Functional central limit theorems for rough volatility

We extend Donsker's approximation of Brownian motion to fractional Brownian motion with Hurst exponent $H \in (0,1)$ and to Volterra-like processes. Some of the most relevant consequences of our `rough Donsker (rDonsker) Theorem' are convergence results for discrete approximations of a large class of rough models. This justifies the validity of simple and easy-to-implement Monte-Carlo methods, for which we provide detailed numerical recipes. We test these against the current benchmark Hybrid scheme \cite{BLP15} and find remarkable agreement (for a large range of values of~$H$). This rDonsker Theorem further provides a weak convergence proof for the Hybrid scheme itself, and allows to construct binomial trees for rough volatility models, the first available scheme (in the rough volatility context) for early exercise options such as American or Bermudan.Comment: 30 pages, 11 figure

Clustering market regimes using the Wasserstein distance

The problem of rapid and automated detection of distinct market regimes is a topic of great interest to financial mathematicians and practitioners alike. In this paper, we outline an unsupervised learning algorithm for clustering financial time-series into a suitable number of temporal segments (market regimes). As a special case of the above, we develop a robust algorithm that automates the process of classifying market regimes. The method is robust in the sense that it does not depend on modelling assumptions of the underlying time series as our experiments with real datasets show. This method -- dubbed the Wasserstein $k$-means algorithm -- frames such a problem as one on the space of probability measures with finite $p^\text{th}$ moment, in terms of the $p$-Wasserstein distance between (empirical) distributions. We compare our WK-means approach with a more traditional clustering algorithms by studying the so-called maximum mean discrepancy scores between, and within clusters. In both cases it is shown that the WK-means algorithm vastly outperforms all considered competitor approaches. We demonstrate the performance of all approaches both in a controlled environment on synthetic data, and on real data

Clustering market regimes using the Wasserstein distance

The problem of rapid and automated detection of distinct market regimes is a topic of great interest to financial mathematicians and practitioners alike. In this paper, we outline an unsupervised learning algorithm for clustering financial time-series into a suitable number of temporal segments (market regimes). As a special case of the above, we develop a robust algorithm that automates the process of classifying market regimes. The method is robust in the sense that it does not depend on modelling assumptions of the underlying time series as our experiments with real datasets show. This method -- dubbed the Wasserstein $k$-means algorithm -- frames such a problem as one on the space of probability measures with finite $p^\text{th}$ moment, in terms of the $p$-Wasserstein distance between (empirical) distributions. We compare our WK-means approach with a more traditional clustering algorithms by studying the so-called maximum mean discrepancy scores between, and within clusters. In both cases it is shown that the WK-means algorithm vastly outperforms all considered competitor approaches. We demonstrate the performance of all approaches both in a controlled environment on synthetic data, and on real data

Diseño de una intervención pedagógica para que los estudiantes aumenten la ingesta de vegetales ofertados en el menú escolar

Analizar la ingesta de vegetales en el comedor escolar antes y después del diseño y aplicación de una intervención pedagógica para aumentar el consumo de los mismos mejorando los hábitos alimenticios. Estudio cuasi experimental prospectivo y longitudinal. A través de una intervención pedagógica de 9 sesiones de duración, se ha promocionado la ingesta de vegetales a un grupo de 25 alumnos de 8 y 9 años que están cursando segundo grado de primaria. Se han utilizado actividades de diferente índole, tratando los aspectos nutricionales, los beneficios para la salud y su gestación en un entorno próximo. La finalidad del trabajo es constatar que, si se promocionan los vegetales con acciones pedagógicas entre la población más joven, el consumo de vegetales aumenta. Los resultados obtenidos constatan que se han cumplido los objetivos establecidos, analizando y comparando el consumo de vegetales de la población estudiada en el pre y pos intervención. Se demuestra que ha habido un aumento de la ingesta de vegetales en la población de estudio, pasando de consumir un 18% de los vegetales ofrecidos en el menú escolar a un 36%. El consumo de frutas ha aumentado ligeramente, de un 36% a un 40%, debido a la repercusión de la acción pedagógica. No se observan aumentos significativos en los otros grupos de alimentos.To measure if there is any increase in the consumption of vegetables while conducting nutrition education intervention during school hours. Quasi-experimental prospective and longitudinal study. Design of a 9-session educational intervention to promote the vegetable intake among the students. Different types of activities have been used to make the students aware of the nutritional aspects and health benefits of the vegetables. The purpose of the study is to measure the increase of vegetables intake, when vegetables are promoted with pedagogical actions among elementary school students. Subjects. Children aged 8-9 years (n = 25 from intervention group). After a 9-session educational program, children (n = 25) in the intervention school had a significant increase in vegetable intake (from 18% to 36% of the school menu). An increase of fruit intake (5%) can be observed as a repercussion of the intervention. No other changes in food or nutrient intake were detected. It is concluded that the intervention group increased vegetable intake due to the 9-session vegetable education program that was implemented.Analitzar la ingesta de vegetals en el menjador escolar abans i després del disseny i aplicació d'una intervenció pedagògica per a augmentar el consum dels mateixos millorant els hàbits alimentosos. Estudi quasi experimental prospectiu i longitudinal. A través d'una intervenció pedagògica de 9 sessions de durada, s'ha promocionat la ingesta de vegetals a un grup de 25 alumnes de 8 i 9 anys que estan cursant segon grau de primària. S'han utilitzat activitats de diferent índole, tractant els aspectes nutricionals, els beneficis per a la salut i la seva gestació en un entorn pròxim. La finalitat del treball és constatar que, si es promocionen els vegetals amb accions pedagògiques entre la població més jove, el consum de vegetals augmenta. Els resultats obtinguts constaten que s'han complert els objectius establerts, analitzant i comparant el consum de vegetals de la població estudiada en el pre i post intervenció. Es demostra que hi ha hagut un augment de la ingesta de vegetals en la població d'estudi, passant de consumir un 18% dels vegetals oferts en el menú escolar a un 36%. El consum de fruites ha augmentat lleugerament, d'un 36% a un 40%, a causa de la repercussió de l'acció pedagògica. No s'observen augments significatius en els altres grups d'aliments

Molecular dynamics simulations of the calmodulin-induced α-helix in the SK2 calcium-gated potassium ion channel

The family of small-conductance Ca2+-activated potassium ion channels (SK channels) is composed of four members (SK1, SK2, SK3, and SK4) involved in neuron-firing regulation. The gating of these channels depends on the intracellular Ca2+ concentration, and their sensitivity to this ion is provided by calmodulin (CaM). This protein binds to a specific region in SK channels known as the calmodulin-binding domain (CaMBD), an event which is essential for their gating. While CaMBDs are typically disordered in the absence of CaM, the SK2 channel subtype displays a small prefolded α-helical region in its CaMBD even if CaM is not present. This small helix is known to turn into a full α-helix upon CaM binding, although the molecular-level details for this conversion are not fully understood yet. In this work, we offer new insights on this physiologically relevant process by means of enhanced sampling, atomistic Hamiltonian replica exchange molecular dynamics simulations, providing a more detailed understanding of CaM binding to this target. Our results show that CaM is necessary for inducing a full α-helix along the SK2 CaMBD through hydrophobic interactions with V426 and L427. However, it is also necessary that W431 does not compete for these interactions; the role of the small prefolded α-helix in the SK2 CaMBD would be to stabilize W431 so that this is the case. In conclusion, our findings provide further insight into a key interaction between CaM and SK channels that is important for channel sensitivity to Ca2+.The authors thank Donostia International Physics Center (DIPC) for providing access to its computational resources. We acknowledge financial support from the Department of Education, Universities, and Research of the Basque Government and the University of the Basque Country (IT1165-19, KK-2020/00110, and IT1707-22), from the Spanish Ministry of Science and Innovation (projects PID2021-128286NB-100, PID2019-105488GB-I00, TED2021-132074B-C32, and RTI2018-097839-B-100) and from FEDER funds