Computing option pricing models under transaction costs

AbstractThis paper deals with the Barles–Soner model arising in the hedging of portfolios for option pricing with transaction costs. This model is based on a correction volatility function Ψ solution of a nonlinear ordinary differential equation. In this paper we obtain relevant properties of the function Ψ which are crucial in the numerical analysis and computing of the underlying nonlinear Black–Scholes equation. Consistency and stability of the proposed numerical method are detailed and illustrative examples are given

Solving random diffusion models with nonlinear perturbations by the Wiener-Hermite expansion method

[EN] This paper deals with the construction of approximate series solutions of random nonlinear diffusion equations where nonlinearity is considered by means of a frank small parameter and uncertainty is introduced through white noise in the forcing term. For the simpler but important case in which the diffusion coefficient is time independent, we provide a Gaussian approximation of the solution stochastic process by taking advantage of the Wiener¿Hermite expansion together with the perturbation method. In addition, approximations of the main statistical functions associated with a solution, such as the mean and variance, are computed. Numerical values of these functions are compared with respect to those obtained by applying the Runge¿Kutta second-order stochastic scheme as an illustrative example.This work was partially supported by the Spanish M.C.Y.T. and FEDER grants MTM2009-08587, TRA2007-68006-C02-02, DPI2010-20891-C02-01 as well as the Universidad Politécnica de Valencia grant PAID-06-09 (ref. 2588).Cortés López, JC.; Romero Bauset, JV.; Roselló Ferragud, MD.; Santamaría Navarro, C. (2011). Solving random diffusion models with nonlinear perturbations by the Wiener-Hermite expansion method. Computers and Mathematics with Applications. 61(8):1946-1950. https://doi.org/10.1016/j.camwa.2010.07.057S1946195061

Do the generalized polynomial chaos and Fröbenius methods retain the statistical moments of random differential equations?

The aim of this paper is to explore whether the generalized polynomial chaos (gPC) and random Fröbenius methods preserve the first three statistical moments of random differential equations. There exist exact solutions only for a few cases, so there is a need to use other techniques for validating the aforementioned methods in regards to their accuracy and convergence. Here we present a technique for indirectly study both methods. In order to highlight similarities and possible differences between both approaches, the study is performed by means of a simple but still illustrative test-example involving a random differential equation whose solution is highly oscillatory. This comparative study shows that the solutions of both methods agree very well when the gPC method is developed in terms of the optimal orthogonal polynomial basis selected according to the statistical distribution of the random input. Otherwise, we show that results provided by the gPC method deteriorate severely. A study of the convergence rates of both methods is also included.This work has been partially supported by the Spanish M.C.Y.T. grants MTM2009-08587, DPI2010-20891-C02-01 as well as the Universitat Politecnica de Valencia grants PAID06-11 (ref. 2070) and PAID00-11 (ref. 2753).Chen Charpentier, BM.; Cortés López, JC.; Romero Bauset, JV.; Roselló Ferragud, MD. (2013). Do the generalized polynomial chaos and Fröbenius methods retain the statistical moments of random differential equations?. Applied Mathematics Letters. 26(5):553-558. doi:10.1016/j.aml.2012.12.013S55355826

Probabilistic solution of random homogeneous linear second-order difference equations

This paper deals with the computation of the first probability density function of the solution of random homogeneous linear second-order difference equations by the Random Variable Transformation method. This approach allows us to generalize the classical solution obtained in the deterministic scenario. Several illustrative examples are provided.This work was sponsored by "Ministerio de Economa y Competitividad" of the Spanish Government in the frame of the Project with Reference TRA2012-36932.Casabán Bartual, MC.; Cortés López, JC.; Romero Bauset, JV.; Roselló Ferragud, MD. (2014). Probabilistic solution of random homogeneous linear second-order difference equations. Applied Mathematics Letters. 34:27-32. https://doi.org/10.1016/j.aml.2014.03.010S27323

A comparative study to the numerical approximation of random Airy differential equation

The aim of this paper is twofold. First, we deal with the extension to the random framework of the piecewise Fröbenius method to solve Airy differential equations. This extension is based on mean square stochastic calculus. Second, we want to explore the capability to provide not only reliable approximations for both the average and the standard deviation functions associated to the solution stochastic process, but also to save computational time as it happens in dealing with the analogous problem in the deterministic scenario. This includes a comparison of the numerical results with respect to those obtained by other commonly used operational methods such as polynomial chaos and Monte Carlo simulations. To conduct this comparative study, we have chosen the Airy random differential equation because it has highly oscillatory solutions. This feature allows us to emphasize differences between all the considered approaches. © 2011 Elsevier Ltd. All rights reserved.This work has been partially supported by the Spanish M.C.Y.T. and FEDER grants MTM2009-08587, DPI2010-20891-C02-01 as well as the Universitat Politecnica de Valencia grant PAID-06-09 (Ref. 2588).Cortés López, JC.; Jódar Sánchez, LA.; Romero Bauset, JV.; Roselló Ferragud, MD. (2011). A comparative study to the numerical approximation of random Airy differential equation. Computers and Mathematics with Applications. 62(9):3411-3417. https://doi.org/10.1016/j.camwa.2011.08.056S3411341762

Using a homogeneous equilibrium model for the study of the inner nozzle flow and cavitation pattern in convergent-divergent nozzles of diesel injectors

[EN] In this paper, the behaviour of the internal nozzle flow and cavitation phenomenon are numerically studied for non-conventional Diesel convergent-divergent nozzles in order to assess their potential in terms of flow characteristics. The used nozzles differs each other in the convergence-divergence level of the orifices but all of them keep the same diameter at the middle of the nozzle orifice. The calculations have been performed using a code previously validated and able to simulate cavitation phenomenon using a homogeneous equilibrium model for the biphasic fluid and using a RANS method (RNG k-ε) as a turbulence modelling approach. For the simulations, one injection pressure and different discharge pressures were used in order to assess the characteristics of nozzles for different Reynolds conditions involving cavitating and non-cavitating conditions. The comparison of the nozzles has been carried out in terms of flow characteristics such as mass flow, momentum flux, effective velocity and other important dimensionless parameters which help to describe the behaviour of the inner flow: discharge coefficient (Cd), area coefficient (Ca) and velocity coefficient (Cv). Additionally, the nozzles have been compared in terms of cavitation inception conditions and cavitation development. The study has shown a high influence on the results of the level of convergence-divergence used in the nozzles. In these nozzles, the vapour originated from cavitation phenomenon came from the throttle of the orifice at the midpoint, and it extended along the whole wall of the divergent nozzle part towards the outlet of the orifice. The main results of the investigation have shown how the different geometries modify the cavitation conditions as well as the discharge coefficient and effective velocity. In particular, the nozzle with highest convergence-divergence level showed cavitation for all the tested conditions while for the nozzle with lowest convergence-divergence level, the cavitation phenomenon could be avoided for high discharge pressures. Additionally, the nozzle with highest convergence-divergence level showed the lowest discharge coefficient values but similar effective injection velocity than the nozzle with lowest level of convergence-divergence level despite of its higher orifice outlet area.This work was partly sponsored by ‘‘Ministerio de Economía y Competitividad’’ of the Spanish Government, in the frame of the project ‘‘Estudio de la interacción chorro-pared em condiciones realistas de motor’’, Reference TRA2015-67679-c2-1- R. This support is gratefully acknowledged by the authors. Mr. Jaramillo’s thesis is supported by ‘‘Conselleria d’Educació, Cultura I Esports’’ of ‘‘Generalitat Valenciana’’ through the program ‘‘Programa VALI+D para investigadores en Formación’’, Reference ACIF/2015/040. The authors would like to express gratitude for the computer resources, technical expertise and assistance provided by the Universidad de Valencia relating to the use of the supercomputer ‘‘Tirant’’.Salvador, FJ.; Jaramillo-Císcar, D.; Romero, J.; Roselló, M. (2017). Using a homogeneous equilibrium model for the study of the inner nozzle flow and cavitation pattern in convergent-divergent nozzles of diesel injectors. Journal of Computational and Applied Mathematics. 309:630-641. https://doi.org/10.1016/j.cam.2016.04.010S63064130

Probabilistic solution of random SI-type epidemiological models using the Random Variable Transformation technique

[EN] This paper presents a full probabilistic description of the solution of random SI-type epidemiological models which are based on nonlinear differential equations. This description consists of determining: the first probability density function of the solution in terms of the density functions of the diffusion coefficient and the initial condition, which are assumed to be independent random variables; the expectation and variance functions of the solution as well as confidence intervals and, finally, the distribution of time until a given proportion of susceptibles remains in the population. The obtained formulas are general since they are valid regardless the probability distributions assigned to the random inputs. We also present a pair of illustrative examples including in one of them the application of the theoretical results to model the diffusion of a technology using real data.This work has been partially supported by the Ministerio de Economia y Competitividad Grants MTM2013-41765-P and TRA2012-36932.Casabán Bartual, MC.; Cortés López, JC.; Romero Bauset, JV.; Roselló Ferragud, MD. (2015). Probabilistic solution of random SI-type epidemiological models using the Random Variable Transformation technique. Communications in Nonlinear Science and Numerical Simulation. 24(1):86-97. https://doi.org/10.1016/j.cnsns.2014.12.016S869724

Some recommendations for applying gPC (generalized polynomial chaos) to modeling: An analysis through the Airy random differential equation

In this paper we study the use of the generalized polynomial chaos method to differential equations describing a model that depends on more than one random input. This random input can be in the form of parameters or of initial or boundary conditions. We investigate the effect of the choice of the probability density functions for the inputs on the output stochastic processes. The study is performed on the Airy¿s differential equation. This equation is a good test case since its solutions are highly oscillatory and errors can develop both in the amplitude and the phase. Several different situations are considered and, finally, conclusions are presented.This work has been partially supported by the Spanish M.C.Y.T. and FEDER Grants MTM2009-08587, DPI2010-20891-C02-01 as well as the Universitat Politecnica de Valencia Grants PAID-00-11 (Ref. 2751) and PAID-06-11 (Ref. 2070).Chen Charpentier, BM.; Cortés López, JC.; Romero Bauset, JV.; Roselló Ferragud, MD. (2013). Some recommendations for applying gPC (generalized polynomial chaos) to modeling: An analysis through the Airy random differential equation. Applied Mathematics and Computation. 219(9):4208-4218. https://doi.org/10.1016/j.amc.2012.11.007S42084218219

EPDR1 up-regulation in human colorectal cancer is related to staging and favours cell proliferation and invasiveness

The finding of novel molecular markers for prediction or prognosis of invasiveness in colorectal cancer (CRC) constitutes an appealing challenge. Here we show the up-regulation of EPDR1 in a prospective cohort of 101 CRC patients, in a cDNA array of 43 patients and in in silico analyses. EPDR1 encodes a protein related to ependymins, a family of glycoproteins involved in intercellular contacts. A thorough statistical model allowed us to conclude that the gene is significantly up-regulated in tumour tissues when compared with normal mucosa. These results agree with those obtained by the analysis of three publicly available databases. EPDR1 up-regulation correlates with the TNM staging parameters, especially T and M. Studies with CRC cell lines revealed that the methylation of a CpG island controls EPDR1 expression. siRNA knocking-down and overexpression of the gene following transient plasmid transfection, showed that EPDR1 favours cell proliferation, migration, invasiveness and adhesion to type I collagen fibres, suggesting a role in epithelial to mesenchymal transition. Both statistical and functional analysis correlated EPDR1 overexpression with invasiveness and dissemination of tumour cells, supporting the inclusion of EPDR1 in panels of genes used to improve molecular subtyping of CRC. Eventually, EPDR1 may be an actionable target.Fil: Gimeno Valiente, F.. No especifíca;Fil: Riffo Campos, Á. L.. Universidad de La Frontera; ChileFil: Ayala, G.. Universidad de Valencia; EspañaFil: Tarazona, N.. Universidad de Valencia; EspañaFil: Gambardella, V.. Universidad de Valencia; EspañaFil: Rodríguez, Fernanda Mariel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Ciencias Veterinarias del Litoral. Universidad Nacional del Litoral. Facultad de Ciencias Veterinarias. Instituto de Ciencias Veterinarias del Litoral; ArgentinaFil: Huerta, M.. Universidad de Valencia; EspañaFil: Martínez-Ciarpaglini, C.. Universidad de Valencia; EspañaFil: Montón Bueno, J.. Universidad de Valencia; EspañaFil: Roselló, S.. Universidad de Valencia; EspañaFil: Roda, D.. Universidad de Valencia; EspañaFil: Cervantes, A.. Universidad de Valencia; EspañaFil: Franco, L.. Universidad de Valencia; EspañaFil: López Rodas, G.. Universidad de Valencia; EspañaFil: Castillo, J.. Universidad de Valencia; Españ