In the context of continuous-time financial market models, we shall concentrate on certain examples of market imperfection: portfolio constraints, incomplete markets, transaction costs. Valuation problems are introduced through stochastic control problems, which we analyse using the Bellman approach. We will look at the face-lifting phenomenon which appears in the superhedging approach, and we will analyse the valuation problem using utility indifference.
- Valuation under portfolio constraints -Modelling: portfolio constraints, examples, fictional markets. Dual formulation: rewriting the problem in a form which generalises the valuation principle for a complete market. Direct resolution in the context of the Gaussian model: log-normal price processes with constrained portfolio in a closed convex set containing the origin.
- Valuation in an incomplete market -Superhedging in a stochastic volatility model: buy-and-hold strategies, the Black-Scholes-Barrenblatt partial derivative equation. Robustness of the Black-Scholes model for convex payoffs. Valuation by utility indifference: Valuation principle, exponential utility.
Valuation by utility indifference in stochastic volatility models: Explicit formula under certain hypotheses about the model. - Valuation in the presence of transaction costs -Modelling: portfolios are now described by transfers. Failure of the superhedging approach: optimality of buy-and-hold strategies.
Choice of portfolio under infinite horizon.
Last Modification : Wednesday 29 July 2009