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11-01-2023

Information loss in volatility measurement with flat price trading

Authors: Peter C. B. Phillips, Jun Yu

Published in: Empirical Economics | Issue 6/2023

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Abstract

A model of financial asset price determination is proposed that incorporates flat trading features into an efficient price process. The model involves the superposition of a Brownian semimartingale process for the efficient price and a Bernoulli process that determines the extent of flat price trading. The approach is related to sticky price modeling and the Calvo pricing mechanism in macroeconomic dynamics. A limit theory for the conventional realized volatility (RV) measure of integrated volatility is developed. The results show that RV is still consistent but has an inflated asymptotic variance that depends on the probability of flat trading. Estimated quarticity is similarly affected, so that both the feasible central limit theorem and the inferential framework suggested in Barndorff-Nielsen and Shephard (J Royal Stat Soc Ser B (Stat Methodol) 64:253–280, 2002) remain valid under flat price trading even though there is information loss due to flat trading effects. The results are related to work by Jacod (J Financ Econom 16:526–569, 2018) and Mykland and Zhang (Ann Stat 34:1931–1963, 2006) on realized volatility measures with random and intermittent sampling, and to ACD models for irregularly spaced transactions data. Extensions are given to include models with microstructure noise. Some simulation results are reported. Empirical evaluations with tick-by-tick data indicate that the effect of flat trading on the limit theory under microstructure noise is likely to be minor in most cases, thereby affirming the relevance of existing approaches.

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Jacod (2018) is a subsequently published version of a working paper that appeared much earlier and was dated as 1993 in later citations of the paper (Delattre and Jacod 1997; Barndorff-Nielsen and Shephard 2002).

Concerns about how to calculate RV based on business time sampling and transaction time sampling have received a great deal of attention in the RV literature; see Hansen and Lunde (2006); Oomen (2006). In particular, business time sampling relates to stopping time approaches and the connection is achieved via the Dubins-Schwarz Theorem. See Yu and Phillips (2001) for an application of the Dubins-Schwarz Theorem to estimate continuous-time models.

Mykland and Zhang (2006) assume the sampling points \(\tau _{j}\) to be deterministic but note later in their paper that the scheme covers the case where the sampling points are random but independent of the observed process. The stopping time scheme (1.10) is random and allows for dependence on past prices.

Note that the values \(L_{i}=0,1,2,\ldots ,\) correspond to realizations \(\xi _{i}=1,\) \(\left\{ \xi _{i}=1,\xi _{i-1}=0\right\} ,\) \(\left\{ \xi _{i}=1,\xi _{i-1}=0,\xi _{i-2}=0\right\} \) with respective probabilities \(\pi ,\) \(\pi \left( 1-\pi \right) ,\pi \left( 1-\pi \right) ^{2},.....\)

In this case we have \(\left\{ m^{-1}\sum _{\ell \le j}D_{m,\ell }<t\right\} , \) which under D2 is asymptotically equivalent to \(\left\{ \int _{0}^{j/m}\mu _{D}\left( s\right) \textrm{d}s<t\right\} .\) The measure \(\mu \left[ 0,t\right] \) is then given by \(\mu \left[ 0,t\right] =r\left( t\right) \) where \(\int _{0}^{r}\mu _{D}\left( s\right) \textrm{d}s=t\) so that \(\mu _{D}\left( r\right) dr=\textrm{d}t.\)

The datasets and dates are arbitrarily selected but are illustrative of heavily traded stocks.

Aït-Sahalia Y, Mykland PA, Zhang L (2005) How often to sample a continuous-time process in the presence of market microstructure noise. Rev Financ Stud 18:351–416

Aït-Sahalia Y, Mykland PA, Zhang L (2011) Ultra high frequency volatility estimation with dependent microstructure noise. J Econom 160:160–175

Andersen TG, Bollerslev T, Diebold FX (2010) “CHAPTER 2 - Parametric and Nonparametric Volatility Measurement,” In: Handbook of financial econometrics: tools and techniques, Aït-Sahalia Y, Hansen LP, San Diego: North-Holland, vol. 1 of Handbooks in Finance, pp 67–137

Andersen TG, Bollerslev T, Diebold FX, Ebens H (2001) The distribution of realized stock return volatility. J Financ Econ 61:43–76

Andersen TG, Bollerslev T, Diebold FX, Labys P (2001) The distribution of realized exchange rate volatility. J Am Stat Assoc 96:42–55

Andersen TG, Bollerslev T, Diebold FX, Labys P (2003) Modeling and Forecasting Realized Volatility. Econometrica 71:579–625

Andersen TG, Bollerslev T, Diebold FX, Wu J (2005) A framework for exploring the macroeconomic determinants of systematic risk. Am Econ Rev 95:398–404

Andersen TG, Bollerslev T, Meddahi N (2005) Correcting the errors: volatility forecast evaluation using high-frequency data and realized volatilities. Econometrica 73:279–296

Back K (1991) Asset pricing for general processes. J Math Econ 20:371–395

Bandi FM, Kolokolov A, Pirino D, Renò R (2020) Zeros. Manage Sci 66:3466–3479

Bandi FM, Pirino D, Renò R (2017) EXcess idle time. Econometrica 85:1793–1846

Bandi FM, Russell JR (2008) Microstructure noise, realized variance, and optimal sampling. Rev Econ Stud 75:339–369

Barndorff-Nielsen OE, Graversen SE, Jacod J, Shephard N (2006) Limit theorems for bipower variation in financial econometrics. Economet Theor 22:677–719

Barndorff-Nielsen OE, Hansen PR, Lunde A, Shephard N (2008) Designing realized Kernels to measure the ex post variation of equity prices in the presence of noise. Econometrica 76:1481–1536

Barndorff-Nielsen OE, Shephard N (2002) Econometric analysis of realized volatility and its use in estimating stochastic volatility models. J Royal Stat Soc Ser B (Statistical Methodology) 64:253–280

Bekaert G, Harvey C, Lundblad C (2007) Liquidity and expected returns: lessons from emerging markets. Rev Financ Stud 20:1783–1831

Berman SM (1962) A law of large numbers for the maximum in a stationary gaussian sequence. Ann Math Stat 33:93–97

Boswijk HP (2001) “Testing for a unit root with near-integrated volatility,” Tinbergen Institute Discussion Paper 01-077/4

Boswijk HP (2005) “Adaptive testing for a unit root with nonstationary volatility,” Tinbergen Institute Discussion Paper 2005/07

Calvo GA (1983) Staggered prices in a utility-maximizing framework. J Monet Econ 12:383–398

Corradi V (2000) Reconsidering the continuous time limit of the GARCH(1,1) process. J Econom 96:145–153

Da R, Xiu D (2021) When moving-average models meet high-frequency data: uniform inference on volatility. Econometrica 89:2787–2825

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

Engle RF (2000) The econometrics of ultra-high-frequency data. Econometrica 68:1–22

Engle RF, Russell JR (1998) Autoregressive conditional duration: a new model for irregularly spaced transaction data. Econometrica 66:1127–1162

Feller W (1951) Two singular diffusion problems. Ann Math 54:173–182

Fleming J, Kirby C, Ostdiek B (2003) The economic value of volatility timing using “realized’’ volatility. J Financ Econ 67:473–509

Ghysels E, Jasiak J (1998) “GARCH for Irregularly Spaced Financial Data: The ACD-GARCH Model,” Studies in Nonlinear Dynamics and Econometrics, 2

Hall P, Heyde CC (1980) Martingale limit theory and its application. Academic Press, New York

Hansen PR, Lunde A (2006) Realized variance and market microstructure noise. J Business Econom Stat 24:127–161

Heston SL (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 6:327–343

Jacod J (2018) Limit of random measures associated with the increments of a Brownian Semimartingale. J Financ Economet 16:526–569

Jacod J, Shiryaev AN (1987) Limit theorems for stochastic processes. Springer-Verlag, New York

Lesmond DA (2005) Liquidity of emerging markets. J Financ Econ 77:411–452

Mankiw NG, Reis R (2002) Sticky information versus sticky prices: a proposal to replace the new Keynesian Phillips curve. Q J Econ 117:1295–1328

Mykland PA, Zhang L (2006) ANOVA for diffusions and Itô processes. Ann Stat 34:1931–1963

Naveau P (2003) Almost sure relative stability of the maximum of a stationary sequence. Adv Appl Probab 35:721–736

Nelson DB (1990) ARCH models as diffusion approximations. J Econom 45:7–38

Oomen RCA (2006) Properties of realized variance under alternative sampling schemes. J Business Econom Stat 24:219–237

Phillips PCB, Yu J (2007) “Information loss in volatility measurement with flat price trading,” Cowles Foundation Discussion Paper No. 1598, Yale University

Protter P (2004) Stochastic integration and differential equations. Springer-Verlag, New York

Reis R (2006) Inattentive producers. Rev Econ Stud 73:793–821

Schilling MF (1990) The longest run of heads. College Math J 21:196–207

Schmidt P (1976) Econometrics. Marcel Dekker, New York

Yu J, Phillips PCB (2001) A Gaussian approach for continuous time models of the short-term interest rate. Economet J 4:210–224

Zhang L (2006) Efficient estimation of stochastic volatility using noisy observations: A multi-scale approach. Bernoulli 12:1019–1043

Zhang L, Mykland PA, Aït-Sahalia Y (2005) A tale of two time scales. J Am Stat Assoc 100:1394–1411

Zhou B (1996) High-frequency data and volatility in foreign-exchange rates. J Business Econom Stat 14:45–52

Title: Information loss in volatility measurement with flat price trading
Authors: Peter C. B. Phillips
Jun Yu
Publication date: 11-01-2023
Publisher: Springer Berlin Heidelberg
Published in: Empirical Economics / Issue 6/2023
Print ISSN: 0377-7332
Electronic ISSN: 1435-8921
DOI: https://doi.org/10.1007/s00181-022-02353-y

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