1,721,096 research outputs found
A reduced basis for option pricing
Full text linkWe introduce a reduced basis method for the efficient numerical solution of partial integro-differential equations which arise in option pricing theory. Our method uses a basis of functions constructed from a sequence of Black-Scholes solutions with different volatilities. We show that this choice of basis leads to a sparse representation of option pricing functions, yielding an approximation whose precision is exponential in the number of basis functions. A Galerkin method using this basis for solving the pricing PDE is presented. Numerical tests based on the CEV diffusion model and the Merton jump diffusion model show that the method has better numerical performance relative to commonly used finite-difference and finite-element methods. We also compare our method with a numerical Proper Orthogonal Decomposition (POD). Finally, we show that this approach may be used advantageously for the calibration of local volatility functions.
Roughness properties of paths and signals
Full text linkFunctions and processes with irregular behaviour in time are ubiquitous in physics, engineering, and finance and have been the focus of various pathwise theories of integration in stochastic analysis, in which the degree of 'roughness' of the function plays an important role. This thesis focuses on various concepts of 'roughness' for continuous functions and processes and their interplay with pathwise integration. We first explore these issues using the concept of pathwise quadratic variation,
then expand results to the more general setting of p-th order variation.
The first chapter discusses some motivations and background for the questions explored in the thesis and provides an overview of the results.
In the second chapter, we study quadratic variation along a sequence of partitions and its dependence with respect to the choice of the partition sequence. We introduce a property which we call quadratic roughness, and show that for H ̈older-continuous paths satisfying this roughness condition, the quadratic variation along ‘balanced’ partitions is invariant with respect to the choice of the partition sequence. Typical paths of Brownian motion satisfy this quadratic roughness property almost-surely along partitions with fine enough mesh. Using these results we derive a formulation of the pathwise F ̈ollmer-Itˆo calculus which is invariant with respect to the partition sequences. Furthermore, we provide an invariance result
for local time under quadratic roughness.
In the third chapter, instead of balanced partition sequences (which is a key condition in Chapter 2) we consider (finitely) refining partition sequences, without any bound on mesh size. We construct a generalized Haar basis along any such finite refining sequence of partitions. We provide a closed-form representation of quadratic variation in terms of Faber-Schauder coefficients along this basis. Further, we construct a class of continuous processes with linear and prescribed quadratic variations along any given finitely refining partition sequence. We provide an example of a rough class of continuous processes with invariant quadratic variations along finitely refining sequences of partitions. Brownian motion belongs
to this ‘rough’ class, but we also give examples of processes with 1/2 -H ̈older continuity in this class. Finally, we extend these constructions to higher dimensions.
In the fourth chapter of the thesis, we consider a more general concept of roughness based on p-th variation and the associated notions of variation and roughness index of a continuous function. We define the normalized p-th variation
of a path and use it to introduce a pathwise estimator to estimate the order of roughness of a signal. We investigate the finite sample performance of our estimator for measuring the roughness of sample paths of stochastic processes
using detailed numerical experiments based on sample paths of fractional Brownian motion and Takagi-Landsberg functions.
In the final chapter we use our ‘roughness’ estimator (discussed in Chapter 4) to investigate the statistical evidence for the use of ‘rough’ fractional processes with Hurst exponent H < 0.5 for the modelling of volatility of financial assets, using a non-parametric, model-free approach. Detailed numerical experiments based on stochastic volatility models show that, even when the instantaneous volatility has diffusive dynamics with the same roughness as Brownian motion, the realized volatility exhibits rough behaviour corresponding to a Hurst exponent significantly smaller than 0.5, which suggests that the origin of the roughness observed in realized volatility time-series lies in the estimation error rather than the volatility process itself. Comparison of roughness estimates for realized and
instantaneous volatility in fractional volatility models with different values of Hurst exponent shows that, irrespective of the value of H, realized volatility always exhibits ‘rough’ behaviour with an apparent Hurst index ˆH < 0.5 but this is not
necessarily indicative of a similar rough behaviour of the spot volatility process which may have H ≥ 1/2
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
Model uncertainty and its impact on the pricing of derivative instruments
Full text linkModel uncertainty, in the context of derivative pricing, can be defined as the uncertainty on the value of a contingent claim resulting from the lack of precise knowledge of the pricing model to be used for its valuation. We introduce here a quantitative framework for defining model uncertainty in option pricing models. After discussing some properties which a quantitative measure of model uncertainty should verify in order to be useful and relevant in the context of risk measurement and management, we propose a method for measuring model uncertainty which verifies these properties and yields numbers which are comparable to other risk measures and compatible with observations of market prices of a set of benchmark derivatives. We illustrate the difference between model uncertainty and the more common notion of "market risk" through examples. Finally, we illustrate the connection between our proposed measure of model uncertainty and the recent literature on coherent and convex risk measures.decision under ambiguity; uncertainty; option pricing; risk measures; mathematical finance
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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