forget money, trading, or any economic approach; he lacks the money or influence to handle those. No, the most effective way is far simpler, he should take his cue from the squirrels.
A squirrel has drastically hurt the stock exchange simply by shorting out the electronics by accident. He could perform far more effective forms of sabotage relatively easily, cutting power to the NY stock exchange for a week or two would be enough to have a small, but real, impact on the exchange; and could be done with simple research into electronics and wire cutters until they get smart enough to guard against your attacks. Even then, you just need to cut the power further away from where their guarding. If you can manage to be not attract attention you can keep power away from them for quite awhile; most of our electronic network isn't really protected or that hard to get to; we just presume no one is malicious like this.
And that's only to start. The best method to kill the stock exchange is fear; any type of fear. Look at what 9/11 did to our economy; that damage wasn't due to lost of the trade buildings; it was due to general fear. Terrorism is the most effective attack.
This is the part where I start expanding on the idea I started writing to see how far I can take the concept in most answers. Unfortunately...as I sit here and start expanding on it I realize how terrifying the most 'optimal' ways of destroying the stock market via fear are. We will not be discussing the best options I think, I prefer not to write a terror-how-to book.
However, lets try one less-horrible example, assuming your character isn't so insane as to do anything too horrible to others (though, really destroying the stock market is Already a horrible thing that will cause massive suffering, even if indirectly). All he needs to do is get people afraid to either work for his old company, or buy his old company's product. Imagine he slips some mild poison into cans of the "Dr MountColaDew" soda the company sells, nothing lethal; but enough to cause sickness, diareha, and general unpleasantness. The media attention would cause even a small number of causes of mild sickness to be blown way out of proportion and cause everyone to avoid the companies product, doing a good amount of harm. This tactic has been used on small scale by crazy people in the past, and has been shown to greatly harm a companies sales figures.
Incidentally, this is all written on the presumption that this character is a sociopath, because frankly he must be. A stock market crash, or even a single company crash, is going to hurt a massive number of people. For starters more then one suicide will likely occur if even one company suddenly crashes. One company going down will bring down the market as a whole, and these sort of things tend to cascade to risk bringing on a full on recession. Weak economy means the government gets less taxes, and thus can't afford to provide social support (food stamps and the like) which leave many poor individuals hungry, sick, or otherwise suffering from lack of government support. The most obvious sociopath effect of destroying a company is that it would force the company to let go employees it can no longer afford to pay, leaving them broke and without jobs. Thus his vengeance for being fired causes many thousands (at least) to also be fired. So being fired is so horrible it requires extreme measures of revenge if done to me, and yet it's perfectly okay to cause it an magnitudes larger scale to others just for my petty revenge. That sort of hypocritical self-centered sort of action sociopaths are known for. Taking down a company due to being fired is the most extreme of disproportionate retribution, in fact it's worse then that since it mostly hurts innocent third parties instead of those responsible for your being fired.
You will need your character to either be unbelievably naive to not realize the repercussions of his actions (at which point it's hard to believe he is competent enough to come up with and successfully pull them off), or a complete sociopath. Plus, only a sociopaths obsession would cause someone to forsake their own livelihood (selling all their possessions and going broke to fund their revenge) just to harm others. It takes a certain type of evil to be willing to destroy yourself just to harm another.