The Annals of Statistics

Nonparametric implied Lévy densities

Likuan Qin and Viktor Todorov

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This paper develops a nonparametric estimator for the Lévy density of an asset price, following an Itô semimartingale, implied by short-maturity options. The asymptotic setup is one in which the time to maturity of the available options decreases, the mesh of the available strike grid shrinks and the strike range expands. The estimation is based on aggregating the observed option data into nonparametric estimates of the conditional characteristic function of the return distribution, the derivatives of which allow to infer the Fourier transform of a known transform of the Lévy density in a way which is robust to the level of the unknown diffusive volatility of the asset price. The Lévy density estimate is then constructed via Fourier inversion. We derive an asymptotic bound for the integrated squared error of the estimator in the general case as well as its probability limit in the special Lévy case. We further show rate optimality of our Lévy density estimator in a minimax sense. An empirical application to market index options reveals relative stability of the left tail decay during high and low volatility periods.

Article information

Ann. Statist., Volume 47, Number 2 (2019), 1025-1060.

Received: November 2017
Revised: February 2018
First available in Project Euclid: 11 January 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation 62M05: Markov processes: estimation
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J75: Jump processes

Itô semimartingale Fourier inversion Lévy density nonparametric density estimation options stochastic volatility


Qin, Likuan; Todorov, Viktor. Nonparametric implied Lévy densities. Ann. Statist. 47 (2019), no. 2, 1025--1060. doi:10.1214/18-AOS1703.

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Supplemental materials

  • Supplement to “Nonparametric implied Lévy densities”. The supplement contains the following items: (1) limit results for the integrated squared error of the nonparametric estimator, (2) lower bounds for the minimax risk of recovering Lévy density from noisy option data with heteroskedastic Gaussian observation errors, and (3) alternative Lévy density estimator based on the second derivatives of the characteristic function of the asset return estimated from the option data.