A study of the Hobson and Rogers volatility model

We firstly examine the model of Hobson and Rogers for the volatility of a financial asset such as a stock or share. The main feature of this model is the specification of volatility in terms of past price returns. The volatility process and the underlying price process share the same source of randomness and so the model is said to be complete. Complete models are advantageous as they allow a unique, preference independent price for options on the underlying price process. One of the main objectives of the model is to reproduce the `smiles' and `skews' seen in the market implied volatilities and this model produces the desired effect. In the first main piece of work we numerically calibrate the model of Hobson and Rogers for comparison with existing literature. We also develop parameter estimation methods based on the calibration of a GARCH model. We examine alternative specifications of the volatility and show an improvement of model fit to market data based on these specifications. We also show how to process market data in order to take account of inter-day movements in the volatility surface. In the second piece of work, we extend the Hobson and Rogers model in a way that better reflects market structure. We extend the model to take into account both first and second order effects. We derive and numerically solve the pde which describes the price of options under this extended model. We show that this extension allows for a better fit to the market data. Finally, we analyse the parameters of this extended model in order to understand intuitively the role of these parameters in the volatility surface.

A new proposition on the martingale representation theorem and on the approximate hedging of contingent claim in mean-variance criterion

In this work we revisit the problem of the hedging of contingent claim using mean-square criterion. We prove that in incomplete market, some probability measure can be identified so that becomes -martingale under .This is in fact a new proposition on the martingale representation theorem. The new results also identify a weight function that serves to be an approximation to the Radon-Nikodým derivative of the unique neutral martingale measure.

Long-term equity anticipation securities and stock market volatility dynamics

Recent empirical findings suggest that the long-run dependence in U.S. stock market volatility is best described by a slowly mean-reverting fractionally integrated process. The present study complements this existing time-series-based evidence by comparing the risk-neutralized option pricing distributions from various ARCH-type formulations. Utilizing a panel data set consisting of newly created exchange traded long-term equity anticipation securities, or leaps, on the Standard and Poor's 500 stock market index with maturity times ranging up to three years, we find that the degree of mean reversion in the volatility process implicit in these prices is best described by a Fractionally Integrated EGARCH (FIEGARCH) model. © 1999 Elsevier Science S.A. All rights reserved.

Latent Stick-Breaking Processes.

We develop a model for stochastic processes with random marginal distributions. Our model relies on a stick-breaking construction for the marginal distribution of the process, and introduces dependence across locations by using a latent Gaussian copula model as the mechanism for selecting the atoms. The resulting latent stick-breaking process (LaSBP) induces a random partition of the index space, with points closer in space having a higher probability of being in the same cluster. We develop an efficient and straightforward Markov chain Monte Carlo (MCMC) algorithm for computation and discuss applications in financial econometrics and ecology. This article has supplementary material online.

Numerical solutions of certain nonlinear models in European options on a distributed computing environment

Financial modelling in the area of option pricing involves the understanding of the correlations between asset and movements of buy/sell in order to reduce risk in investment. Such activities depend on financial analysis tools being available to the trader with which he can make rapid and systematic evaluation of buy/sell contracts. In turn, analysis tools rely on fast numerical algorithms for the solution of financial mathematical models. There are many different financial activities apart from shares buy/sell activities. The main aim of this chapter is to discuss a distributed algorithm for the numerical solution of a European option. Both linear and non-linear cases are considered. The algorithm is based on the concept of the Laplace transform and its numerical inverse. The scalability of the algorithm is examined. Numerical tests are used to demonstrate the effectiveness of the algorithm for financial analysis. Time dependent functions for volatility and interest rates are also discussed. Applications of the algorithm to non-linear Black-Scholes equation where the volatility and the interest rate are functions of the option value are included. Some qualitative results of the convergence behaviour of the algorithm is examined. This chapter also examines the various computational issues of the Laplace transformation method in terms of distributed computing. The idea of using a two-level temporal mesh in order to achieve distributed computation along the temporal axis is introduced. Finally, the chapter ends with some conclusions.

On the Viability of Microservers for Financial Analytics

Energy consumption and total cost of ownership are daunting challenges for Datacenters, because they scale disproportionately with performance. Datacenters running financial analytics may incur extremely high operational costs in order to meet performance and latency requirements of their hosted applications. Recently, ARM-based microservers have emerged as a viable alternative to high-end servers, promising scalable performance via scale-out approaches and low energy consumption. In this paper, we investigate the viability of ARM-based microservers for option pricing, using the Monte Carlo and Binomial Tree kernels. We compare an ARM-based microserver against a state-of-the-art x86 server. We define application-related but platform-independent energy and performance metrics to compare those platforms fairly in the context of datacenters for financial analytics and give insight on the particular requirements of option pricing. Our experiments show that through scaling out energyefficient compute nodes within a 2U rack-mounted unit, an ARM-based microserver consumes as little as about 60% of the energy per option pricing compared to an x86 server, despite having significantly slower cores. We also find that the ARM microserver scales enough to meet a high fraction of market throughput demand, while consuming up to 30% less energy than an Intel server

Iso-Quality of Service: Fairly Ranking Servers for Real-Time Data Analytics

We present a mathematically rigorous Quality-of-Service (QoS) metric which relates the achievable quality of service metric (QoS) for a real-time analytics service to the server energy cost of offering the service. Using a new iso-QoS evaluation methodology, we scale server resources to meet QoS targets and directly rank the servers in terms of their energy-efficiency and by extension cost of ownership. Our metric and method are platform-independent and enable fair comparison of datacenter compute servers with significant architectural diversity, including micro-servers. We deploy our metric and methodology to compare three servers running financial option pricing workloads on real-life market data. We find that server ranking is sensitive to data inputs and desired QoS level and that although scale-out micro-servers can be up to two times more energy-efficient than conventional heavyweight servers for the same target QoS, they are still six times less energy efficient than high-performance computational accelerators.

Methods and Metrics for Fair Server Assessment under Real-Time Financial Workloads

We present a rigorous methodology and new metrics for fair comparison of server and microserver platforms. Deploying our methodology and metrics, we compare a microserver with ARM cores against two servers with ×86 cores running the same real-time financial analytics workload. We define workload-specific but platform-independent performance metrics for platform comparison, targeting both datacenter operators and end users. Our methodology establishes that a server based on the Xeon Phi co-processor delivers the highest performance and energy efficiency. However, by scaling out energy-efficient microservers, we achieve competitive or better energy efficiency than a power-equivalent server with two Sandy Bridge sockets, despite the microserver's slower cores. Using a new iso-QoS metric, we find that the ARM microserver scales enough to meet market throughput demand, that is, a 100% QoS in terms of timely option pricing, with as little as 55% of the energy consumed by the Sandy Bridge server.

Influência do risco de incumprimento nos retornos de capital

Sempre foi do interesse das instituições financeiras de crédito determinar o risco de incumprimento associado a uma empresa por forma a avaliar o seu perfil. No entanto, esta informação é útil a todos os stakeholders de uma empresa, já que também estes comprometem uma parte de si ao interagirem com esta. O aumento do número de insolvências nos últimos anos tem reafirmado a necessidade de ampliar e aprofundar a pesquisa sobre o stress financeiro. A identificação dos fatores que influenciam a determinação do preço dos ativos sempre foi do interesse de todos os stakeholders, por forma a antecipar a variação dos retornos e agir em sua conformidade. Nesta dissertação será estudada a influência do risco de incumprimento sobre os retornos de capital, usando como indicador do risco de incumprimento a probabilidade de incumprimento obtida segundo o modelo de opções de Merton (1974). Efetuou-se esta análise durante o período de Fevereiro de 2002 a Dezembro de 2011, utilizando dados de empresas Portuguesas, Espanholas e Gregas. Os resultados evidenciam uma relação negativa do risco de incumprimento com os retornos de capital, que é devida a um efeito momentum e à volatilidade. A par disso, também se demonstra que o tamanho e o book-to-market não são representativos do risco de incumprimento na amostra aqui utilizada, ao contrário do que Fama & French (1992; 1996) afirmavam.

Essays on Applications of the Factor Model

Thesis (Ph.D.)--University of Washington, 2013

Three essays on the term structure of interest rates

A PhD Dissertation, presented as part of the requirements for the Degree of Doctor of Philosophy from the NOVA - School of Business and Economics

Latent Variable Models for Stochastic Discount Factors.

Latent variable models in finance originate both from asset pricing theory and time series analysis. These two strands of literature appeal to two different concepts of latent structures, which are both useful to reduce the dimension of a statistical model specified for a multivariate time series of asset prices. In the CAPM or APT beta pricing models, the dimension reduction is cross-sectional in nature, while in time-series state-space models, dimension is reduced longitudinally by assuming conditional independence between consecutive returns, given a small number of state variables. In this paper, we use the concept of Stochastic Discount Factor (SDF) or pricing kernel as a unifying principle to integrate these two concepts of latent variables. Beta pricing relations amount to characterize the factors as a basis of a vectorial space for the SDF. The coefficients of the SDF with respect to the factors are specified as deterministic functions of some state variables which summarize their dynamics. In beta pricing models, it is often said that only the factorial risk is compensated since the remaining idiosyncratic risk is diversifiable. Implicitly, this argument can be interpreted as a conditional cross-sectional factor structure, that is, a conditional independence between contemporaneous returns of a large number of assets, given a small number of factors, like in standard Factor Analysis. We provide this unifying analysis in the context of conditional equilibrium beta pricing as well as asset pricing with stochastic volatility, stochastic interest rates and other state variables. We address the general issue of econometric specifications of dynamic asset pricing models, which cover the modern literature on conditionally heteroskedastic factor models as well as equilibrium-based asset pricing models with an intertemporal specification of preferences and market fundamentals. We interpret various instantaneous causality relationships between state variables and market fundamentals as leverage effects and discuss their central role relative to the validity of standard CAPM-like stock pricing and preference-free option pricing.

Asymmetric Smiles, Leverage Effects and Structural Parameters.

In this paper, we characterize the asymmetries of the smile through multiple leverage effects in a stochastic dynamic asset pricing framework. The dependence between price movements and future volatility is introduced through a set of latent state variables. These latent variables can capture not only the volatility risk and the interest rate risk which potentially affect option prices, but also any kind of correlation risk and jump risk. The standard financial leverage effect is produced by a cross-correlation effect between the state variables which enter into the stochastic volatility process of the stock price and the stock price process itself. However, we provide a more general framework where asymmetric implied volatility curves result from any source of instantaneous correlation between the state variables and either the return on the stock or the stochastic discount factor. In order to draw the shapes of the implied volatility curves generated by a model with latent variables, we specify an equilibrium-based stochastic discount factor with time non-separable preferences. When we calibrate this model to empirically reasonable values of the parameters, we are able to reproduce the various types of implied volatility curves inferred from option market data.

Modélisation du carnet d'ordres limites et prévision de séries temporelles

Le contenu de cette thèse est divisé de la façon suivante. Après un premier chapitre d’introduction, le Chapitre 2 est consacré à introduire aussi simplement que possible certaines des théories qui seront utilisées dans les deux premiers articles. Dans un premier temps, nous discuterons des points importants pour la construction de l’intégrale stochastique par rapport aux semimartingales avec paramètre spatial. Ensuite, nous décrirons les principaux résultats de la théorie de l’évaluation en monde neutre au risque et, finalement, nous donnerons une brève description d’une méthode d’optimisation connue sous le nom de dualité. Les Chapitres 3 et 4 traitent de la modélisation de l’illiquidité et font l’objet de deux articles. Le premier propose un modèle en temps continu pour la structure et le comportement du carnet d’ordres limites. Le comportement du portefeuille d’un investisseur utilisant des ordres de marché est déduit et des conditions permettant d’éliminer les possibilités d’arbitrages sont données. Grâce à la formule d’Itô généralisée il est aussi possible d’écrire la valeur du portefeuille comme une équation différentielle stochastique. Un exemple complet de modèle de marché est présenté de même qu’une méthode de calibrage. Dans le deuxième article, écrit en collaboration avec Bruno Rémillard, nous proposons un modèle similaire mais cette fois-ci en temps discret. La question de tarification des produits dérivés est étudiée et des solutions pour le prix des options européennes de vente et d’achat sont données sous forme explicite. Des conditions spécifiques à ce modèle qui permettent d’éliminer l’arbitrage sont aussi données. Grâce à la méthode duale, nous montrons qu’il est aussi possible d’écrire le prix des options européennes comme un problème d’optimisation d’une espérance sur en ensemble de mesures de probabilité. Le Chapitre 5 contient le troisième article de la thèse et porte sur un sujet différent. Dans cet article, aussi écrit en collaboration avec Bruno Rémillard, nous proposons une méthode de prévision des séries temporelles basée sur les copules multivariées. Afin de mieux comprendre le gain en performance que donne cette méthode, nous étudions à l’aide d’expériences numériques l’effet de la force et la structure de dépendance sur les prévisions. Puisque les copules permettent d’isoler la structure de dépendance et les distributions marginales, nous étudions l’impact de différentes distributions marginales sur la performance des prévisions. Finalement, nous étudions aussi l’effet des erreurs d’estimation sur la performance des prévisions. Dans tous les cas, nous comparons la performance des prévisions en utilisant des prévisions provenant d’une série bivariée et d’une série univariée, ce qui permet d’illustrer l’avantage de cette méthode. Dans un intérêt plus pratique, nous présentons une application complète sur des données financières.

Exercise sheet 4

Exercises and solutions in LaTex