Application of the Beck model to stock markets: Value-at-Risk and portfolio risk assessment

We apply the Beck model, developed for turbulent systems that exhibit scaling properties, to stock markets. Our study reveals that the Beck model elucidates the properties of stock market returns and is applicable to practical use such as the Value-at-Risk estimation and the portfolio analysis. We perform empirical analysis with daily/intraday data of the S&P500 index return and find that the volatility fluctuation of real markets is well-consistent with the assumptions of the Beck model: The volatility fluctuates at a much larger time scale than the return itself and the inverse of variance, or “inverse temperature”, β obeys Γ-distribution. As predicted by the Beck model, the distribution of returns is well-fitted by q-Gaussian distribution of Tsallis statistics. The evaluation method of Value-at-Risk (VaR), one of the most significant indicators in risk management, is studied for q-Gaussian distribution. Our proposed method enables the VaR evaluation in consideration of tail risk, which is underestimated by the variance–covariance method. A framework of portfolio risk assessment under the existence of tail risk is considered. We propose a multi-asset model with a single volatility fluctuation shared by all assets, named the single β model, and empirically examine the agreement between the model and an imaginary portfolio with Dow Jones indices. It turns out that the single β model gives good approximation to portfolios composed of the assets with non-Gaussian and correlated returns.