Joint tails impact in stochastic volatility portfolio selection models
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Joint tails impact in stochastic volatility portfolio selection models Marco Bonomelli1 · Rosella Giacometti1
· Sergio Ortobelli Lozza1
© Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper examines the impact of the joints tails of the portfolio return and its empirical volatility on the optimal portfolio choices. We assume that the portfolio return and its volatility dynamic is approximated by a bivariate Markov chain constructed on its historical distribution. This allows the introduction of a non parametric stochastic volatility portfolio model without the explicit use of a GARCH type or other parametric stochastic volatility models. We describe the bi-dimensional tree structure of the Markov chain and discuss alternative portfolio strategies based on the maximization of the Sharpe ratio and of a modified Sharpe ratio that takes into account the behaviour of a market benchmark. Finally, we empirically evaluate the impact of the portfolio and its stochastic volatility joint tails on optimal portfolio choices. In particular, we examine and compare the out of sample wealth obtained optimizing the portfolio performances conditioned on the joint tails of the proposed stochastic volatility model. Keywords Markov chain · Sharpe ratio · Stochastic dominance · Stochastic volatility
1 Introduction There is a general consensus (Engle 1982; Bollerslev 1986) that the variance of the financial asset returns is time variant and a great amount of efforts are directing to realize mathematical models which, by choosing the variance dynamics as the model corner-stone, should be effectively able to model financial prices. Surely the GARCH model is a reference instrument to study the volatility dynamics, and among its advantages there is its high flexibility to be suitable to capture the most important features of the financial variables. As Glosten
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Rosella Giacometti [email protected] Marco Bonomelli [email protected] Sergio Ortobelli Lozza [email protected]
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Department SAEMQ, University of Bergamo, Via dei Caniana, 2, 24127 Bergamo, Italy
123
Annals of Operations Research
et al. (1993), and Nelson (1991) explain many GARCH type models and in particular the GARCH(1,1) model can be represented as a bivariate Markovian system (i.e., the state of the process is uniquely represented by price and variance states). This feature allows to approximate GARCH type models by a discrete Markov chain. The Markovian and semiMarkovian models have been used in different fields of the financial literature typically in option pricing and credit risk (see, among others, Duan and Simonato 2001; D’Amico and Di Biase 2009; D’Amico et al. 2009, 2010), and in portfolio theory (see Angelelli and Ortobelli Lozza 2009; Iaquinta and Ortobelli Lozza 2008). Elliott and Siu (2010) and Canakoglu and Ozekici (2008) model the economic phases as a discrete Markov chain. Duan and Simonato (2001) proposed a methodology based on a Markov chain process to approximate the asset price distribution
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