Title:

Robust pricing and hedging beyond one marginal

The robust pricing and hedging approach in Mathematical Finance, pioneered by Hobson (1998), makes statements about nontraded derivative contracts by imposing very little assumptions about the underlying financial model but directly using information contained in traded options, typically call or put option prices. These prices are informative about marginal distributions of the asset. Mathematically, the theory of Skorokhod embeddings provides one possibility to approach robust problems. In this thesis we consider mostly robust pricing and hedging problems of Lookback options (options written on the terminal maximum of an asset) and Convex Vanilla Options (options written on the terminal value of an asset) and extend the analysis which is predominately found in the literature on robust problems by two features: Firstly, options with multiple maturities are available for trading (mathematically this corresponds to multiple marginal constraints) and secondly, restrictions on the total realized variance of asset trajectories are imposed. Probabilistically, in both cases, we develop new optimal solutions to the Skorokhod embedding problem. More precisely, in Part I we start by constructing an iterated AzemaYor type embedding (a solution to the nmarginal Skorokhod embedding problem, see Chapter 2). Subsequently, its implications are presented in Chapter 3. From a Mathematical Finance perspective we obtain explicitly the optimal superhedging strategy for Barrier/Lookback options. From a probability theory perspective, we find the maximum maximum of a martingale which is constrained by finitely many intermediate marginal laws. Further, as a byproduct, we discover a new class of martingale inequalities for the terminal maximum of a cadlag submartingale, see Chapter 4. These inequalities enable us to rederive the sharp versions of Doob's inequalities. In Chapter 5 a different problem is solved. Motivated by the fact that in some markets both Vanilla and Barrier options with multiple maturities are traded, we characterize the set of market models in this case. In Part II we incorporate the restriction that the total realized variance of every asset trajectory is bounded by a constant. This has been previously suggested by Mykland (2000). We further assume that finitely many put options with one fixed maturity are traded. After introducing the general framework in Chapter 6, we analyse the associated robust pricing and hedging problem for convex Vanilla and Lookback options in Chapters 7 and 8. Robust pricing is achieved through construction of appropriate Root solutions to the Skorokhod embedding problem. Robust hedging and pathwise duality are obtained by a careful development of dynamic pathwise superhedging strategies. Further, we characterize existence of market models with a suitable notion of arbitrage.
