Continuous financial models

We turn to discuss continuous financial models. These models in general involve infinite dimensional spaces and are more complex. Our focus here is to use relatively simple models to illustrate the convex duality between the price of a contingent claim and the process of cash borrowed in delta hedging. This reveals the root of the convexity in contingent claims. Interestingly, when hedging with a contingent claim instead of the underlying, a similar duality in the sense of generalized Fenchel conjugate holds. Correspondingly, this generalized duality leads to the generalized convexity of the contingent claims with many interesting applications. Much of the material presented in this chapter appear here for the first time.