Fabio Bellini

Bio: Fabio Bellini is an academic researcher from University of Milano-Bicocca. The author has contributed to research in topics: Quantile & Esscher transform. The author has an hindex of 16, co-authored 59 publications receiving 1460 citations. Previous affiliations of Fabio Bellini include Bilkent University & University of Milan.

Generalized Quantiles as Risk Measures

Abstract: In the statistical and actuarial literature several generalizations of quantiles have been considered, by means of the minimization of a suitable asymmetric loss function. All these generalized quantiles share the important property of elicitability, that is recently receiving a lot of attention since it corresponds to the existence of a natural backtesting methodology. In this paper we investigate the case of M-quantiles, defined as the minimizers of an asymmetric convex loss function, in contrast to Orlicz quantiles, that have been considered in Bellini and Rosazza Gianin (2012). We discuss their properties as risk measures and point out the connection with the zero utility premium principle and with shortfall risk measures introduced by Follmer and Schied (2002). In particular, we show that the only M-quantiles that are coherent risk measures are the expectiles, introduced by Newey and Powell (1987) as the minimizers of an asymmetric quadratic loss function. We provide their dual and Kusuoka representations and discuss their relationship with CVaR. We analyze their asymptotic properties and show that for very heavy tailed distributions expectiles are more conservative than the usual quantiles. Finally, we show their robustness in the sense of lipschitzianity with respect to the Wasserstein metric.

On the Existence of Minimax Martingale Measures

Abstract: Embedding asset pricing in a utility maximization framework leads naturally to the concept of minimax martingale measures. We consider a market model where the price process is assumed to be an R-semimartingale X and the set of trading strategies consists of all predictable, X-integrable, R-valued processes H for which the stochastic integral (H.X) is uniformly bounded from below. When the market is free of arbitrage, we show that a sufficient condition for the existence of the minimax measure is that the utility function u: R - R is concave and nondecreasing. We also show the equivalence between the no free lunch with vanishing risk condition, the existence of a separating measure, and a properly defined notion of viability.

On the Existence of Minimax Martingale Measures

Abstract: Embedding asset pricing in a utility maximization framework leads naturally to the concept of minimax martingale measures. We consider a market model where the price process is assumed to be an d-semimartingale X and the set of trading strategies consists of all predictable, X-integrable, d-valued processes H for which the stochastic integral (H.X) is uniformly bounded from below. When the market is free of arbitrage, we show that a sufficient condition for the existence of the minimax measure is that the utility function u : is concave and nondecreasing. We also show the equivalence between the no free lunch with vanishing risk condition, the existence of a separating measure, and a properly defined notion of viability.

Risk management with expectiles

Abstract: Expectiles (EVaR) are a one-parameter family of coherent risk measures that have been recently suggested as an alternative to quantiles (VaR) and to expected shortfall (ES). In this work we review their known properties, we discuss their financial meaning, we compare them with VaR and ES and we study their asymptotic behaviour, refining some of the results in Bellini et al. [(2014). “Generalized Quantiles as Risk Measures.” Insurance: Mathematics and Economics, 54:41–48]. Moreover, we present a real-data example for the computation of expectiles by means of simple Garch(1,1) models and we assess the accuracy of the forecasts by means of a consistent loss function as suggested by Gneiting [(2011). “Making and Evaluating Point Forecast.” Journal of the American Statistical Association, 106 (494): 746–762]. Theoretical and numerical results indicate that expectiles are perfectly reasonable alternatives to VaR and ES risk measures.

Generalized quantiles as risk measures

Abstract: In the statistical and actuarial literature several generalizations of quantiles have been considered, by means of the minimization of a suitable asymmetric loss function. All these generalized quantiles share the important property of elicitability , which has received a lot of attention recently since it corresponds to the existence of a natural backtesting methodology. In this paper we investigate the case of M -quantiles as the minimizers of an asymmetric convex loss function, in contrast to Orlicz quantiles that have been considered in Bellini and Rosazza Gianin (2012). We discuss their properties as risk measures and point out the connection with the zero utility premium principle and with shortfall risk measures introduced by Follmer and Schied (2002). In particular, we show that the only M -quantiles that are coherent risk measures are the expectiles , introduced by Newey and Powell (1987) as the minimizers of an asymmetric quadratic loss function. We provide their dual and Kusuoka representations and discuss their relationship with CVaR. We analyze their asymptotic properties for α → 1 and show that for very heavy tailed distributions expectiles are more conservative than the usual quantiles. Finally, we show their robustness in the sense of lipschitzianity with respect to the Wasserstein metric.

Putting order in risk measures

Abstract: This paper introduces a set of axioms that define convex risk measures. Duality theory provides the representation theorem for these measures and the link with pricing rules.

Comparison Methods for Stochastic Models and Risks

Abstract: (2003). Comparison Methods for Stochastic Models and Risks. Technometrics: Vol. 45, No. 4, pp. 370-371.

The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets

Abstract: Let χ be a family of stochastic processes on a given filtered probability space (Ω, F, (Ft)t∈T, P) with T⊆R+. Under the assumption that the set Me of equivalent martingale measures for χ is not empty, we give sufficient conditions for the existence of a unique equivalent martingale measure that minimizes the relative entropy, with respect to P, in the class of martingale measures. We then provide the characterization of the density of the minimal entropy martingale measure, which suggests the equivalence between the maximization of expected exponential utility and the minimization of the relative entropy.