Student-like models for risky asset with dependence

We present a new construction of the Student and Student-like fractal activity time model for risky asset. The construction uses the diffusion processes and their superpositions and allows for specified exact Student or Student-like marginal distributions of the returns and for exible and tractable dependence structure. The fractal activity time is asymptotically self-similar, which is a desired feature seen in practice

Option Pricing in Multivariate Stochastic Volatility Models of OU Type

We present a multivariate stochastic volatility model with leverage, which is flexible enough to recapture the individual dynamics as well as the interdependencies between several assets while still being highly analytically tractable. First we derive the characteristic function and give conditions that ensure its analyticity and absolute integrability in some open complex strip around zero. Therefore we can use Fourier methods to compute the prices of multi-asset options efficiently. To show the applicability of our results, we propose a concrete specification, the OU-Wishart model, where the dynamics of each individual asset coincide with the popular Gamma-OU BNS model. This model can be well calibrated to market prices, which we illustrate with an example using options on the exchange rates of some major currencies. Finally, we show that covariance swaps can also be priced in closed form.Comment: 28 pages, 5 figures, to appear in SIAM Journal on Financial Mathematic

The Variance Gamma (VG) Model with Long Range Dependence

This thesis mainly builds on the Variance Gamma (VG) model for financial assets over time of Madan & Seneta (1990) and Madan, Carr & Chang (1998), although the model based on the t distribution championed in Heyde & Leonenko (2005) is also given attention. The primary contribution of the thesis is the development of VG models, and the extension of t models, which accommodate a dependence structure in asset price returns. In particular it has become increasingly clear that while returns (log price increments) of historical financial asset time series appear as a reasonable approximation of independent and identically distributed data, squared and absolute returns do not. In fact squared and absolute returns show evidence of being long range dependent through time, with autocorrelation functions that are still significant after 50 to 100 lags. Given this evidence against the assumption of independent returns, it is important that models for financial assets be able to accommodate a dependence structure

Fractal analysis of the EEG and clinical applications

2010/2011Most of the knowledge about physiological systems has been learned using linear system theory. The randomness of many biomedical signals has been traditionally ascribed to a noise-like behavior. An alternative explanation for the irregular behavior observed in systems which do not seem to be inherently stochastic is provided by one of the most striking mathematical developments of the past few decades, i.e., chaos theory. Chaos theory suggests that random-like behavior can arise in some deterministic nonlinear systems with just a few degrees of freedom. One of the most evocative aspects of deterministic chaos is the concept of fractal geometry. Fractal structure, characterized by self-similarity and noninteger dimension, is displayed in chaotic systems by a subset of the phase space known as strange attractor. However, fractal properties are observed also in the unpredictable time evolution and in the 1/f^β power-law of many biomedical signals. The research activities carried out by the Author during the PhD program are concerned with the analysis of the fractal-like behavior of the EEG. The focus was set on those methods which evaluate the fractal geometry of the EEG in the time domain, in the hope of providing physicians and researchers with new valuable tools of low computational cost for the EEG analysis. The performances of three widely used techniques for the direct estimation of the fractal dimension of the EEG were compared and the accuracy of the fBm scaling relationship, often used to obtain indirect estimates from the slope of the spectral density, was assessed. Direct estimation with Higuchi's algorithm turned out to be the most suitable methodology, producing correct estimates of the fractal dimension of the electroencephalogram also on short traces, provided that minimum sampling rate required to avoid aliasing is used. Based on this result, Higuchi's fractal dimension was used to address three clinical issues which could involve abnormal complexity of neuronal brain activity: 1) the monitoring of carotid endarterectomy for the prevention of intraoperative stroke, 2) the assessment of the depth of anesthesia to monitor unconsciousness during surgery and 3) the analysis of the macro-structural organization of the EEG in autism with respect to mental retardation. The results of the clinical studies suggest that, although linear spectral analysis still represents a valuable tool for the investigation of the EEG, time domain fractal analysis provides additional information on brain functioning which traditional analysis cannot achieve, making use of techniques of low computational cost.La maggior parte delle conoscenze acquisite sui sistemi fisiologici si deve alla teoria dei sistemi lineari. Il comportamento pseudo stocastico di molti segnali biomedici è stato tradizionalmente attribuito al concetto di rumore. Un'interpretazione alternativa del comportamento irregolare rilevato in sistemi che non sembrano essere intrinsecamente stocastici è fornita da uno dei più sorprendenti sviluppi matematici degli ultimi decenni: la teoria del caos. Tale teoria suggerisce che una certa componente casuale può sorgere in alcuni sistemi deterministici non lineari con pochi gradi di libertà. Uno degli aspetti più suggestivi del caos deterministico è il concetto di geometria frattale. Strutture frattali, caratterizzate da auto-somiglianza e dimensione non intera, sono rilevate nei sistemi caotici in un sottoinsieme dello spazio delle fasi noto con il nome di attrattore strano. Tuttavia, caratteristiche frattali possono manifestarsi anche nella non prevedibile evoluzione temporale e nella legge di potenza 1/f^β tipiche di molti segnali biomedici. Le attività di ricerca svolte dall'Autore nel corso del dottorato hanno riguardato l'analisi del comportamento frattale dell'EEG. L'attenzione è stata rivolta a quei metodi che affrontano lo studio della geometria frattale dell'EEG nel dominio del tempo, nella speranza di fornire a medici e ricercatori nuovi strumenti utili all'analisi del segnale EEG e caratterizzati da bassa complessità computazionale. Sono state messe a confronto le prestazioni di tre tecniche largamente utilizzate per la stima diretta della dimensione frattale dell'EEG e si è valutata l'accuratezza della relazione di scaling del modello fBm, spesso utilizzata per ottenere stime indirette a partire dalla pendenza della densità spettrale di potenza. Il metodo più adatto alla stima della dimensione frattale dell'elettroencefalogramma è risultato essere l'algoritmo di Higuchi, che produce stime accurate anche su segmenti di breve durata a patto che il segnale sia campionato alla minima frequenza di campionamento necessaria ad evitare il fenomeno dell'aliasing. Sulla base di questo risultato, la dimensione frattale di Higuchi è stata utilizzata per esaminare tre questioni cliniche che potrebbero coinvolgere una variazione della complessità dell'attività neuronale: 1) il monitoraggio dell'endoarterectomia carotidea per la prevenzione dell'ictus intraoperatorio, 2) la valutazione della profondità dell'anestesia per monitorare il livello di incoscienza durante l'intervento chirurgico e 3) l'analisi dell'organizzazione macro-strutturale del EEG nell'autismo rispetto alla condizione di ritardo mentale. I risultati degli studi clinici suggeriscono che, sebbene l'analisi spettrale rappresenti ancora uno strumento prezioso per l'indagine dell'EEG, l'analisi frattale nel dominio del tempo fornisce informazioni aggiuntive sul funzionamento del cervello che l'analisi tradizionale non è in grado di rilevare, con il vantaggio di impiegare tecniche a basso costo computazionale.XXIV Ciclo198

Probability Theory Compatible with the New Conception of Modern Thermodynamics. Economics and Crisis of Debts

We show that G\"odel's negative results concerning arithmetic, which date back to the 1930s, and the ancient "sand pile" paradox (known also as "sorites paradox") pose the questions of the use of fuzzy sets and of the effect of a measuring device on the experiment. The consideration of these facts led, in thermodynamics, to a new one-parameter family of ideal gases. In turn, this leads to a new approach to probability theory (including the new notion of independent events). As applied to economics, this gives the correction, based on Friedman's rule, to Irving Fisher's "Main Law of Economics" and enables us to consider the theory of debt crisis.Comment: 48p., 14 figs., 82 refs.; more precise mathematical explanations are added. arXiv admin note: significant text overlap with arXiv:1111.610

Gravitational Lenses as High-Resolution Telescopes

The inner regions of active galaxies host the most extreme and energetic phenomena in the universe including, relativistic jets, supermassive black hole binaries, and recoiling supermassive black holes. However, many of these sources cannot be resolved with direct observations. I review how strong gravitational lensing can be used to elucidate the structures of these sources from radio frequencies up to very high energy gamma rays. The deep gravitational potentials surrounding galaxies act as natural gravitational lenses. These gravitational lenses split background sources into multiple images, each with a gravitationally-induced time delay. These time delays and positions of lensed images depend on the source location, and thus, can be used to infer the spatial origins of the emission. For example, using gravitationally-induced time delays improves angular resolution of modern gamma-ray instruments by six orders of magnitude, and provides evidence that gamma-ray outbursts can be produced at even thousands of light years from a supermassive black hole, and that the compact radio emission does not always trace the position of the supermassive black hole. These findings provide unique physical information about the central structure of active galaxies, force us to revise our models of operating particle acceleration mechanisms, and challenge our assumptions about the origin of compact radio emission. Future surveys, including LSST, SKA, and Euclid, will provide observations for hundreds of thousands of gravitationally lensed sources, which will allow us to apply strong gravitational lensing to study the multi-wavelength structure for large ensembles of sources. This large ensemble of gravitationally lensed active galaxies will allow us to elucidate the physical origins of multi-wavelength emissions, their connections to supermassive black holes, and their cosmic evolution.Comment: Invited (Accepted) review for Physics Report

Cascades and transitions in turbulent flows

Turbulence is characterized by the non-linear cascades of energy and other inviscid invariants across a huge range of scales, from where they are injected to where they are dissipated. Recently, new experimental, numerical and theoretical works have revealed that many turbulent configurations deviate from the ideal 3D/2D isotropic cases characterized by the presence of a strictly direct/inverse energy cascade, respectively. We review recent works from a unified point of view and we present a classification of all known transfer mechanisms. Beside the classical cases of direct and inverse cascades, the different scenarios include: split cascades to small and large scales simultaneously, multiple/dual cascades of different quantities, bi-directional cascades where direct and inverse transfers of the same invariant coexist in the same scale-range and finally equilibrium states where no cascades are present, including the case when a condensate is formed. We classify all transitions as the control parameters are changed and we analyse when and why different configurations are observed. Our discussion is based on a set of paradigmatic applications: helical turbulence, rotating and/or stratified flows, MHD and passive/active scalars where the transfer properties are altered as one changes the embedding dimensions, the thickness of the domain or other relevant control parameters, as the Reynolds, Rossby, Froude, Peclet, or Alfven numbers. We discuss the presence of anomalous scaling laws in connection with the intermittent nature of the energy dissipation in configuration space. An overview is also provided concerning cascades in other applications such as bounded flows, quantum, relativistic and compressible turbulence, and active matter, together with implications for turbulent modelling. Finally, we present a series of open problems and challenges that future work needs to address.Comment: accepted for publication on Physics Reports 201

Fractal activity time risky asset models with dependence

The paradigm Black-Scholes model for risky asset prices has occupied a central place in asset-liability management since its discovery in 1973. While the underlying geometric Brownian motion surely captured the essence of option pricing (helping spawn a multi-billion pound derivatives industry), three decades of statistical study has shown that the model departs significantly from the realities of returns (increments in the logarithm of risky asset price) data. To remedy the shortcomings of the Black-Scholes model, we present the fractal activity time geometric Brownian motion model proposed by Chris Heyde in 1999. This model supports the desired empirical features of returns including no correlation but dependence, and distributions with heavier tails and higher peaks than Gaussian. In particular, the model generalises geometric Brownian motion whereby the standard Brownian motion is evaluated at random activity time instead of calendar time. There are also strong suggestions from literature that the activity time process here is approximately self-similar. Thus we require a way to accommodate both the desired distributional and dependence features as well as the property of asymptotic self-similarity. In this thesis, we describe the construction of this fractal activity time based on chi-square type processes, through Ornstein-Uhlenbeck processes driven by Levy noise, and via diffusion-type processes. Once we validate the model by fitting real data, we endeavour to state a new explicit formula for the price of a European option. This is made possible as Heyde's model remains within the Black-Scholes framework of option pricing, which allows us to use their engendered arbitrage-free methodology. Finally, we introduce an alternative to the previously considered approach. The motivation for which comes from the understanding that activity time cannot be exactly self-similar. We provide evidence that multi-scaling occurs in financial data and outline another construction for the activity time process