Multifractal Spectral Analysis of the 1987 Stock Market Crash

The multifractal model of asset returns captures the volatility persis-tence of many financial time series. Its multifractal spectrum computed from wavelet modulus maxima lines provides the spectrum of irregular-ities in the distribution of market returns over time and thereby of the kind of uncertainty or "randomness " in a particular market. Changes in this multifractal spectrum display distinctive patterns around substantial market crashes or “drawdowns. " In other words, the kinds of singularities and the kinds of irregularity change in a distinct fashion in the periods immediately preceding and following major market drawdowns. This pa-per focuses on these identifiable multifractal spectral patterns surround-ing the stock market crash of 1987. Although we are not able to find a uniquely identifiable irregularity pattern within the same market pre-ceding different crashes at different times, we do find the same uniquely identifiable pattern in various stock markets experiencing the same crash at the same time. Moreover, our results suggest that all such crashes are preceded by a gradual increase in the weighted average of the values of the Lipschitz regularity exponents, under low dispersion of the multifrac-tal spectrum. At a crash, this weighted average irregularity value drops to a much lower value, while the dispersion of the spectrum of Lipschitz exponents jumps up to a much higher level after the crash. Our most striking result, however, is that the multifractal spectra of stock market returns are not stationary. Also, while the stock market returns show a global Hurst exponent of slight persistence 0.5 < H < 0.7, these spectra tend to be skwed towards anti-persistence in the returns. 1