8 research outputs found
Exact simulation pricing with Gamma processes and their extensions
Exact path simulation of the underlying state variable is of great practical
importance in simulating prices of financial derivatives or their sensitivities
when there are no analytical solutions for their pricing formulas. However, in
general, the complex dependence structure inherent in most nontrivial
stochastic volatility (SV) models makes exact simulation difficult. In this
paper, we present a nontrivial SV model that parallels the notable Heston SV
model in the sense of admitting exact path simulation as studied by Broadie and
Kaya. The instantaneous volatility process of the proposed model is driven by a
Gamma process. Extensions to the model including superposition of independent
instantaneous volatility processes are studied. Numerical results show that the
proposed model outperforms the Heston model and two other L\'evy driven SV
models in terms of model fit to the real option data. The ability to exactly
simulate some of the path-dependent derivative prices is emphasized. Moreover,
this is the first instance where an infinite-activity volatility process can be
applied exactly in such pricing contexts.Comment: Forthcoming The Journal of Computational Financ
Appendix to "Approximating perpetuities"
An algorithm for perfect simulation from the unique solution of the
distributional fixed point equation is constructed, where
and are independent and is uniformly distributed on . This
distribution comes up as a limit distribution in the probabilistic analysis of
the Quickselect algorithm. Our simulation algorithm is based on coupling from
the past with a multigamma coupler. It has four lines of code
A statistical view on exchanges in Quickselect
In this paper we study the number of key exchanges required by Hoare's FIND
algorithm (also called Quickselect) when operating on a uniformly distributed
random permutation and selecting an independent uniformly distributed rank.
After normalization we give a limit theorem where the limit law is a perpetuity
characterized by a recursive distributional equation. To make the limit theorem
usable for statistical methods and statistical experiments we provide an
explicit rate of convergence in the Kolmogorov--Smirnov metric, a numerical
table of the limit law's distribution function and an algorithm for exact
simulation from the limit distribution. We also investigate the limit law's
density. This case study provides a program applicable to other cost measures,
alternative models for the rank selected and more balanced choices of the pivot
element such as median-of- versions of Quickselect as well as further
variations of the algorithm.Comment: Theorem 4.4 revised; accepted for publication in Analytic
Algorithmics and Combinatorics (ANALCO14
Quantile clocks
Quantile clocks are defined as convolutions of subordinators , 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
, such that their marginal distributions have a desired law in the class of
generalized -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 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
Exact Simulation of the Extrema of Stable Processes
We exhibit an exact simulation algorithm for the supremum of a stable process
over a finite time interval using dominated coupling from the past (DCFTP). We
establish a novel perpetuity equation for the supremum (via the representation
of the concave majorants of L\'evy processes) and apply it to construct a
Markov chain in the DCFTP algorithm. We prove that the number of steps taken
backwards in time before the coalescence is detected is finite. We analyse
numerically the performance of the algorithm (the code, written in Julia 1.0,
is available on GitHub).Comment: 26 pages, 3 figures, Julia implementation of the exact simulation
algorithm is in the GitHub repository:
https://github.com/jorgeignaciogc/StableSupremum.j
The Double CFTP method
Abstract. We consider the problem of the exact simulation of random variables Z that satisfy the distributional identity Z L = V Y + (1 − V)Z, where V ∈ [0, 1] and Y are independent, and L = denotes equality in distribution. Equivalently, Z is the limit of a Markov chain driven by that map. We give an algorithm that can be automated under the condition that we have a source capable of generating independent copies of Y, and that V has a density that can be evaluated in a black box format. The method uses a doubling trick for inducing coalescence in coupling from the past. Applications include exact samplers for many Dirichlet means, some two-parameter Poisson–Dirichlet means, and a host of other distributions related to occupation times of Bessel bridges that can be described by stochastic fixed point equations. Keywords and phrases. Random variate generation. Perpetuities. Coupling from the past. Random partitions. Stochastic recurrences. Stochastic fixed point equations. Distribution theory. Markov chain Monte Carlo. Simulation. Expected time analysis. Bessel bridge. Poisson-Dirichlet. Dirichlet means