Quantile clocks

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.Comment: Published in at http://dx.doi.org/10.1214/10-AAP752 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Multivariate subordination using generalised Gamma convolutions with applications to Variance Gamma processes and option pricing

“…Other possible extensions of the V G 1 -class comprise a range of possible sample path behaviour. In [45] the univariate CGM Y -processes have been identified to be subordinated Brownian motions, and it is also known that the associated subordinator is a GGC 1 -subordinator (see [31], their Example 8.2). As perceived in [13], Blumenthal-Getoor indices of CGM Yprocesses exhaust the whole of the interval (0, 2).…”

The Normal-Generalised Gamma-Pareto process: A novel pure-jump Lévy process with flexible tail and jump-activity properties

“…τ > σ, due to the mixture form (8), the GGP distribution arises as the marginal distribution of a quantile clock process (see Theorem 3.1 byJames and Zhang (2011)) with parameters (R, L)…”

Sequential sampling for CGMY processes via decomposition of their time changes

“…(π−U ) 1/2 I{0<U <π}+ξI{0≤U ≤π,γ<1} (1+(π/2) 1/2 )γ 1/2 /ζ+z until U < π and Z := W ρ ≤ 1let a = A(U ), m = (bM λ 1/Y /a) Y , δ = (mY /a) 1/2 , a 1 = δ(π/2) 1/2 , a 2 = δ a 3 = z/a, s = a 1 + a 2 + a 3 generate V ′ uniformly on [0, 1] if V ′ < a 1 /s then generate N ′ ∼ Normal(0,1) and let X ′ ← m − δ|N ′ | else if V ′ < (a 1 + a 2 )/s then generate X ′ uniformly on [m, m + δ] else generate E ′ ∼ Exponential(1) and let X ′ ← m + δ + E ′ a 3 let E = − log(Z) until X ′ ≥ 0 and a(X ′ − m) + M λ 1/Y (X ′−b − m −b ) − N ′2 2 I{X ′ < m} − E ′ I{X ′ > m + δ} ≤ E return λ 1/Y /X ′b A.2.The Double CFTP SamplerWe present the double CFTP algorithm (cf [18]…”

“…The Double CFTP SamplerWe present the double CFTP algorithm (cf [18](18) in Section 5.1. Recall that the density function h(•) ofB is bounded from below on [0, 1] by a constant c h > 0 and 0 < Q ≤ c Q < ∞.…”