Crash Discovery in Stock and Option Markets

Abstract

This article investigates, both theoretically and empirically, the economics of stock market crashes. Using more than 100 years of daily data on the DJIA (and shorter series on NASDAQ, IBM, and Caterpillar), we first document empirically that (a) the probability of a daily stock market decline in excess of 5% is non-negligible (about 0.25%); (b) stock market crashes are not only relatively more likely to occur than rallies (higher crash arrival rates), but substantially more brutal; (c) the pre-1945 crash valuation measures depart radically from the post-1945 counterpart with the left tail decaying to zero much slower than the right tail; and (d) the motion of large percentage price declines and rises conforms closely with the characteristics of the Frechet distribution (asymptotically). To realistically model the empirical properties of crashes and extremes, we propose a family of Markov processes for which the density of the maximum percentage price drop can also be derived. The objective probability of the crash is found to be related, in an intuitive manner, to higher order moments of the return distribution. Examination of this model suggests that the implied probabilities are not at odds with the empirical counterparts. To assess the implications of our findings for real-life investment analysis, we generated buy/sell signals contingent on the crash probability. Investment trading rules relying on the model's prediction outperform traditional ones (e.g., buy and hold). Our implementation methods are sufficiently versatile to discover crash/rally information embedded in option markets. Exploiting more than 17,000 out-of-money option prices, the framework quantifies three dimensions of crash discovery (i) time-variations in Arrow-Debreu security price on the extreme, (ii) the structure of jump-fear levels, and (iii) the term structure of forward jump-risks. This paper provides a unified treatment for discovering crashes in stock and option markets.

JEL Classification: G10, G12, G13