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Four Essays in Financial Mathematics
Full text linkThe first Chapter of the Thesis presents a general and abstract framework for the analysis of mean-variance portfolio optimization problems. Under a minimal no-arbitrage condition, we consider a whole range of quadratic optimization problems, which are solved in a unified way. We give general and model-independent characterizations of the optimal solutions as well as abstract generalizations of classical results from financial economics such as two-fund separation results, mean-variance efficiency and a CAPM-type formula. Finally, we apply our general results to the valuation of contingent claims according to several mean-variance indifference valuation rules.
The second Chapter considers a general reduced-form credit risk model, where the default time is modeled as a doubly stochastic random time with default intensity driven by a diffusion affine process. We characterize the family of all locally equivalent probability measures which preserve the affine structure of the model by giving necessary and sufficient conditions on their density process. We illustrate the usefulness of our results first in the context of a jump-to-default extension of the popular Heston (1993) stochastic volatility model and then in the context of a more general hybrid equity/credit risk multifactor model, providing applications of interest in view of risk management as well as pricing purposes.
The third Chapter deals with general diffusion-based models and shows that, even in the absence of an Equivalent Local Martingale Measure, the financial market may still be viable, in the sense that strong forms of arbitrage are ruled out. Relying partly on the recent literature, we provide necessary and sufficient conditions for market viability in terms of the market price of risk process and martingale deflators. Regardless of the existence of a martingale measure, we show that the financial market may still be complete and contingent claims can be valued under the original (real-world) probability measure, provided we use as numéraire the Growth-Optimal Portfolio.
Finally, the fourth Chapter deals with no-arbitrage conditions which are weaker than the classical \No Free Lunch with Vanishing Risk (NFLVR) criterion, providing necessary and sufficient conditions for their validity in terms of the characteristics of the discounted price process. We study the stability of weak no-arbitrage conditions with respect to changes of numéraire, absolutely continuous changes of the reference probability measure and restrictions/enlargements of the reference filtration. In particular, we prove that weak no-arbitrage conditions, unlike the classical No Arbitrage (NA) and NFLVR criteria, are in general robust with respect to these changes. Finally, we provide a general characterization of attainable contingent claims and market completeness without relying on the NFLVR condition.Il primo Capitolo di questa Tesi contiene un approccio generale e astratto a problemi di ottimizzazione di portafoglio secondo un criterio media-varianza. In particolare, vengono studiati e risolti congiuntamente diversi problemi di ottimizzazione in media-varianza, assumendo unicamente una condizione minimale di non-arbitraggio. Le soluzioni ottime a tali problemi vengono descritte esplicitamente, senza alcuna ipotesi sulle caratteristiche del modello sottostante. Inoltre, vengono presentate generalizzazioni di risultati classici dell'economia finanziaria, come il teorema di separazione in due fondi, la frontiera efficiente media-varianza e una formula di tipo CAPM. Infine, i risultati generali ottenuti vengono applicati alla valutazione di strumenti finanziari.
Il secondo Capitolo è dedicato allo studio di un modello generale a forma ridotta per il rischio di credito, in cui il tempo di fallimento viene modellizzato come un tempo aleatorio doppiamente stocastico la cui intensità è funzione di un processo diffusivo di tipo affine. Si ottiene una caratterizzazione completa della famiglia di tutte le misure di probabilità localmente equivalenti che preservano la struttura affine del modello, formulando condizioni necessarie e sufficienti sul processo densità. L'utilità di questi risultati generali viene illustrata prima nel contesto di un modello a volatilità stocastica di Heston (1993) con l'aggiunta di un possibile fallimento e succesivamente nel contesto di un modello multi-fattoriale più generale che consente di modellizzare congiuntamente il rischio di credito e il rischio di mercato. Si considerano applicazioni di interesse per la valutazione di strumenti derivati come anche per il risk management.
Il terzo Capitolo è dedicato allo studio di modelli basati su processi diffusivi. In particolare, viene mostrato che, anche in assenza di una Misura Martingala Locale Equivalente, il mercato finanziario può essere privo di forme forti di arbitraggio. Basandoci in parte sulla letteratura recente, vengono fornite condizioni necessarie e sufficienti per l'assenza di forme forti di arbitraggio. Tali condizioni coinvolgono il prezzo di mercato del rischio e processi martingale deflator. Indipendentemente dall'esistenza di una misura martingala, si dimostra che il mercato finanziario può essere completo e strumenti derivati possono essere valutati rispetto alla misura di probabilità del mondo reale, utilizzando come numéraire il Growth-Optimal Portfolio.
Infine, il quarto Capitolo contiene uno studio delle condizioni di non-arbitraggio più deboli del classico criterio No Free Lunch with Vanishing Risk (NFLVR). Vengono fornite condizioni necessarie e sufficienti per la validità di tali condizioni deboli di non-arbitraggio, espresse rispetto alle caratteristiche del processo che rappresenta il prezzo scontato degli asset. Viene anche studiata la stabilità delle condizioni deboli di non-arbitraggio rispetto a cambiamenti di numéraire, cambiamenti assolutamente continui della misura di probabilità di riferimento e restrizioni/allargamenti della filtrazione di riferimento. In particolare, si dimostra che le condizioni deboli di non-arbitraggio considerate nel presente lavoro godono di buone proprietà di stabilità, al contrario di quanto accade per le classiche condizioni di Non Arbitraggio (NA) e NFLVR. Infine, presentiamo una caratterizzazione generale dei titoli finanziari che possono essere replicati, dimostrando che il mercato finanziario può essere completo anche in assenza della condizione NFLVR
Arbitrage concepts under trading restrictions in discrete-time financial markets
Full text linkIn a discrete-time setting, we study arbitrage concepts in the presence of convex trading constraints. We show that solvability of portfolio optimization problems is equivalent to absence of arbitrage of the first kind, a condition weaker than classical absence of arbitrage opportunities. We center our analysis on this characterization of market viability and derive versions of the fundamental theorems of asset pricing based on portfolio optimization arguments. By considering specifically a discrete-time setup, we simplify existing results and proofs that rely on semimartingale theory, thus allowing for a clear understanding of the foundational economic concepts involved. We exemplify these concepts, as well as some unexpected situations, in the context of one-period factor models with arbitrage opportunities under borrowing constraints
CBI-time-changed Lévy processes
No full textWe introduce and study the class of {CBI-time-changed Lévy processes} (CBITCL), obtained by time-changing a Lévy process with respect to an integrated continuous-state branching process with immigration (CBI). We characterize CBITCL processes as solutions to a certain stochastic integral equation and relate them to affine stochastic volatility processes. We provide a complete analysis of the time of explosion of exponential moments of CBITCL processes and study their asymptotic behavior. In addition, we show that CBITCL processes are stable with respect to a suitable class of equivalent changes of measure. As illustrated by some examples, CBITCL processes are flexible and tractable processes with a significant potential for applications in finance
Valuation of general GMWB annuities in a low interest rate environment
Full text linkVariable annuities with Guaranteed Minimum Withdrawal Benefits (GMWB) entitle the policy holder to periodic withdrawals together with a terminal payoff linked to the performance of an equity fund. In this paper, we consider the valuation of a general class of GMWB annuities, allowing for step-up, bonus and surrender features, taking also into account mortality risk and death benefits. When dynamic withdrawals are allowed, the valuation of GMWB annuities leads to a stochastic optimal control problem, which we address here by dynamic programming techniques. Adopting a Hull-White interest rate model, correlated with the equity fund, we propose an efficient tree-based algorithm. We perform a thorough analysis of the determinants of the market value of GMWB annuities and of the optimal withdrawal strategies. In particular, we study the impact of a low/negative interest rate environment. Our findings indicate that low/negative rates profoundly affect the optimal withdrawal behaviour and, in combination with step-up and bonus features, increase significantly the fair values of GMWB annuities, which can only be compensated by large management fees
Martingale spaces and representations under absolutely continuous changes of probability
Full text linkIn a fully general setting, we study the relation between martingale spaces under two locally absolutely continuous probabilities and prove that the martingale represen- tation property (MRP) is always stable under locally absolutely continuous changes of probability. Our approach relies on minimal requirements, is constructive and, as shown by a simple example, enables us to study situations which cannot be covered by the existing theory
Simplified mean-variance portfolio optimisation
No full textWe propose a simplified approach to mean-variance portfolio problems by changing their parametrisation from trading strategies to final positions. This allows us to treat, under a very mild no-arbitrage-type assumption, a whole range of quadratic optimisation problems by simple mathematical tools in a unified and model-independent way. We pro- vide explicit formulas for optimal positions and values, connections between the solutions to the different problems, two-fund separation results, and explicit expressions for indifference values
Credit risk and incomplete information: filtering and EM parameter estimation.
No full textWe consider a reduced-form credit risk model where default intensities and interest rate are functions of a not fully observable Markovian factor process, thereby introducing an information-driven default contagion effect among defaults of different issuers. We determine arbitrage-free prices of /OTC/ products coherently with information from the financial market, in particular yields and credit spreads and this can be accomplished via a filtering approach coupled with an EM-algorithm for parameter estimation
On the Non-Intrusive Load Monitoring in dwellings: a feasibility perspective
No full textThe oncoming modernization process of the power grids, driven above all by decarbonisation objectives and the continuous improvement of digital technologies, is encouraging active participation in the electricity market by consumers through the Demand-Response mechanism. From this perspective, the introduction of smart meters and energy consumption monitoring devices plays a fundamental role, being able to give benefits to consumers, suppliers and the electricity grid itself. This paper proposes a supervised method of non-intrusive load monitoring (NILM) based on the recognition of patterns in the time domain with the Dynamic Time Warping algorithm which is suitable for low-cost smart metering applications in dwelling. The technique has been tested with the Natural Dataset of the LIT-Dataset showing a recognition success rate of 100%
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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