Financial bubbles analysis with a cross-sectional estimator

We highlight a very simple statistical tool for the analysis of financial bubbles, which has already been studied in [1]. We provide extensive empirical tests of this statistical tool and investigate analytically its link with stocks correlation structure. Introduction Forecasting the burst of financial bubbles would be incredibly useful for many players in stock exchanges, including regulators, portfolio managers and investment banks. Fundamental indicators relying on economic analysis can be monitored. But is it possible to find some statistical regularity in market crashes? Several authors, including [2, 3, 4], have already attempted to answer this question. In this paper, we focus on a very simple statistical tool, first introduced in [1], and study analytically its link with stocks correlation structure. This approach is similar to the one studied in [2, 4] although different through the statistical object under consideration. 1 A spatial survival function In [1], an unusual and interesting statistical tool is introduced in order to study market crashes. Given N stocks on a market place, a reference date tref and the current date t, we set SN (z) = 1 N N ∑ i=1 1{Xi(tref ,t)>z} where Xi(tref , t) := Si(t) Si(tref ) is the performance of stock i over [tref , t]. In the following, we shall leave the time argument for notational simplicity. Intuitively, SN (z) can be seen as the proportion of stocks displaying a greater performance than z, i.e. the survival function of stocks on day t. From this point of view, it is a measure of stocks dispersion: a slow decreasing SN (z) indicates broadly distributed performances, thus reflecting an important dispersion. Since tref is supposed to be as close as possible to the onset of the bubble, Xi(tref , t) might be of the order of monthly or yearly returns. Short notes on similar statistical objects can be found in [5, 6]. This is very different from looking at daily returns as in [2, 4] and might be more relevant for bubble detection since it often takes long time for a bubble to build up and for bubbling stocks to disperse. Statistical properties of SN (z) are interestingly robust. The main features of this quantity are: • for z ∼ +∞, SN (z) ∼ z ; • the variance of the Xi’s increases dramatically before crashes. These two features are robust with respect to the choice of the arbitrary reference date tref and are valid over several financial crashes. We illustrate these two facts on figures 1 and 2. We use daily close prices of three different sets of stocks: the stocks composing the Australian Stock Exchange All Ordinaries index (AORD, 500 stocks); the stocks composing the New York Stock Exchange Composite index (NYA, 1800 stocks); the stocks composing the Shanghai Composite index (SSE, 900 stocks). We choose the first trading day of 2003 as our date tref . As for the first fact, figure 1 shows three examples of distributions of Xi’s at random dates. It appears that the power-law tail is indeed a good fit as the normalized prices grow. As for the second fact, we plot on figure 2 the timeseries of BNP Paribas Chair of Quantitative Finance, Ecole Centrale Paris, MAS Laboratory. Natixis, Equity Derivatives and Arbitrage. E-mail: [email protected]. The authors would like to thank the members of Natixis’ equity derivatives quantitative R&D team for fruitful discussions. supposed to be close to the onset of a financial bubble 1 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 -3 10 -2 10 -1 10