Stochastic Processes and Financial Mathematics
(part one)
1.4 Modelling discussion
Our proof that the arbitrage free value for
Happily, this is precisely the type of situation where mathematics can help. What is needed is a systematic way of calculating arbitrage free prices, that always works. In order to find one, we’ll need to first develop several key concepts from probability theory. More precisely:
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• We need to be able to express the idea that, as time passes, we gain information.
For example, in our market, at time
we don’t know how the stock price will change. But at time , it has already changed and we do know. Of course, real markets have more than one time step, and we only gain information gradually. -
• We need stochastic processes.
Our stock price process
, with its two branches, is too simplistic. Real stock prices have a ‘jagged’ appearance (see Figure 1.1). What we need is a library of useful stochastic processes, to build models out of.
In fact, these two requirements are common to almost all stochastic modelling. For this reason, we’ll develop our probabilistic tools based on a wide range of examples. We’ll return to study (exclusively) financial markets in Chapter 5, and again in Chapters 15-19.