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  Issue 25 | Archive November 2015

Jumps in Equity Returns Before and During the Recent Financial Crisis: A Bayesian Analysis

A paper by Steven Kou (National University of Singapore), Cindy Yu (Iowa State University), and Haowen Zhong (Columbia University)

RMI's Director Prof. Steven Kou, who is also the Provost's Chair Professor of Mathematics at NUS, wrote a paper with his co-authors Associate Professor Cindy Yu and Dr. Haowen Zhong titled, " Jumps in Equity Index Returns Before and During the Recent Financial Crisis: A Bayesian Analysis." The paper is set to appear in Management Science in 2016.

The paper attempts to answer two questions: (i) How did jumps in equity returns change after the 2008-2009 financial crisis; in particular, were there significant changes in jump rates or in jump sizes, or both? And (ii) Can the performance of affine jump-diffusion models be improved if jump sizes are larger, i.e. jumps with tails heavier than those of the normal distribution?

Their answer to the second question is affirmative: The performance of affine jump diffusion models can be significantly improved if the jump sizes are larger, i.e. jumps have heavier tails than those of the normal distribution. More precisely, they find that a simple affine jump-diffusion model with both stochastic volatility and double-exponential jump sizes in returns fits both the S&P 500 and the Nasdaq 100 daily returns well. In fact, the model outperforms existing ones (in particular models with the variance gamma jumps, affine jump-diffusion models with normal jump sizes, and models with jumps in volatility) for the returns during the crisis, and is comparable to the returns before the crisis.

There are some intuitive explanations for this finding, related to the second question. (1) There is a drawback of using normal distribution with a negative mean to model jump sizes because such a distribution does not have a monotone decreasing density for negative jumps. For example, if the jump mean is -3%, then with the normal distribution it is more likely that one can see a -2% negative jump than a -0.5% negative jump. This lack of monotonicity may lead to a poor fitting for small jumps, which is intuitively why previously people found that Levy-type jumps fit the data better than the affine jump diffusion models with normal jump sizes. However, with the double-exponential distribution for jump sizes, the density is monotonically decreasing for negative jumps, resulting in a potentially better fit for small jumps. (2) The heavy-tail feature of the double-exponential distribution also helps fit large jumps during the crisis period. (3) More complex structure of volatilities, such as jumps in volatility, may not necessarily result in better performance in models with double-exponential jumps. This is partly because, as volatility cannot rise forever, to compensate sudden jumps in volatility the mean-reverting speed in stochastic volatility tends to be higher for models with jumps in volatility. However, higher mean-reverting speed may hinder the model capability of generating enough volatility clustering effects to match those observed in the data during the 2008 financial crisis.

For the first question, based on the best fitted model in the paper, for both the S&P 500 and the Nasdaq 100, they find that (a) during the crisis, negative-jump rates increased significantly while there is little change in the average negative-jump sizes; (b) jump rates can decrease even when there is no change in volatility. It is necessary to be very cautious about the empirical conclusions aforementioned of the changes in the jump rates and jump sizes, as there are only about 5 years (2008 to 2013) worth of data after the financial crisis of 2008. However, point (b) above tentatively suggests, from the previous findings, that jump rate is positively correlated with volatility, does not necessarily hold after the crisis of 2008.

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Published quarterly by Risk Management Institute, NUS
Editor: Shivani Nakhare (rminsr@nus.edu.sg)