Financial market instruments can be divided into two distinct species. There are the ‘underlying’ stocks: shares, bonds, commodities, foreign currencies; and their ‘derivatives’, claims that promise some payment or delivery in the future contingent on an underlying stock's behaviour. Derivatives can reduce risk — by enabling a player to fix a price for a future transaction now, for example — or they can magnify it. A costless contract agreeing to pay off the difference between a stock and some agreed future price lets both sides ride the risk inherent in owning stock without needing the capital to buy it outright.

In form, one species depends on the other — without the underlying (stock) there could be no future claims — but the connection between the two is sufficiently complex and uncertain for both to trade fiercely in the same market. The apparently random nature of stocks filters through to the claims — they appear random too.

Yet mathematicians have known for a while that to be random is not necessarily to be without some internal structure — put crudely, things are often random in non-random ways. The study of probability and expectation shows one way of coping with randomness and this book will build on probabilistic foundations to find the strongest possible links between claims and their random underlying stocks. The current state of truth is, however, unfortunately complex and there are many false trails through this zoo of the new. Of these, one is particularly tempting.