Optimal portfolio problems under model ambiguity

Abstract

The topic of this thesis is portfolio optimization under model ambiguity, i.e. a situation when the probability distribution of the events in the sample space is not known. The financial market studied is driven by a Brownian motion: a continuous driving element, and a doubly stochastic Poisson random process: a discontinuous driving element. What separates the doubly stochastic Poisson random process from the standard Poisson case, is that the jump intensity is a stochastic process. From a modeling point of view, this adds more flexibility in capturing hidden random effects. Models of this type appear in the literature of credit risk and in financial price modeling in the class of stochastic volatility models. See e.g. "On Cox processes and credit risky securities" by D. Lando and "Stochastic volatility for Lèvy Processes" by P. Carr et al., respectively. In this thesis we assume that the investing agent is ambiguity averse, i.e. the agent does not take any risk with respect to the uncertainty of the probability distribution, and thus relates to the worst case probability distribution. A dynamic risk measure that respects the agents ambiguity aversion is applied to quantify the risk of a hedging strategy, and the agent wishes to make the risk vanish at all times. The optimization problem takes the form of a stochastic differential game, in which the agent minimizes the risk of the hedging strategy, while the opponent drives in the opposite way proposing a probability distribution yielding the worst case scenario. This thesis entails two methods of solving this stochastic differential game. First, through backward stochastic differential equations (BSDEs), and then using the maximum principle. In this thesis, the approach with BSDEs will give a solution of a price process at all times in the give horizon, while the approach using the maximum principle will give a solution at the initial time. This approach of pricing is of great interest to insurers, as it gives a supplement to Value-at-Risk and Estimated Shortfall calculations, and gives a benchmark for capital needed to withstand extreme scenarios.