Summary

In this chapter we introduce some basic definitions from finance and investigate the problem of pricing financial instruments in the context of a very crude model. We suppose the market to be observed at just two times: zero, when we enter into a financial contract; and T, the time at which the contract expires. We further suppose that the market can only be in one of a finite number of states at time T. Although simplistic, this model reveals the importance of the central paradigm of modern finance: the idea of a perfect hedge. It is also adequate for a preliminary discussion of the notion of ‘complete market’ and its importance if we are to find a ‘fair’ price for our financial contract.

The proofs in §1.5 can safely be omitted, although we shall from time to time refer back to the statements of the results.

Some definitions from finance

Financial market instruments can be divided into two types. 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 – 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 a stock, without needing the capital to buy it outright.