[Delta]-VaR and [Delta]-TVaR for portfolios with mixture of elliptic distributions risk factors and DCC

This paper generalizes the [Delta]-VaR and [Delta]-TVaR method from portfolios with normally distributed risk factors to portfolios with mixture of elliptically distributed ones, when the volatility is governed by an elliptic MGARCH. Special attention is given to the particular case of a mixture of multivariate t-distributions with the elliptic dynamic conditional correlation (E-DCC).Capital allocation Dynamic volatility Risk management Solvency II VaR TVaR MGARCH Mixture of elliptic distributions

Decomposition method for the Camassa–Holm equation

International audienceThe Adomian decomposition method is applied to the Camassa–Holm equation. Approximate solutions are obtained for three smooth initial values. These solutions are weak solutions with some peaks. We plot those approximate solutions and find that they are very similar to the peaked soliton solutions. Also, one single and two anti-peakon approximate solutions are presented. Compared with the existing method, our procedure just works with the polynomial and algebraic computations for the CH equation

Approximation of multiple integrals over hyperboloids with application to a quadratic portfolio with options

An application involving a financial quadratic portfolio, where the joint underlying log-returns follow a multivariate elliptic distribution, is considered. This motivates the need for methods for the approximation of multiple integrals over hyperboloids. Transformations are used to reduce the hyperboloid integrals to products of integrals which can be approximated with appropriate numerical methods. The application of these methods is demonstrated using some financial applications examples.

Fuzzy value-at-risk and expected shortfall for portfolios with heavy-tailed returns

International audienceThis paper is concerned with linear portfolio value-at-risk (VaR) and expected shortfall (ES) computation when the portfolio risk factors are leptokurtic, imprecise and/or vague. Following Yoshida (2009), the risk factors are modeled as fuzzy random variables in order to handle both their random variability and their vagueness. We discuss and extend the Yoshida model to some non-Gaussian distributions and provide associated ES. Secondly, assuming that the risk factors' degree of imprecision changes over time, original fuzzy portfolio VaR and ES models are introduced. For a given subjectivity level fixed by the investor, these models allow the computation of a pessimistic and an optimistic estimation of the value-at-risk and of the expected shortfall. Finally, some empirical examples carried out on three portfolios constituted by some chosen French stocks, show the effectiveness of the proposed methods

Capital asset pricing model with fuzzy returns and hypothesis testing First draft

Abstract Over the last four decades, several estimation issues of the beta have been discussed extensively in a large literature. An emerging consensus is that the betas are time-varying and their estimates are impacted upon the return interval and the length of the estimation period. These findings lead to the prominence of the practical implementation of the Capital Asset Pricing Model. Our goal in this paper is two-fold: After studying the impact of the return interval on the beta estimates, we analyze the sample size effects on the preceding estimation. Working in the framework of fuzzy set theory, we first associate the returns based on closing prices with the intraperiod volatility for the representation by the means of a fuzzy random variable in order to incorporate the effect of the interval period over which the returns are measured in the analysis. Next, we use these fuzzy returns to estimate the beta via fuzzy least square method in order to deal efficiently with outliers in returns, often caused by structural breaks and regime switches in the asset prices. A bootstrap test and an asymptotic test are carried out to determine whether there is a linear relationship between the market portfolio fuzzy return and the given asset fuzzy return. Finally, the empirical results on French stocks reveal that our beta estimates seem to be more stable than the ordinary least square (OLS) estimates when the return intervals and the sample size change