We present a model for stock price and volume behaviour during market crashes. The model incorporates a market mechanism for the share exchange between buyers and sellers while taking into account their cash balances. Using an analytical approach and the Monte-Carlo technique for the simulation of the trading volume, we analyzed the dynamics of the stock price and trading volume during market crashes. The trading volume was simulated through the trading exchange process using Monte-Carlo technique. We found that trading volume is inversely proportional to the square of the stock price in the case of the sharp price declines. This result is empirically supported in price and volume data for major recent US stock bankruptcies and market crashes, including Lehman Brothers Inc, Enron, Wachovia, Washington Mutual, Citigroup, Merrill Lynch, and MF Global. The results of the analytical approach may be used for marketing analysis of the sales in the case of the shocking market conditions.
Keywords: stock price, trading volume, market crash, panic, behaviour, Monte-Carlo simulation
(ProQuest: ... denotes formulae omitted.)
Stock price modeling has an extensive empirical and theoretical research history. Existing mathematical models have mainly focused on the empirical investigation of the time-series of price returns and volatility. Two main fundamental assumptions of stock price dynamics are important. The assumption of the random walk of the return movement was suggested by Bachelier in 1900. Stock returns were considered as a series of independent "shocks", which, by the Central Limit theorem, leads to the Gaussian distribution of returns. The Efficient Markets Hypothesis (EMH) plays a key role in the assumption of the random walk, which allows to widely use the Black-Scholes paradigm for the valuation of the derivative instruments (Black & Scholes, 1973). The EMH assumes that market participants possess homogeneous information, and therefore the hypothesis can be applied in limited real-life settings. Numerous empirical studies have shown that the stock market does not follow a random walk. Lo and MacKinlay (1988) used the Dickey-Fuller test to demonstrate that the Random Walk Hypothesis is violated over short horizons. Mandelbrot (1963) demonstrated that price crashes occur much more frequently than what would be predicted by a log-Normal distribution. Mandelbrot suggested that the exchange of money as an economic interaction can be considered by analogy as the exchange of energy between gas-phase molecules. Chatterjee and Chakrabarti (2007) empirically showed that the income and wealth distribution was close to the Gibbs distribution of energy of an ideal gas. Although there are similarities between the collision theory of the ideal gas and the Random Walk price dynamics of the trading market, the interaction between agents in the trading market can deviate from the collision theory because of the properties of the self-organized market and uncertain human behaviour.
The fat tails of the return distribution were intensively investigated by Fisher and Tippett (1928) using Extreme Value Theory. The fat tails of the return distribution are usually parameterized by Generalized Pareto distribution. Obviously, Extreme Value Theory cannot describe the non-random stock behavior with negative trend of the stock price. Moreover, the application of the complicated mathematical functions, for example Pareto distribution, for fitting of the return distribution cannot explain the nature of the market panic phenomena.
The value of the trading volume is the most important indicator of the stock market, which quantifies the supply and demand intersection for the price equilibrium. The dynamics of the trading volume are widely used in the technical analysis. Blume, Easly and O'Hara (1994) investigated the application of the trading volume in the technical analysis from the ability of technical analysts to predict the liquidity and stability of the equity market. …