The most popular of these positions are covered calls and protective puts. This chapter also concerns what can be said of the valuation and replication of standard calls and puts relying only on the assumptions of no riskless arbitrage opportunities and perfect markets. In particular, until Section 3.4, no assumptions are made concerning the nature of the uncertain movements of the underlying asset price or the riskless return. Our most important result in this general setting is the put-call parity relation joining the values of otherwise identical European puts and calls. Other results take the form of inequalities governing the relation of the current call value with its underlying asset price, the price of another call, and the prices of two other calls.
Similar results are available for puts.
We can use this analysis to make general predictions about the conditions under which the early exercise of American options is desirable. The key insight behind modern option pricing theory is the third fundamental theorem of financial economics. Applied to options, it implies that it is possible to replicate the payoff of an option by managing a self-financing dynamic trading strategy utilizing only the underlying asset and cash. Valuation and replication are inextricably linked since the current value of an option is equal to the current cost of constructing its replicating portfolio. For each option position, we can deduce general qualitative features of this portfolio.