استفاده از مدل بک برای بازارهای سهام: ارزش در معرض خطر و ارزیابی ریسک پرتفوی

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”, ββ obeys View the MathML sourceΓ-distribution. As predicted by the Beck model, the distribution of returns is well-fitted by qq-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 qq-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 ββ model, and empirically examine the agreement between the model and an imaginary portfolio with Dow Jones indices. It turns out that the single ββ model gives good approximation to portfolios composed of the assets with non-Gaussian and correlated returns.

Nowadays, the improvement of risk management systems becomes a crucial issue for financial institutions. The fundamental feature of risk management is an estimation of risk and control of its amount within a limited risk buffer. The assets exposed to risk, such as stocks, bonds, and loans are called risk assets. Risk assets sometimes fluctuate unpredictably but such fluctuations are mutually not independent, but correlated. The amount of risk corresponding to risk assets is measured by statistical methods such as Value-at-Risk explained below. Based on the estimated risk, financial institutions have to reserve a capital with which the loss would be compensated when the risk is realized. As a capital, or a risk buffer, is limited, control of the total amount of risk and the allocation strategy of the risk buffer between risk assets to maximize earnings become important problems. In order to handle such problems the portfolio theory offers an established framework.

In this paper, we applied the Beck model, which has been developed for turbulent flows, to stock markets. Since the Beck model is a model with fluctuating temperature, or volatility in finance, and thus consistent with heteroskedasticity observed in financial markets, the application of the Beck model to markets seems worthwhile. We investigated the adequacy of representing stock markets as the Beck model with data of S&P500 index, from the viewpoint of a relaxation time, distributions of volatility, and distributions of returns. As the result of empirical analysis, we confirmed that the time constant of volatility is of the order of months and that the inverse temperature ββ approximately obeys View the MathML sourceΓ-distribution, which supports the assumption of the Beck model. Also, returns obey qq-Gaussian distribution in broad time scales, as expected by the Beck model.