On pathwise functional Itô calculus and its applications to mathematical finance

In recent years, pathwise Itô calculus has been particularly popular in mathematical finance and economics. This is due to the fact that the results derived with the help of the pathwise Itô calculus are robust with respect to model risk that might stem from a misspecification of probabilistic dynamics. In this sense, there is also a close link to robust statistics. The only assumption on the underlying paths is that they admit the quadratic variation in the sense of Föllmer. In this thesis, we will be particularly interested in the functional extension of Föllmer’s pathwise calculus, since it is natural to assume that randomness impacts the current situation not simply by influencing the current state of the process but through its entire past evolution. In a first part, we derive the associativity property of the pathwise Itô integral in a functional setting for continuous integrators. This allows us to establish existence and uniqueness results for a class of linear functional Itô differential equations. With this at hand, we turn to financial applications. First, we use functional pathwise Itô calculus to prove a strictly pathwise version of the master formula in Fernholz’ stochastic portfolio theory. This adds a new case study in which continuous-time trading strategies can be constructed in a probability-free manner by means of pathwise Itô calculus. Moreover, the portfolio generating function may depend on the entire history of the asset trajectories and on an additional continuous trajectory of bounded variation. Our results are illustrated by several examples and shown to work on empirical market data. Second, we consider a strictly pathwise setting for Delta hedging exotic options, based on Föllmer’s pathwise Itô calculus. Price trajectories are d-dimensional continuous functions whose pathwise quadratic variations and covariations are determined by a given local volatility matrix. The existence of Delta hedging strategies in this pathwise setting is established via existence results for recursive schemes of parabolic Cauchy problems and via the existence of functional Cauchy problems on path space. Our main results establish the nonexistence of pathwise arbitrage opportunities in classes of strategies containing these Delta hedging strategies and under relatively mild conditions on the local volatility matrix