Quantile clocks

TL;DR

Quantile clocks are a flexible modeling tool for asset prices, allowing for continuous, volatility-clustering processes with marginals matching popular exponential Lévy models, and enabling efficient sampling for option pricing.

Contribution

This paper introduces the concept of quantile clocks, demonstrating their ability to produce desired marginal distributions and their practical advantages in financial modeling.

Findings

Quantile clocks can be strictly increasing and continuous.

They can replicate marginals of popular models like VG, CGMY, NIG.

Efficient sampling methods are developed for certain subclasses.

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 L\'{e}vy 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.