There is some research on this area. Overall, the effect of natural disasters on stock markets depend on type of disaster, industry, and country.
For example, this paper studies 30 natural disasters, from a variety of countries (including Hurricane Katrina, in the US). It concludes:
We find that different natural disasters have different effects on stock markets and industries. Our evidence suggests that while earthquakes, hurricanes and tornadoes could negatively affect market returns several weeks after the events, other disasters such as floods, tsunamis and volcanic eruptions have limited impact on stock markets. We also find that construction and materials industry is generally positively affected by natural disasters but non-life and travel industries are likely to suffer negative effects.
This other paper has the following abstract:
This paper investigates the impact of natural disasters on the insurance sector as well as on the composite stock market in Japan and the US. GARCH models are employed to capture both wealth and risk effects of natural disasters. There are no wealth effects in the US and Japan composite stock markets, indicating that these markets can well diversify away the impact of natural disasters on stock return, but there are significant wealth effects in the US and Japan insurance sectors. While US investors in the insurance sector lose, those in Japan gain. All markets except the composite stock market in Japan face risk effects of natural disasters.
There are several more studies. For example, this one focuses solely on earthquakes, and this one studies Australia only.