An exotic instrument is basically any contract more complicated than a simple put or call option.
For a slightly fictionalized version of why these might be useful in the real world, let's say a Finnish life insurer is selling life insurance in Hungary. So, they're doing their accounting (the numerare) in Markka (FIM), taking in payments in Forints (HUF), and exposed to changes in Hungarian mortality rates. So, for the next 30 years, this insurance company wants some secondary protection against adverse shifts in the Hungarian mortality curve, but to make that insurance less expensive, they only want protection if those losses in HUF, converted to FIM exceed 100 million FIM.
So, this insurance company goes to the structured products desk of large multi-national firm and gets a very bespoke option created. The multi-national sells this single contract to the insurance company, and hedges the individual FX components using FX options and the actuarial risk with actuarial derivatives. There's basically no market for this stuff, so the large multi-national basically commits to hedging out the other side of this contract for the next 30 years.
I've worked with optimizing risk modeling for exotics, because it turns out that for some of the more complex exotics, the daily computational costs get very expensive for basically re-calculating a first-order multidimensional Taylor approximation (the risk components) of these contracts. Every day, we then take these risk calculations and buy/sell a bunch of simple options to bring the first-order partial derivatives (risk components) back close to zero.
I don't know much about the structured product particulars or the details of the hedging. I'm not sure if the actuarial derivatives are mostly options or futures. I think they're mostly over-the-counter products, and rather large contract sizes. So, I think trading actuarial derivatives involves passing some certifications, getting a few million dollars, and cold-calling a bunch of large financial institutions to see what they currently have on offer, and convincing them that you're not going to go bankrupt in the next 30 years.
But, I don't really know. I know a bit about making the computers go fast, and I talked with the structurers enough to know very generally what was going on.
Funny story poking around exotic financial models: a big part of the collateral a shipping company can offer is its ships. I saw "cape size" show up in a model for the value of a ship, and imagined ships dressed up like Superman or the Green Lantern. It turns out that for the same carrying capacity, ships that are dimensioned to fit through the Panama and Suez canals are worth significantly more than those that are forced to round the Cape of Good Hope or Cape Horn ("cape size").
Don't misconstrue this as investment advice, but be careful about taking the short side. If you happen to randomly be short Hungarian lives, and genocide breaks out, and you make 3 billion dollars out of pure happenstance, there will be double-digit percentages of the population believing in conspiracy theories about you for hundreds of years. Very few people will want to live in the same city or even sell a hamburger to the alleged architect of the Hungarian Genocide. This isn't investment advice; it's life advice. It's possible to be 100% right, for the wrong reason, and lose 1000%.
I should add that I'm pretty sure the actuarial derivatives pay out based on population-level official government statistics. They aren't based on the death rate of the population actually insured by one insurer, and the insurer has to hope that their insured population is similar to the population as a whole, or is at least predictably and consistently offset from the general population so that the hedges are strongly correlated with expenses.
e.g. A custom tailored interest-rate swap[1] to hedge very specific circumstances between two parties. Or if you like cool names, maybe a gigantic Bermudan swaption[2]. Or maybe two traders want to take opposite ends of bizarre 10 leg option with unlisted strike prices...
