Joint value at risk: a new conditional risk measure

In this PhD thesis a new firm level conditional risk measure is developed. It is named Joint Value at Risk (JVaR) and is defined as a quantile of a conditional distribution of interest, where the conditioning event is a latent upper tail event. It addresses the problem of how risk changes under extreme volatility scenarios. The properties of JVaR are studied based on a stochastic volatility representation of the underlying process. We prove that JVaR is leverage consistent, i.e. it is an increasing function of the dependence parameter in the stochastic representation. A feasible class of nonparametric M-estimators is introduced by exploiting the elicitability of quantiles and the stochastic ordering theory. Consistency and asymptotic normality of the two stage M-estimator are derived, and a simulation study is reported to illustrate its finite-sample properties. Parametric estimation methods are also discussed. The relation with the VaR is exploited to introduce a volatility contribution measure, and a tail risk measure is also proposed. The analysis of the dynamic JVaR is presented based on asymmetric stochastic volatility models. Empirical results with S&P500 data show that accounting for extreme volatility levels is relevant to better characterize the evolution of risk. The work is complemented by a review of the literature, where we provide an overview on quantile risk measures, elicitable functionals and several stochastic orderings.

Stochastic models and dynamic measures for the characterization of bistable circuits

During the last few years, a great deal of interest has risen concerning the applications of stochastic methods to several biochemical and biological phenomena. Phenomena like gene expression, cellular memory, bet-hedging strategy in bacterial growth and many others, cannot be described by continuous stochastic models due to their intrinsic discreteness and randomness. In this thesis I have used the Chemical Master Equation (CME) technique to modelize some feedback cycles and analyzing their properties, including experimental data. In the first part of this work, the effect of stochastic stability is discussed on a toy model of the genetic switch that triggers the cellular division, which malfunctioning is known to be one of the hallmarks of cancer. The second system I have worked on is the so-called futile cycle, a closed cycle of two enzymatic reactions that adds and removes a chemical compound, called phosphate group, to a specific substrate. I have thus investigated how adding noise to the enzyme (that is usually in the order of few hundred molecules) modifies the probability of observing a specific number of phosphorylated substrate molecules, and confirmed theoretical predictions with numerical simulations. In the third part the results of the study of a chain of multiple phosphorylation-dephosphorylation cycles will be presented. We will discuss an approximation method for the exact solution in the bidimensional case and the relationship that this method has with the thermodynamic properties of the system, which is an open system far from equilibrium.In the last section the agreement between the theoretical prediction of the total protein quantity in a mouse cells population and the observed quantity will be shown, measured via fluorescence microscopy.

Price manipulation: theoretical models and empirical investigation

Understanding why market manipulation is conducted, under which conditions it is the most profitable and investigating the magnitude of these practices are crucial questions for financial regulators. Closing price manipulation induced by derivatives’ expiration is the primary subject of this thesis. The first chapter provides a mathematical framework in continuous time to study the incentive to manipulate a set of securities induced by a derivative position. An agent holding a European-type contingent claim, depending on the price of a basket of underlying securities, is considered. The agent can affect the price of the underlying securities by trading on each of them before expiration. The elements of novelty are at least twofold: (1) a multi-asset market is considered; (2) the problem is solved by means of both classic optimisation and stochastic control techniques. Both linear and option payoffs are considered. In the second chapter an empirical investigation is conducted on the existence of expiration day effects on the UK equity market. Intraday data on FTSE 350 stocks over a six-year period from 2015-2020 are used. The results show that the expiration of index derivatives is associated with a rise in both trading activity and volatility, together with significant price distortions. The expiration of single stock options appears to have little to no impact on the underlying securities. The last chapter examines the existence of patterns in line with closing price manipulation of UK stocks on option expiration days. The main contributions are threefold: (1) this is one of the few empirical studies on manipulation induced by the options market; (2) proprietary equity orderbook and transaction data sets are used to define manipulation proxies, providing a more detailed analysis; (3) the behaviour of proprietary trading firms is studied. Despite the industry concerns, no evidence is found of this type of manipulative behaviour.

Non-Markovian stochastic processes and their applications: from anomalous diffusion to time series analysis

This work provides a forward step in the study and comprehension of the relationships between stochastic processes and a certain class of integral-partial differential equation, which can be used in order to model anomalous diffusion and transport in statistical physics. In the first part, we brought the reader through the fundamental notions of probability and stochastic processes, stochastic integration and stochastic differential equations as well. In particular, within the study of H-sssi processes, we focused on fractional Brownian motion (fBm) and its discrete-time increment process, the fractional Gaussian noise (fGn), which provide examples of non-Markovian Gaussian processes. The fGn, together with stationary FARIMA processes, is widely used in the modeling and estimation of long-memory, or long-range dependence (LRD). Time series manifesting long-range dependence, are often observed in nature especially in physics, meteorology, climatology, but also in hydrology, geophysics, economy and many others. We deepely studied LRD, giving many real data examples, providing statistical analysis and introducing parametric methods of estimation. Then, we introduced the theory of fractional integrals and derivatives, which indeed turns out to be very appropriate for studying and modeling systems with long-memory properties. After having introduced the basics concepts, we provided many examples and applications. For instance, we investigated the relaxation equation with distributed order time-fractional derivatives, which describes models characterized by a strong memory component and can be used to model relaxation in complex systems, which deviates from the classical exponential Debye pattern. Then, we focused in the study of generalizations of the standard diffusion equation, by passing through the preliminary study of the fractional forward drift equation. Such generalizations have been obtained by using fractional integrals and derivatives of distributed orders. In order to find a connection between the anomalous diffusion described by these equations and the long-range dependence, we introduced and studied the generalized grey Brownian motion (ggBm), which is actually a parametric class of H-sssi processes, which have indeed marginal probability density function evolving in time according to a partial integro-differential equation of fractional type. The ggBm is of course Non-Markovian. All around the work, we have remarked many times that, starting from a master equation of a probability density function f(x,t), it is always possible to define an equivalence class of stochastic processes with the same marginal density function f(x,t). All these processes provide suitable stochastic models for the starting equation. Studying the ggBm, we just focused on a subclass made up of processes with stationary increments. The ggBm has been defined canonically in the so called grey noise space. However, we have been able to provide a characterization notwithstanding the underline probability space. We also pointed out that that the generalized grey Brownian motion is a direct generalization of a Gaussian process and in particular it generalizes Brownain motion and fractional Brownain motion as well. Finally, we introduced and analyzed a more general class of diffusion type equations related to certain non-Markovian stochastic processes. We started from the forward drift equation, which have been made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation has been interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time-evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the same memory kernel K(t). We developed several applications and derived the exact solutions. Moreover, we considered different stochastic models for the given equations, providing path simulations.

Model reduction of stochastic groundwater flow and transport processes

This work presents a comprehensive methodology for the reduction of analytical or numerical stochastic models characterized by uncertain input parameters or boundary conditions. The technique, based on the Polynomial Chaos Expansion (PCE) theory, represents a versatile solution to solve direct or inverse problems related to propagation of uncertainty. The potentiality of the methodology is assessed investigating different applicative contexts related to groundwater flow and transport scenarios, such as global sensitivity analysis, risk analysis and model calibration. This is achieved by implementing a numerical code, developed in the MATLAB environment, presented here in its main features and tested with literature examples. The procedure has been conceived under flexibility and efficiency criteria in order to ensure its adaptability to different fields of engineering; it has been applied to different case studies related to flow and transport in porous media. Each application is associated with innovative elements such as (i) new analytical formulations describing motion and displacement of non-Newtonian fluids in porous media, (ii) application of global sensitivity analysis to a high-complexity numerical model inspired by a real case of risk of radionuclide migration in the subsurface environment, and (iii) development of a novel sensitivity-based strategy for parameter calibration and experiment design in laboratory scale tracer transport.

Estimation of Quasi-Rational DSGE Models

Small-scale dynamic stochastic general equilibrium have been treated as the benchmark of much of the monetary policy literature, given their ability to explain the impact of monetary policy on output, inflation and financial markets. One cause of the empirical failure of New Keynesian models is partially due to the Rational Expectations (RE) paradigm, which entails a tight structure on the dynamics of the system. Under this hypothesis, the agents are assumed to know the data genereting process. In this paper, we propose the econometric analysis of New Keynesian DSGE models under an alternative expectations generating paradigm, which can be regarded as an intermediate position between rational expectations and learning, nameley an adapted version of the "Quasi-Rational" Expectatations (QRE) hypothesis. Given the agents' statistical model, we build a pseudo-structural form from the baseline system of Euler equations, imposing that the length of the reduced form is the same as in the `best' statistical model.

Statistical analysis of the error associated with the simplification of the stratigraphy in geotechnical models

The uncertainties in the determination of the stratigraphic profile of natural soils is one of the main problems in geotechnics, in particular for landslide characterization and modeling. The study deals with a new approach in geotechnical modeling which relays on a stochastic generation of different soil layers distributions, following a boolean logic – the method has been thus called BoSG (Boolean Stochastic Generation). In this way, it is possible to randomize the presence of a specific material interdigitated in a uniform matrix. In the building of a geotechnical model it is generally common to discard some stratigraphic data in order to simplify the model itself, assuming that the significance of the results of the modeling procedure would not be affected. With the proposed technique it is possible to quantify the error associated with this simplification. Moreover, it could be used to determine the most significant zones where eventual further investigations and surveys would be more effective to build the geotechnical model of the slope. The commercial software FLAC was used for the 2D and 3D geotechnical model. The distribution of the materials was randomized through a specifically coded MatLab program that automatically generates text files, each of them representing a specific soil configuration. Besides, a routine was designed to automate the computation of FLAC with the different data files in order to maximize the sample number. The methodology is applied with reference to a simplified slope in 2D, a simplified slope in 3D and an actual landslide, namely the Mortisa mudslide (Cortina d’Ampezzo, BL, Italy). However, it could be extended to numerous different cases, especially for hydrogeological analysis and landslide stability assessment, in different geological and geomorphological contexts.

Models and Algorihtm for the Optimization of Real-World Routing and Logistics Problems

Logistics involves planning, managing, and organizing the flows of goods from the point of origin to the point of destination in order to meet some requirements. Logistics and transportation aspects are very important and represent a relevant costs for producing and shipping companies, but also for public administration and private citizens. The optimization of resources and the improvement in the organization of operations is crucial for all branches of logistics, from the operation management to the transportation. As we will have the chance to see in this work, optimization techniques, models, and algorithms represent important methods to solve the always new and more complex problems arising in different segments of logistics. Many operation management and transportation problems are related to the optimization class of problems called Vehicle Routing Problems (VRPs). In this work, we consider several real-world deterministic and stochastic problems that are included in the wide class of the VRPs, and we solve them by means of exact and heuristic methods. We treat three classes of real-world routing and logistics problems. We deal with one of the most important tactical problems that arises in the managing of the bike sharing systems, that is the Bike sharing Rebalancing Problem (BRP). We propose models and algorithms for real-world earthwork optimization problems. We describe the 3DP process and we highlight several optimization issues in 3DP. Among those, we define the problem related to the tool path definition in the 3DP process, the 3D Routing Problem (3DRP), which is a generalization of the arc routing problem. We present an ILP model and several heuristic algorithms to solve the 3DRP.

Modelling spillover effects in spatial stochastic frontier analysis

In the last two decades, authors have begun to expand classical stochastic frontier (SF) models in order to include also some spatial components. Indeed, firms tend to concentrate in clusters, taking advantage of positive agglomeration externalities due to cooperation, shared ideas and emulation, resulting in increased productivity levels. Until now scholars have introduced spatial dependence into SF models following two different paths: evaluating global and local spatial spillover effects related to the frontier or considering spatial cross-sectional correlation in the inefficiency and/or in the error term. In this thesis, we extend the current literature on spatial SF models introducing two novel specifications for panel data. First, besides considering productivity and input spillovers, we introduce the possibility to evaluate the specific spatial effects arising from each inefficiency determinant through their spatial lags aiming to capture also knowledge spillovers. Second, we develop a very comprehensive spatial SF model that includes both frontier and error-based spillovers in order to consider four different sources of spatial dependence (i.e. productivity and input spillovers related to the frontier function and behavioural and environmental correlation associated with the two error terms). Finally, we test the finite sample properties of the two proposed spatial SF models through simulations, and we provide two empirical applications to the Italian accommodation and agricultural sectors. From a practical perspective, policymakers, based on results from these models, can rely on precise, detailed and distinct insights on the spillover effects affecting the productive performance of neighbouring spatial units obtaining interesting and relevant suggestions for policy decisions.

Vehicle routing and location routing problems with minimum latency: algorithms and models

Latency can be defined as the sum of the arrival times at the customers. Minimum latency problems are specially relevant in applications related to humanitarian logistics. This thesis presents algorithms for solving a family of vehicle routing problems with minimum latency. First the latency location routing problem (LLRP) is considered. It consists of determining the subset of depots to be opened, and the routes that a set of homogeneous capacitated vehicles must perform in order to visit a set of customers such that the sum of the demands of the customers assigned to each vehicle does not exceed the capacity of the vehicle. For solving this problem three metaheuristic algorithms combining simulated annealing and variable neighborhood descent, and an iterated local search (ILS) algorithm, are proposed. Furthermore, the multi-depot cumulative capacitated vehicle routing problem (MDCCVRP) and the multi-depot k-traveling repairman problem (MDk-TRP) are solved with the proposed ILS algorithm. The MDCCVRP is a special case of the LLRP in which all the depots can be opened, and the MDk-TRP is a special case of the MDCCVRP in which the capacity constraints are relaxed. Finally, a LLRP with stochastic travel times is studied. A two-stage stochastic programming model and a variable neighborhood search algorithm are proposed for solving the problem. Furthermore a sampling method is developed for tackling instances with an infinite number of scenarios. Extensive computational experiments show that the proposed methods are effective for solving the problems under study.