finance. In: Abergel, F., Chakrabarti, B.K., Chakraborti, A., Mitra, M. (eds) Econophysics of Order-driven Markets. New Economic Windows. Springer, Milano. https://doi.org/10.1007/978-88-470-1766-5_14
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DOI: https://doi.org/10.1007/978-88-470-1766-5_14
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