U.S. Stock Market Crash Risk, 1926-2006 [report]

David Bates
2009 unpublished
This paper applies the Bates (RFS, 2006) methodology to the problem of estimating and filtering timechanged Lévy processes, using daily data on U.S. stock market excess returns over 1926-2006. In contrast to density-based filtration approaches, the methodology recursively updates the associated conditional characteristic functions of the latent variables. The paper examines how well time-changed Lévy specifications capture stochastic volatility, the "leverage" effect, and the substantial
more » ... s occasionally observed in stock market returns. The paper also finds that the autocorrelation of stock market excess returns varies substantially over time, necessitating an additional latent variable when analyzing historical data on stock market returns. The paper explores option pricing implications, and compares the results with observed prices of options on S&P 500 futures. What is the risk of stock market crashes? Answering this question is complicated by two features of stock market returns: the fact that conditional volatility evolves over time, and the fat-tailed nature of daily stock market returns. Each issue affects the other. What we identify as outliers depends upon that day's assessment of conditional volatility. Conversely, our estimates of current volatility from past returns can be disproportionately affected by outliers such as the 1987 crash. In standard GARCH specifications, for instance, a 10% daily change in the stock market has 100 times the impact on conditional variance revisions of a more typical 1% move. This paper explores whether recently proposed continuous-time specifications of timechanged Lévy processes are a useful way to capture the twin properties of stochastic volatility and fat tails. The use of Lévy processes to capture outliers dates back at least to Mandelbrot's (1963) use of the stable Paretian distribution, and there have been many specifications proposed; e.g., Merton's (1976) jump-diffusion, Madan and Seneta's (1990) variance gamma; Eberlein, Keller and Prause's (1998) hyperbolic Lévy; and Carr, Madan, Geman and Yor's (2002) CGMY process. As all of these distributions assume identical and independently distributed returns, however, they are unable to capture stochastic volatility. More recently, Carr, Geman, Madan and Yor (2003) and Carr and Wu (2004) have proposed combining Lévy processes with a subordinated time process. The idea of randomizing time dates back to at least to Clark (1973). Its appeal in conjunction with Lévy processes reflects the increasing focus in finance -especially in option pricing -on representing probability distributions by their associated characteristic functions. Lévy processes have log characteristic functions that are linear in time. If the time randomization depends on underlying variables that have an analytic conditional characteristic function, the resulting conditional characteristic function of time-changed Lévy processes is also analytic. Conditional probability densities, distributions, and option prices can then be numerically computed by Fourier inversion of simple functional transforms of this characteristic function. Thus far, empirical research on the relevance of time-changed Lévy processes for stock market returns has largely been limited to the special cases of time-changed versions of Brownian Carr, Peter and Liuren Wu (
doi:10.3386/w14913 fatcat:6h7gu4okqjaprcr64k7pcx6wve