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What exactly is a “derivative”?
A derivative is a financial instrument of a special kind, the kind “whose price depends on, or is derived from, another asset”.  This definition is from John Hull, Options, Futures and Other Derivatives – a book definitely worth to own if you are curious about this, you can easily find old copies for a few dollars. The first point is that a derivative is a financial instrument, like credits, or insurances, the second point is that its price depends closely from the price of something else, the mentioned asset. In most cases derivatives can be understood as financial insurances against some risk bound to the asset. In the sequel I give a small list of derivatives and highlight the assets and the risk they can be bound to.  And first, let me point out that the definition is (marginally) wrong because some derivatives depend on things which are not assets, nor do they have a price, like temperature, sunlight, or even your own life in the case of mortgages.  But before going in this list, let me go through the remaining points of your question. What is the basic idea and concept behind a derivative? As already noted, in most cases, a derivative can be understood as a financial insurance compensating from a risk of some sort.  In a classical insurance contract, one party of the contract is an insurance company, but in the broader case of a derivative, that counterparty can be pretty anything: an insurance, a bank, a government, a large company, and most probably market makers. How is it really used, and how does this deviate from the first point? Briefly, how does is it affecting people, and how is it causing problems? An important point with derivatives is that it can be arbitrarily complicated to compute their prices.  Actually what is hidden in the attempt of giving a definition for derivatives, is that they are products whose price Y is a measurable function of one or several random variables X_1, X_2, … X_n on which we can use the theory of arbitrage pricing to get hints on the actual price Y of the asset – this is what the depends on means in technical terms.  In the most favorable case, we obtain an easy formula linking Y to the X_is which tells us what is the price of our financial instrument. But in practice, it can be very difficult, if at all possible, to determine a price for derivatives. This has two implications: Persons possessing sophisticated techniques to compute the price of derivatives have a strategic advantage on derivatives market, in comparison to less advanced actors on the market. Organisation owning assets they cannot price cannot compute their bilan anymore, so that they cannot know for sure their financial situation. They are somehow playing roulette. But wait, if derivatives are insurances they should help to mitigate some financial risk, which precisely means that they should help their owners to more accurately see their financial situation! How is this not a contradiction? Some persons with sophisticated techniques to compute the price of derivatives are actually selling complicated derivatives to less knowledgeable persons.  For instance, many communes in France and Germany have contracted credits whose reimbursements have a fixed interest part, like in a classical credit, and a variable interest part whose rate is computed against a complicated formula involving the value of the Swiss frank at each quarter starting from the inception of the credit.  (So, for a 25 years running credit of theis type, the price Y of the credit at its inception depends on 100 Xs, which are the uncertain prices for the Swiss frank each quarter of the 25 next years.)  Some of these communes can be quite small, with 5.000 inhabitants, and needless to say, do not have the required expertise to analyse the risks bound to such instruments, which in that special case led the court call the credit a swindling and to cancel the credit.  But what chain of events leads a 5.000 inhabitants city in France to own a credit whose reimbursements depends on the Swiss frank?  After the credit crunch in 2007 and the fall of Lehman Brothers in 2008, it has begun to be very hard to organise funding, which basically means to conclude credits running long in time on large amounts of money.  So, the municipality needs a 25 years credit of 10.000.000 EUROS and goes to its communal bank.  The communal bank has hundreds or thousands of municipalities looking for credits and needs itself a financing.  So the communal bank goes to one of the five largest financial institutions in the world, which insists on selling a huge credit whose reimbursements have a variable part depending on hundred of values the Swiss frank will have in the 25 next years. Since the the big bank has better computation techniques than the small bank it makes a big profit.  Since the small bank has no idea, how to compute the correct price of the credit it bought, it cuts this in pieces and sell it in the same form to the various communes it works with.  If we were to attribute this kind of intentions to the largest five banks, we could ask about the possibility that they designed the credit to take advantage of the primitive evaluation methods of the small bank. We could also ask if they organised a cartel to force communal banks to buy their bermudean snowballs. And we could also ask, if they are so influent that they eventually can manipulate the Swiss frank to secure an even higher profit.  But I will not go into this.  To the best of my understanding, the subprime crisis is a play along the same plot, with different actors, but I know this latter subject only by what I could read in French newspapers. So much for the “How is it causing problems?” part. What is some of the terminology in relation to derivatives (and   there meanings of course)? Answering this question is basically the purpose of the 7 first chapters of the book by Hull, along with deriving some important mathematical principles.  And I will not copy these seven chapters here! How would someone get started dealing in derivatives (I'm playing a     realistic stock market simulation, so it doesn't matter if your     answer to this costs me money)? If you ask the question, I understand that you are not a professional, so that your are actually trying to become the one that has money and zero knowledge in the play I outlined above. I would recommand not doing this.  That said, if you have a good mathematical background and can program well, once you are confindent with the books of Hull and Joshi, you can have fun implementing various market models and implementing trading strategies. Once you are confident with this, you can also read the articles on quantitative finance on arXiv.org. And once you are done with this, you can decide for yourself if you want to play the same market as the guys writing these articles. (And yes, even for the simplest options, they have better models than you have and will systematically outperform you in the long run, even if some random successes will give you the feeling that you do well and could do better.) (indeed, I've made it a personal goal to somehow lose every last cent of my money) You know your weapons! :) Two parties agree today on a price for one to deliver a commodity to the other at some future instant.  This is a classical future contract, it can be modified in every imaginable way, usually by embedding options.  For instance one party could have the option to choose between different delivery points or delivery days. Two parties write today a contract allowing the one party to buy at some future time a commodity to the the second party.  The price is written today, as part of the contract.  (There is the corresponding option entitling the owner to sell something.)  Unlike the future contract, only one party can be obliged to do something, the other jas a right but no obligation. If you buy and option, your are buying some sort of insurance against a change of price on some asset. This is the most familiar to anybody.  Credits can come in many different flavours, especially the formula to compute interests, or also embed options.  Common options are early settlement options or restructuration options. While this is not completely inutitive, the credit works like an insurance.  This is most easily understood from the side of the organisation lending the money, that speculates that the ratio of creanciers going bankrupt will be low enough for her to make profit, just like a fire insurance company speculates that the ratio of fire accidents will be low enough for her to make a profit. This is like a mortgage on a financial institution.  Two parties agree that one will recive an upfront today and give a compensation to the second one if some third party defaults. Here this is an explicit insurance against the unfortuante event, where a creancier goes bankrupt. One finds here more or less standard options on electricity. But electricity have delicious particularities as it can practically not be stored, and fallout is also (usually) avoided. As for classical options, these are insurances against price moves. A swap is like two complementary credits on the same amount of money, so that it ends up in the two parties not actually exchanging the credit nominal and only paying interest one to the other — which makes only sense if these interests are computed with different formulas. Typical example are fixed rate vs. EURIBOR on some given maturity, which we interpret as an insurance against fluctuations of the EURIBOR, or a fixed rate vs. the exchange ratio between two currencies, which we interpret as an insurance against the two currencies decorrelating. Swaps are the richest and the most generic category of financial derivatives.  The off-the-counter market features very imaginative, very customised insurance products. The most basic form is the insurance against drought, but you can image different dangers, and once you have it you can put it in options, in a swap, etc.  For instance, a restaurant with a terrasse could enter in a weather insurance, paying each year a fixed amount of money and becoming in return an amount of money based on the amount of rainy day in a year. Actually, this list is virtually without limits!