A decomposition formula for option prices in the Heston model and applications to option pricing approximation

By means of classical Itô's calculus we decompose option prices as the sum of the classical Black-Scholes formula with volatility parameter equal to the root-mean-square future average volatility plus a term due by correlation and a term due to the volatility of the volatility. This decomposition allows us to develop first and second-order approximation formulas for option prices and implied volatilities in the Heston volatility framework, as well as to study their accuracy. Numerical examples are given.Stochastic Volatility, Heston Model, Itô's Calculus.

A general decomposition formula for derivative prices in stochastic volatility models

We see that the price of an european call option in a stochastic volatility framework can be decomposed in the sum of four terms, which identify the main features of the market that affect to option prices: the expected future volatility, the correlation between the volatility and the noise driving the stock prices, the market price of volatility risk and the difference of the expected future volatility at different times. We also study some applications of this decomposition.Continuous-time option pricing model, stochastic volatility, Ito's formula, incomplete markets

A generalization of Hull and White formula and applications to option pricing approximation

By means of Malliavin Calculus we see that the classical Hull and White formula for option pricing can be extended to the case where the noise driving the volatility process is correlated with the noise driving the stock prices. This extension will allow us to construct option pricing approximation formulas. Numerical examples are presented.Continuous-time option pricing model, stochastic volatility, Malliavin calculus

A note on the Malliavin differentiability of the Heston volatility

We show that the Heston volatility or equivalently the Cox-Ingersoll-Ross process is Malliavin differentiable and give an explicit expression for the derivative. This result assures the applicability of Malliavin calculus in the framework of the Heston stochastic volatility model and the Cox-Ingersoll-Ross model for interest rates.Malliavin calculus, stochastic volatility models, Heston model, Cox-Ingersoll-Ross process

Probiotics, Prebiotics and the Nervous System

Treballs Finals de Grau de Farmàcia, Facultat de Farmàcia i Ciències de l'Alimentació, Universitat de Barcelona, 2018. Tutora: Magdalena Rafecas.[eng] The relationship between the gut microbiota and the central nervous system (the microbiota-gut-brain axis) is an area of increasing interest and research. Studies based on germ-free models have provided a big amount of evidence of the connection between the gut and the brain. This research has done a great step forward in recent years, due to the application of metagenomics and bioinformatics. We now know that the human gut microbiome can be classified into three enterotypes, characterized by the variation in three genera: Bacteroides, Bacteroidetes, and Prevotella. Type of birth, formula feeding, and antibiotic intake are among the main factors that impact on infant microbiome assembly. Moreover, the composition of the gut microbiota is strongly associated with diet. A review of the bibliographical evidence connecting the alterations in the microbiome and some central nervous system disorders (as Alzheimer disease, Parkinson disease or autism spectrum disorder) shows us that the levels of Prevotella and the Firmicutes/Bacteroidetes ratio are altered in these pathologies. Accumulating data reveals that the microbiota-gut-brain axis can be modulated by the administration of probiotics and prebiotics. Moreover, some traditional fermented food has been seen to have probiotic properties and high-fiber containing diets have been associated with a lower Firmicutes/Bacteroidetes ratio and higher levels of Prevotella.[cat] La relació entre la microbiota intestinal i el sistema nerviós central (l'eix microbiota-intestí-cervell) és una àrea d'interès i investigació creixents. Els estudis basats en models lliures de gèrmens han proporcionat gran quantitat d’evidències de la connexió entre l'intestí i el cervell. Aquesta investigació ha fet un gran avenç en els últims anys gràcies a la metagenòmica i la bioinformàtica. Ara sabem que el microbioma intestinal humà es pot classificar en tres enterotips, caracteritzats per la variació en tres gèneres: Bacteroides, Bacteroidetes i Prevotella. Els tipus de naixement, el tipus d’alletament i la ingesta d'antibiòtics són els principals factors que afecten el desenvolupament del microbioma infantil. A més, la composició de la microbiota intestinal està fortament relacionada amb la dieta. Una revisió de les evidències bibliogràfiques que connecten les alteracions en el microbioma i alguns trastorns del sistema nerviós central (com la malaltia d'Alzheimer, la malaltia de Parkinson o el trastorn de l'espectre autista) ens mostra que els nivells de Prevotella i la relació Firmicutes/Bacteroidetes estan alterats en aquestes patologies. Cada cop hi ha més dades que revelen que l'eix microbiota-intestí-cervell pot ser modulat per l'administració de probiòtics i prebiòtics. D'altra banda, s'ha observat que alguns aliments fermentats tradicionals tenen propietats probiòtiques i que les dietes amb alt contingut de fibra s’associen a nivells inferiors de la raó Firmicutes /Bacteroidetes i superiors de Prevotella

Stochastic partial differential equations with Dirichlet white-noise boundary conditions

ABSTRACT. – The paper is devoted to one-dimensional nonlinear stochastic partial differential equations of parabolic type with non homogeneous Dirichlet boundary conditions of white-noise type. We formulate a set of conditions that a random field must satisfy to solve the equation. We show that a unique solution exists and that we can write it in terms of the stochastic kernel related to the problem. This formulation allows us to study the basic properties of the solution, as the continuity and the boundary-layer behavior, by means of Malliavin calculus. 2002 Éditions scientifiques et médicales Elsevier SAS AMS classification: 60H15; 60H07 RÉSUMÉ. – Cet article est consacré à l’étude d’équations aux dérivées partielles stochastiques non linéaires paraboliques en dimension un avec conditions aux bord de type Dirichlet non homogènes. Nous formulons des conditions qu’un champ aléatoire doit satisfaire pour resoudre l’EDPS. Nous montrons qu’il existe une solution unique et qu’elle s’exprime à l’aide d’un noyau stochastique relié au problème. Cette formulation nous permet d’étudier les propriétés de base de la solution, telles que la continuité et le comportement au bord, en utilisant le calcul de Malliavin. 2002 Éditions scientifiques et médicales Elsevier SAS 1

On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility

In this paper we use Malliavin calculus techniques to obtain an expression for the short-time behavior of the at-the-money implied volatility skew for a generalization of the Bates model, where the volatility does not need to be neither a difussion, nor a Markov process as the examples in section 7 show. This expression depends on the derivative of the volatility in the sense of Malliavin calculus.Black-Scholes formula, derivative operator, Itô's formula for the Skorohod integral, jump-diffusion stochastic volatility model

A Hull and White formula for a general stochastic volatility jump-diffusion model with applications to the study of the short-time behavior of the implied volatility

In this paper, generalizing results in Alòs, León and Vives (2007b), we see that the dependence of jumps in the volatility under a jump-diffusion stochastic volatility model, has no effect on the short-time behaviour of the at-the-money implied volatility skew, although the corresponding Hull and White formula depends on the jumps. Towards this end, we use Malliavin calculus techniques for Lévy processes based on Løkka (2004), Petrou (2006), and Solé, Utzet and Vives (2007).Hull and White formula, Malliavin calculus, Ito’s formula for the Skorohod integral, jumpdiffusion stochastic volatility models