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  • Torben G. Andersen (a1), Nicola Fusari (a2), Viktor Todorov (a1) and Rasmus T. Varneskov (a1)

We develop parametric inference procedures for large panels of noisy option data in a setting, where the underlying process is of pure-jump type, i.e., evolves only through a sequence of jumps. The panel consists of options written on the underlying asset with a (different) set of strikes and maturities available across the observation times. We consider an asymptotic setting in which the cross-sectional dimension of the panel increases to infinity, while the time span remains fixed. The information set is augmented with high-frequency data on the underlying asset. Given a parametric specification for the risk-neutral asset return dynamics, the option prices are nonlinear functions of a time-invariant parameter vector and a time-varying latent state vector (or factors). Furthermore, no-arbitrage restrictions impose a direct link between some of the quantities that may be identified from the return and option data. These include the so-called jump activity index as well as the time-varying jump intensity. We propose penalized least squares estimation in which we minimize the L2 distance between observed and model-implied options. In addition, we penalize for the deviation of the model-implied quantities from their model-free counterparts, obtained from the high-frequency returns. We derive the joint asymptotic distribution of the parameters, factor realizations and high-frequency measures, which is mixed Gaussian. The different components of the parameter and state vector exhibit different rates of convergence, depending on the relative (asymptotic) informativeness of the high-frequency return data and the option panel.

Corresponding author
*Address correspondence to Viktor Todorov, Department of Finance, Kellogg School of Management, Northwestern University, Evanston, IL 60208, USA; e-mail:
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Andersen and Varneskov gratefully acknowledge support from CREATES, Center for Research in Econometric Analysis of Time Series (DNRF78), funded by the Danish National Research Foundation. The work is partially supported by NSF Grant SES-1530748. We would like to thank the Editor (Peter C.B. Phillips), Co-Editor (Dennis Kristensen), and anonymous referees for many useful comments and suggestions.

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Ait-Sahalia, Y. & Jacod, J. (2009) Estimating the degree of activity of jumps in high frequency financial data. Annals of Statistics 37, 22022244.
Andersen, T.G., Fusari, N., & Todorov, V. (2015) Parametric inference and dynamic state recovery from option panels. Econometrica 83, 10811145.
Andersen, T.G., Fusari, N., Todorov, V., & Varneskov, R.T. (2018) Unified inference for nonlinear factor models from panels with fixed and large time span. Journal of Econometrics, forthcoming.
Barndorff-Nielsen, O.E. & Shephard, N. (2001) Non-gaussian ornstein-uhlenbeck-based models and some of their applications in financial economics. Journal of the Royal Statistical Society: Series B 63, 167241.
Bernanke, B., Boivin, J., & Eliasz, P. (2005) Measuring the effects of monetary policy: A Factor-Augmented Vector Autoregressive (FAVAR) Approach. Quarterly Journal of Economics 120, 387422.
Bull, A. (2016) Near-optimal estimation of jump activity in semimartingales. Annals of Statistics 44, 5886.
Carr, P., Geman, H., Madan, D., & Yor, M. (2002) The fine structure of asset returns: An empirical investigation. Journal of Business 75, 305332.
Carr, P., Geman, H., Madan, D.B., & Yor, M. (2003) Stochastic volatility for Lévy processes. Journal of Business 75, 345382.
Carr, P. & Madan, D. (2001) Optimal positioning in derivative securities. Quantitiative Finance 1, 1937.
Carr, P. & Wu, L. (2003) The finite moment log stable process and option pricing. Journal of Finance LVIII, 753778.
Carr, P. & Wu, L. (2004) Time-changed Lévy processes and option pricing. Journal of Financial Economics 17, 113141.
Duffie, D. (2001) Dynamic Asset Pricing Theory, 3rd ed. Princeton University Press.
Duffie, D., Pan, J., & Singleton, K. (2000) Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 13431376.
Fama, E. (1963) Mandelbrot and the stable paretian hypothesis. Journal of Business 36, 420429.
Fama, E. & Roll, R. (1968) Some properties of symmetric stable distributions. Journal of the American Statistical Association 63, 817836.
Filipović, D. (2001) A general characterization of one factor affine term structure models. Finance Stochastic 5, 389412.
Hounyo, U. & Varneskov, R.T. (2017) A local stable bootstrap for power variations of pure-jump semimartingales and activity index estimation. Journal of Econometrics 198, 1028.
Jacod, J. & Protter, P. (2012) Discretization of Processes. Springer-Verlag.
Jacod, J. & Shiryaev, A.N. (2003) Limit Theorems For Stochastic Processes, 2nd ed. Springer-Verlag.
Jacod, J. & Todorov, V. (2018) Limit theorems for integrated local empirical characteristic exponents from noisy high-frequency data with application to volatility and jump activity estimation. Annals of Applied Probability, forthcoming.
Jing, B., Kong, X., & Liu, Z. (2011) Estimating the jump activity index under noisy observations using high-frequency data. Journal of the American Statistical Association 106, 558568.
Jing, B., Kong, X., & Liu, Z. (2012) Modeling high-frequency financial data by pure jump processes. Annals of Statistics 40, 759784.
Jing, B., Kong, X., Liu, Z., & Mykland, P. (2012) On the jump activity index of semimartingales. Journal of Econometrics 166, 213223.
Kalnina, I. & Xu, D. (2017) Nonparametric estimation of the leverage effect: A trade-off between robustness and efficiency. Journal of the American Statistical Association 112, 384396.
Kong, X., Liu, Z., & Jing, B. (2015) Testing for pure-jump processes for high-frequency data. Annals of Statistics 43, 847877.
Madan, D. & Milne, F. (1991) Option pricing with VG martingale components. Mathematical Finance 1, 3956.
Madan, D. & Seneta, E. (1990) The variance gamma (VG) model for share market returns. Journal of Business 63, 511524.
Mandelbrot, B. (1961) Stable paretian random functions and the multiplicative variation of income. Econometrica 29, 517543.
Mandelbrot, B. (1963) The variation of certain speculative prices. Journal of Business 36, 394419.
Park, J.Y. & Phillips, P.C.B. (1988) Statistical inference in regressions with integrated processes: Part 1. Econometric Theory 4, 468497.
Park, J.Y. & Phillips, P.C.B. (1989) Statistical inference in regressions with integrated processes: Part 2. Econometric Theory 5, 95132.
Phillips, P.C.B. (1988) Weak convergence of sample covariance matrices to stochastic integrals via martingale approximations. Econometric Theory 4, 528533.
Qin, L. & Todorov, V. (2018) Nonparametric implied Lévy densities. Annals of Statistics, forthcoming.
Rosinski, J. (2007) Tempering stable processes. Stochastic Processes and Applications 117, 677707.
Sato, K. (1999) Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.
Sims, C., Stock, J., & Watson, M. (1990) Inference in linear time series models with some unit roots. Econometrica 58, 113144.
Todorov, V. (2015) Jump activity estimation for pure-jump semimartingales via self-normalized statistics. Annals of Statistics 43, 18311864.
Todorov, V. & Tauchen, G. (2011a) Limit theorems for power variations of pure-jump processes with application to activity estimation. The Annals of Applied Probability 21, 546588.
Todorov, V. & Tauchen, G. (2011b) Volatility jumps. Journal of Business and Economic Statistics 29, 356371.
Todorov, V. & Tauchen, G. (2012) Realized laplace transforms for pure-jump semimartingales. Annals of Statistics 40, 12331262.
Woerner, J. (2003) Variational sums and power variation: A unifying approach to model selection and estimation in semimartingale models. Statistics and Decisions 21, 4768.
Woerner, J. (2007) Inference in Lévy-type stochastic volatility models. Advances in Applied Probability 39, 531549.
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Econometric Theory
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