A unified framework for robust modelling of financial markets in discrete time

We unify and establish equivalence between the pathwise and the quasi-sure approaches to robust modelling of financial markets in finite discrete time. In particular, we prove a fundamental theorem of asset pricing and a superhedging theorem which encompass the formulations of Bouchard and Nutz [12] and Burzoni et al. [13]. In bringing the two streams of literature together, we examine and compare their many different notions of arbitrage. We also clarify the relation between robust and classical ℙ-specific results. Furthermore, we prove when a superhedging property with respect to the set of martingale measures supported on a set $\Omega $ Ω of paths may be extended to a pathwise superhedging on $\Omega $ Ω without changing the superhedging price.

Arbitrage Pricing

The chapter starts with a detailed discussion of the bank account in discrete and continuous time. The Black–Scholes model is then introduced, and using the principle of no arbitrage we study the problem of pricing an arbitrary financial derivative within this model. Using the classical delta hedging approach we derive the Black–Scholes PDE for the pricing problem and using Feynman–Kač we also derive the corresponding risk neutral valuation formula and discuss the connection to martingale measures. Some concrete examples are studied in detail and the Black–Scholes formula is derived. We also discuss forward and futures contracts, and we derive the Black-76 futures option formula. We finally discuss the concepts and roles of historic and implied volatility.

The Esscher Transform and the Minimal Martingale Measure

In this chapter we present two ways to choose a unique martingale measure in an incomplete market. The first way is to use an extended version of the Esscher transform, which implies that we restrict the class of martingale measures. The second way is to use the minimal martingale measure, that is, the measure which minimizes the norm of the associated Girsanov kernel. We exemplify the two methods and discuss the economic significance.

Currency Derivatives

In this chapter we develop a theory for derivatives based on the exchange rate between two (or more) currencies. This is initially done using classical delta hedging methods, but the main part of the theory is developed using martingale methods. We discuss the foreign and the domestic martingale measures and the relations between these measures, and in particular we show that the likelihood ratio between the measures equals the ratio between the foreign and the domestic stochastic discount factors. Option pricing formulas are also derived, and we discuss the Siegel paradox.

On the support of extremal martingale measures with given marginals: the countable case

We investigate the supports of extremal martingale measures with prespecified marginals in a two-period setting. First, we establish in full generality the equivalence between the extremality of a given measure Q and the denseness in $L^1(Q)$ of a suitable linear subspace, which can be seen in a financial context as the set of all semistatic trading strategies. Moreover, when the supports of both marginals are countable, we focus on the slightly stronger notion of weak exact predictable representation property (WEP) and provide two combinatorial sufficient conditions, called the ‘2-link property’ and ‘full erasability’, on how the points in the supports are linked to each other for granting extremality. When the support of the first marginal is a finite set, we give a necessary and sufficient condition for the WEP to hold in terms of the new concepts of 2-net and deadlock. Finally, we study the relation between cycles and extremality.

A characterization of the set of local martingale measures

We generalize the results of [1] to continuous time case by stating necessary and sufficient conditions on a set of probability measures to be the set of local martingale measures for a vector valued, locally bounded and adapted process.