arXiv subjects

Lorenzo Torricelli

Publications and source records attributed to Lorenzo Torricelli.

8 recordsLinked to original sources

arXiv2025

Thorin processes and their subordination

A Thorin process is a stochastic process with independent and stationary increments whose laws are weak limits of finite convolutions of gamma distributions. Many popular Lévy processes fall under this class. The Thorin class can be characterized by a representing triplet that conveys more information on the process compared to the Lévy triplet. In this paper we investigate some relationships between the Thorin structure and the process properties, and find that the support of the Thorin measure characterizes the existence of the critical exponential moment, as well as the asymptotic equivalence between the Lévy tail function and the complementary distribution function. Furthermore, it is illustrated how univariate Brownian subordination with respect to Thorin subordinators produces Thorin processes whose representing measure is given by a pushforward with respect to a hyperbolic function, leading to arguably easier formulae compared to the Bochner integral determining the Lévy measure. We provide a full account of the theory of multivariate Thorin processes, starting from the Thorin--Bondesson representation for the characteristic exponent, and highlight the roles of the Thorin measure in the analysis of density functions, moments, path variation and subordination. Various old and new examples are discussed. We finally detail a treatment of subordination of gamma processes with respect to negative binomial subordinators which is made possible by the Thorin-Bondesson representation.

math.PR↗

arXiv2024

Time-changed Markov processes and space-time coupled non-local equations

In this paper we study coupled fully non-local equations, where a linear non-local operator jointly acts on the time and space variables. We establish existence and uniqueness of the solution. A maximum principle is proved and used to derive uniqueness. Existence is established by providing a stochastic representation based on anomalous processes constructed as a time change via the undershooting of an independent subordinator. This leads to general non-stepped processes with intervals of constancy representing a sticky or trapping effect. Our theory allows these intervals to be dependent on the immediately subsequent jump. These processes include scaling limit of suitable coupled continuous time random walks previously studied in applications, in particular in the context of anomalous diffusion and option pricing. Here we exploit our general theory to obtain a non-local analog of the Black and Scholes equation, addressing the problem of determining the seasoned price of a derivative security, in case the price fluctuations are described by a process whose jumps are dependent on the previous interval.

math.PR↗

arXiv2024

On the convolution equivalence of tempered stable distributions on the real line

We show the convolution equivalence property of univariate tempered stable distributions in the sense of Rosińsky (2007). This makes rigorous various classic heuristic arguments on the asymptotic similarity between the probability and Lévy densities of such distributions. Some specific examples from the literature are discussed.

math.PR↗

arXiv2023

Tempered geometric stable distributions and processes

We introduce a notion of geometric tempering using exponentially-dampened Mittag-Leffler tempering functions and closely investigate the univariate case. Characteristic exponents and cumulants are calculated, as well as spectral densities. Absolute continuity relations are shown, and short and long time scaling limits of the associated Lévy processes analyzed.

math.PR↗

arXiv2022

Tempered positive Linnik processes and their representations

This paper analyzes various classes of processes associated with the tempered positive Linnik (TPL) distribution. We provide several subordinated representations of TPL Lévy processes and in particular establish a stochastic self-similarity property with respect to negative binomial subordination. In finite activity regimes we show that the explicit compound Poisson representations gives rise to innovations following Mittag-Leffler type laws which are apparently new. We characterize two time-inhomogeneous TPL processes, namely the Ornstein-Uhlenbeck (OU) Lévy-driven processes with stationary distribution and the additive process determined by a TPL law. We finally illustrate how the properties studied come together in a multivariate TPL Lévy framework based on a novel negative binomial mixing methodology. Some potential applications are outlined in the contexts of statistical anti-fraud and financial modelling.

math.PR↗

arXiv2020

Anomalous diffusions in option prices: connecting trade duration and the volatility term structure

Anomalous diffusions arise as scaling limits of continuous-time random walks (CTRWs) whose innovation times are distributed according to a power law. The impact of a non-exponential waiting time does not vanish with time and leads to different distribution spread rates compared to standard models. In financial modelling this has been used to accommodate for random trade duration in the tick-by-tick price process. We show here that anomalous diffusions are able to reproduce the market behaviour of the implied volatility more consistently than usual Lévy or stochastic volatility models. We focus on two distinct classes of underlying asset models, one with independent price innovations and waiting times, and one allowing dependence between these two components. These two models capture the well-known paradigm according to which shorter trade duration is associated with higher return impact of individual trades. We fully describe these processes in a semimartingale setting leading no-arbitrage pricing formulae, and study their statistical properties. We observe that skewness and kurtosis of the asset returns do not tend to zero as time goes by. We also characterize the large-maturity asymptotics of Call option prices, and find that the convergence rate is slower than in standard Lévy regimes, which in turn yields a declining implied volatility term structure and a slower decay of the skew.

q-fin.PR↗

arXiv2015

Valuation of asset and volatility derivatives using decoupled time-changed Lévy processes

In this paper we propose a general derivative pricing framework which employs decoupled time-changed (DTC) Lévy processes to model the underlying asset of contingent claims. A DTC Lévy process is a generalized time-changed Lévy process whose continuous and pure jump parts are allowed to follow separate random time scalings; we devise the martingale structure for a DTC Lévy-driven asset and revisit many popular models which fall under this framework. Postulating different time changes for the underlying Lévy decomposition allows to introduce asset price models consistent with the assumption of a correlated pair of continuous and jump market activities; we study one illustrative DTC model having this property by assuming that the instantaneous activity rates follow the the so-called Wishart process. The theory developed is applied to the problem of pricing claims depending not only on the price or the volatility of an underlying asset, but also to more sophisticated derivatives that pay-off on the joint performance of these two financial variables, like the target volatility option (TVO). We solve the pricing problem through a Fourier-inversion method; numerical computations validating our technique are provided.

q-fin.PR↗

arXiv2012

Pricing joint claims on an asset and its realized variance under stochastic volatility models

In a stochastic volatility framework, we find a general pricing equation for the class of payoffs depending on the terminal value of a market asset and its final quadratic variation. This allows a pricing tool for European-style claims paying off at maturity a joint function of the underlying and its realised volatility/variance. We study the solution under different stochastic volatility models, give a formula for the computation of the Delta and Gamma of these claims, and introduce some new interesting payoffs that can be priced through this equation. Numerical results are given and compared to those from plain vanilla derivatives.

q-fin.PR↗