The Annals of Applied Probability

Quantile clocks

Lancelot F. James and Zhiyuan Zhang

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Abstract

Quantile clocks are defined as convolutions of subordinators L, with quantile functions of positive random variables. We show that quantile clocks can be chosen to be strictly increasing and continuous and discuss their practical modeling advantages as business activity times in models for asset prices. We show that the marginal distributions of a quantile clock, at each fixed time, equate with the marginal distribution of a single subordinator. Moreover, we show that there are many quantile clocks where one can specify L, such that their marginal distributions have a desired law in the class of generalized s-self decomposable distributions, and in particular the class of self-decomposable distributions. The development of these results involves elements of distribution theory for specific classes of infinitely divisible random variables and also decompositions of a gamma subordinator, that is of independent interest. As applications, we construct many price models that have continuous trajectories, exhibit volatility clustering and have marginal distributions that are equivalent to those of quite general exponential Lévy price models. In particular, we provide explicit details for continuous processes whose marginals equate with the popular VG, CGMY and NIG price models. We also show how to perfectly sample the marginal distributions of more general classes of convoluted subordinators when L is in a sub-class of generalized gamma convolutions, which is relevant for pricing of European style options.

Article information

Source
Ann. Appl. Probab. Volume 21, Number 5 (2011), 1627-1662.

Dates
First available in Project Euclid: 25 October 2011

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1319576605

Digital Object Identifier
doi:10.1214/10-AAP752

Zentralblatt MATH identifier
05994506

Mathematical Reviews number (MathSciNet)
MR2884047

Subjects
Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60G09: Exchangeability

Keywords
Generalized gamma convolutions Lévy processes perfect sampling self-decomposable laws time changed price processes

Citation

James, Lancelot F.; Zhang, Zhiyuan. Quantile clocks. The Annals of Applied Probability 21 (2011), no. 5, 1627--1662. doi:10.1214/10-AAP752. http://projecteuclid.org/euclid.aoap/1319576605.


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