Stochastic Processes and Financial Mathematics
(part one)
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1.4 Modelling discussion
Our proof that the arbitrage free value for \(K\) was \(s(1+r)\) is mathematically correct, but it is not ideal. We relied on discovering specific trading strategies that (eventually) resulted in arbitrage. If we tried to
price a more complicated contract, we might fail to find the right trading strategies and hence fail to find the right prices. In real markets, trading complicated contracts is common.
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 \(t=0\) we don’t know how the stock price will change. But at time \(t=1\), it has already changed and we do know. Of course, real markets have more than one time step, and
we only gain information gradually.
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• We need stochastic processes.
Our stock price process \(S_0\mapsto S_1\), 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.