Don't worry I understand mathematics: ito's calc, martingales, etc. I am just curious what options I should test, and from what indices. Is there stuff I can test from the 2008 crash to measure their effects, or from COVID-19. Or data preloaded into Matlab. What type of options would be best to model their behavior using a Merton jump-diffusion?
The bellwether Indices for testing, are NASDAQ, Technology sector, S & P 500 Big 500 capital weighted Stocks, Russell 2000, MID sector stocks and some small stocks. It is better to use the data fro their relative, ETF's eg. QQQ, SPY, IWM. The Dow is covered by the Nasdaq and the S&P 500, it is the 20 biggest stocks on the market and is not useful. Getting historic option data costs money, it is not free like stocks, futures and commodities data. You will have to create your own Jump diffusion model in matlab to apply on the data, you can refer to Espen Haug's complete Book of Option Pricing models, he implements the formula in a concise algorithm in VBA in the book, but it should be easier in Matlab, on the disk that accompanied the 2003, & 2007 editions he did all the models in C++ as well.
Short-term power options have very high volatility and kurtosis. If you can find power data from, say, 2001... that would have jumps. However, those data are likely difficult to come by.
If you are going to model jumps, you might do better by looking into work by Todorov and Tauchen. They find jumps in both the underlier and in volatility itself. Their work will also show you data to look at with jumps; and, that would allow you to compare results.