Robust utility maximization without compactness of the model-uncertainty set
Julio Daniel Backhoff Veraguas (Humboldt Universität zu Berlin, Germany)
Joint work with JoaquĆ­n Fontbona

Wednesday June 4, 14:30-15:00 | session 5.2 | Portfolio Optimization | room CD

In this work we extend the convex analysis approach to robust utility maximization of final wealth in a financial market model, to the case where the set of uncertain measures (model uncertainty) does not belong to a compact set, which is the standing assumption in the existing literature. The motivation of this comes from considering a maximizing agent who is uncertain of the real-world measure but knows certain (linear) statistics of it, such as moments, probabilities or correlations pertaining the price process. We provide a full answer (attainability issues, min-max equality) in the complete market case, where we fix an Orlicz Space which permits to connect the properties of the uncertainty set to that of the utility function and its conjugate. Further we prove that the same approach never works beyond the complete market case, as the mentioned Orlicz Space has to be replaced by a Modular Space, which is not rich enough in the incomplete case. In the motivating set-up of linear statistics we apply general results of entropy minimization to give a new representation of the worst-case measure within the uncertainty set.

Financial bubbles in incomplete markets
Martin Schweizer (ETH Zurich, Switzerland)
Joint work with Martin Herdegen

Wednesday June 4, 15:00-15:30 | session 5.2 | Portfolio Optimization | room CD

The presence of bubbles in financial markets raises many important and nontrivial questions. How can one model a bubble? How can one recognise a bubble? What is the connection between bubbles and arbitrage? How do bubbles affect valuation formulas for options? What happens to put-call parity? How are bubbles connected to strict local martingales? Especially for incomplete markets, most of these questions do not seem to have found clear answers so far.
Our goal is to first develop a notion of financial markets with bubbles that is motivated from economic considerations and works equally well for complete and incomplete markets. We also want concepts that will not change if we switch from one valuation measure ('the market measure') to another. We therefore use a numeraire-independent framework for our financial market, and we say (in the spirit of Robert C. Merton) that the market contains a bubble if there is some simple buy-and-hold strategy in the underlying assets that can be improved without risk if we trade dynamically over time instead of only statically. We then give dual characterisations of such bubble markets, and we show in particular that they automatically have a close connection to strict local martingales. We explain how this approach is related to existing work on bubbles, and we then continue (in a second part) to explore its consequences for economically consistent valuation rules.

Economically consistent valuation for incomplete markets with bubbles
Martin Herdegen (ETH Zürich, Switzerland)
Joint work with Martin Schweizer

Wednesday June 4, 15:30-16:00 | session 5.2 | Portfolio Optimization | room CD

Building on a framework for modelling financial bubbles in incomplete markets (presented by my co-author in another talk), we propose a valuation rule for incomplete financial markets with bubbles. This rule only depends on primary properties of the market, which have a direct economic interpretation, and not on dual objects like equivalent local martingale measures. The main idea is that in order to consistently value a contingent claim, the market extended by the value process of the contingent claim should satisfy the same good properties as the original market.
Using our valuation rule, we fully characterise all economically consistent values for a contingent claim. We show that simply taking the expectation of a contingent claim under an equivalent local martingale measure is economically inconsistent for bubble markets; one has to add a local martingale correction term. This term is bounded by a minimal and a maximal local martingale corresponding to the contingent claim and the equivalent local martingale measure. Perhaps surprisingly, for complete markets, an economically consistent value of a contingent claim is in general not unique. However, uniqueness of economically consistent values for call and put options holds for complete markets if the bank account is maximal, in the sense that the buy-and-hold strategy of the bank account cannot be improved without risk by trading dynamically over time. Even without maximality of the bank account, our approach produces option values in the model which satisfy put-call parity; this also extends to incomplete markets. Finally, we compare our results to existing ideas and approaches and explain among others which of the two existing formulas for the call price in the CEV model (Hull 2003, Heston et al. 2007) is economically consistent in our sense.