A contingent contract in finance generally refers to a contract in which the amount of money one agent pays to another in the future will differ depending on the realization of some future event. A simple example of a contingent contract would be a contract which gave the bearer of that contract nothing if it rains next Thursday but one dollar if it does not rain. These kinds of contracts are more common than you might believe at first glance. A farmer's crop may depend rather heavily or whether or not it rains. If it does rain, he has a healthy crop which he can sell on the market. If it does not rain the crop will be ruined and the farmer will having nothing. The farmer can minimize this risk by buying many of these contingent contracts. If the farmer buys the contingent contracts and it does not rain, his crop will be worthless but he will get $1 for each contract he holds. Of course, if it does rain his crop will be valuable, but he'll also have paid money for contingent contracts which are now worthless. If the farmer buys enough of the contracts, he can insure that he receives the same amount of money no matter what the weather does. This sort of risk-minimization is known as hedging and is used quite frequently, particularly in finance.
From an informational standpoint, contingent contracts (also known as "contingent claims") are very nice because they tell how likely the market thinks some event will happen. Suppose our $1 if it doesn't rain and $0 if it does rain contingent contract is selling for 70 cents. This implies that the market believes that there is a 70% chance it will not rain and a 30% chance that it will. This is because we believe that 70% of the time the contingent contract will be worth $1 and 30% of the time the contingent contract will be worth nothing. So on average, we'd expect the contingent contract to be worth 70 cents. Now suppose a number of people in the "rain" market got a new piece of information (say satellite photos) and now believed that the chance of it not raining on Thursday is now 90%. This would cause them to value the contract at 90 cents, but the price is currently at 70 cents. So they would buy these contingent contracts as they'd expect to make 20 cents on average. The increase in demand for the contracts will cause the price to rise and if enough people in the market believed the chance of a lack of precipitation was 90%, we'd expect to see the value of the contingent contract to rise to 90 cents.
To see a good example of price changes and contingent contracts, we'll look at the world of baseball.
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