An Introduction to Financial Markets

Chapter Thirteen Option Pricing: Complete Markets

We have introduced vanilla options in Section 1.2.6.3 and, in Section 2.3.4, we have seen how the no-arbitrage principle can be used to find an option price in the simple single-step binomial setting. The approach relies on a replication argument: It is possible to replicate the option payoff in both states of the world by a portfolio consisting of two primary assets, since the market is complete. We have also seen that lack of arbitrage is related to the existence of a risk-neutral probability measure, and that this measure is unique in a complete market. Hence, we may use that measure for pricing purposes, and risk aversion does not play any role, since any payoff may be replicated exactly, state by state. When markets are not complete, the pricing measure is not unique anymore, and risk aversion cannot be disregarded. From a more practical viewpoint, market completeness is a somewhat paradoxical assumption. If markets were complete, options would be redundant assets, and it is hard to see why an option market should exist in the first place. Another paradoxical consequence of market completeness would be that we could always get rid of any risk, since we could synthesize any payoff we wish. Clearly, this is too good to be true and, in fact, markets are not complete. In real life, we have to cope with issues related to incompleteness and residual risk. Nevertheless, a reasonably deep understanding of the simpler case of complete ...