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Essays on Quantization in Financial Mathematics
Full text linkThis thesis is devoted to the study of some applications of quantization to Financial Mathematics, especially to option pricing and calibration of financial data. Quantization is a technique that comes originally from numerical probability, and consists in approximating random variables and stochastic processes taking infinitely many values, with a discrete version of them, in order to simplify the quadrature algorithms for the computation of expected values.
The purpose of this thesis is to show the great flexibility that quantization can have in the area of numerical probability and option pricing. In the literature, often there are ad hoc methods for a particular type of model or derivative, but no general framework seems to exist. Finite difference methods are heavily affected by the curse of dimensionality, while Monte Carlo methods need intense computational effort in order to have good precision, and are not designed for calibration purposes. Quantization can give an alternative methodology for a broad class of models and deriva- tives.
The aim of the thesis is twofold: first, the extension of the literature about quantization to a broad class of models, namely local and stochastic volatility models, affine, pure jumps and polynomial processes, is an interesting theoretical exercise in itself. In fact, every time we deal with a different model we have to take in consideration the properties of the process and therefore the quantization algorithm must be adapted. Second, it is important to consider the computational results of the new types of quantization introduced. Indeed, the algorithms that we have developed turn out to be fast and numerically stable, and these aspects are very relevant, as we can overcome some of the issues present in literature for other types of approach. The first line of research deals with a technique called Recursive Marginal Quantization. Introduced in Pagès and Sagna (2015), this methodology exploits the conditional distribution of the Euler scheme of a one dimensional stochastic differential equation in order to construct a step-by-step approximation of the process. In this thesis we deal with the generalization of this technique to systems of stochastic differential equations, in particular to the case of stochastic volatility models. The Recursive Marginal Quantization of multidimensional stochastic process allows us to price European and path dependent options, in particular American options, and to perform calibration on financial data, giving then an alternative, and sometimes overcoming, to the usual Monte Carlo techniques.
The second line of research takes a different perspective on quantization. Instead of using discretization schemes in order to compute the distribution of a stochastic process, we exploit the properties of the characteristic function and of the moment generating function for a broad class of processes. We consider the price process at maturity as a random variable, and we focus on the quantization of the stochastic variable, instead of focusing on the quantization of the whole stochastic process. This gives a faster and more precise technology for the pricing of options, and allows the quantization of a huge set of models for which the Recursive Marginal Quantization cannot be applied or is not numerically competitive
Quantized calibration in local volatility
No full textPricing of a derivative should be fast and accurate, otherwise it cannot be calibrated efficiently. Here, Giorgia Callegaro,
Lucio Fiorin and Martino Grasselli apply a fast quantization methodology, in a local volatility context, to the pricing of vanilla
and barrier options that overcomes the numerical problems in existing method
Pricing via recursive Quantization in Stochastic Volatility Models
No full textWe provide the first recursive quantization-based approach for pricing options in the presence of
stochastic volatility. This method can be applied to any model for which an Euler scheme is available
for the underlying price process and it allows to price vanillas, as well as exotics, thanks to the
knowledge of the transition probabilities for the discretized stock process.We apply the methodology
to some celebrated stochastic volatility models, including the Stein and Stein [Rev. Financ. Stud. 1991,
(4), 727–752] model and the SABR model introduced in Hagan et al. [Wilmott Mag., 2002, 84–108].
A numerical exercise shows that the pricing of vanillas turns out to be accurate; in addition, when
applied to some exotics like equity-volatility options, the quantization-based method overperforms
by far the Monte Carlo simulation
Pricing and Calibration in Local Volatility Models Via Fast Quantization
No full textIn this paper we propose the first calibration exercise based on quantization methods. Pricing and calibration are typically difficult tasks to accomplish: pricing should be fast and accurate, otherwise calibration cannot be performed efficiently. We apply in a local volatility context the recursive marginal quantization methodology to the pricing of vanilla and barrier options. A successful calibration of the Quadratic Normal Volatility model is performed in order to show the potentiality of the method in a concrete example, while a numerical exercise on barrier options shows that quantization overcomes Monte-Carlo methods
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
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Full text linkWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
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