The aim of this course is to present some recent problems about the non-arbitrage condition and portfolio choice. To do so, we'll look at modeling in mathematical finance and mathematical economics.

We deal with the modelling of :

• Financial markets
• Absence of arbitrage opportunities
• Prices
• Economic agents' choices
• Optimal portfolios.

We present the models and discuss their limitations. We will also discuss current problems linked to the robustness of models and Keynesian uncertainty.

1. Modeling financial markets and the fundamental theorem of asset pricing
2. Modeling the notion of price: super-replication price (incomplete market)
3. Modeling choices under uncertainty
4. Utility maximization problem in the case of one belief
5. Multi-priors utility maximization problem

We begin with an introduction to financial markets, and then to their probabilistic modeling. We will discuss the characterization of the no-arbitrage condition, a fundamental assumption of mathematical finance. We will discuss the consequences in terms of pricing.

We will then discuss the problem of choice under uncertainty in economics, and in particular Von Neumann and Morgenstern's (1947) theory of utility expectation. This representation of agents' preferences has been extensively studied and used in economics and finance.

We present two methods for solving the problem of expected utility: the primal approach and the dual approach. We will also introduce so-called utility prices, which allow us to take into account the agents' choices in the valuation of derivatives.

We will also discuss Keynesian model uncertainty (a set of probabilities instead of a single belief). We will discuss non-arbitrage and the existence of a strategy that maximizes utility expectation in the case where the worst belief occurs among all those modeled by Q.