Large markets: asymptotic arbitrage and portfolio optimisation

In this thesis, we aim to shed some light on the intricate behaviour of large, correlated financial markets, the existence or absence of asymptotic arbitrage in such a model and its connection to optimal investing. Therefore we will approximate these finite large markets by infinite-sized markets, and derive strategies describing how to invest optimally, based on the modelling coefficients. Our correlated finite real world market, spanning n ∈ ℕ single secu- rities as well as at most the fixed number of J ∈ ℕ, global assets is similar to the approach known from the well-established Capital Asset Pricing Model (CAPM) and can be viewed as an increasing sequence of smaller sub-markets, which can be interpreted as a collection of sectors or a set of different economies, where each is endowed with its own market risk factor. The core problem we want to tackle in this thesis is to find out how an agent can optimally distribute his positive initial endowment over a large or infinite number of securities in order to build the optimal portfolio which can maximise his expected utility of terminal wealth. We will give both general as well as explicit solutions for a power utility function using optimal control methods. If we are going to approximate large markets by infinite ones, we need to carefully consider the limiting arbitrage conditions, namely the asymptotic arbitrage. Our main result provides a new link between the two concepts, no asymptotic arbitrage and the existence of a finite limiting value function in the infinite market. We succeed to prove that, for measurable and adapted coefficients, there is no asymptotic arbitrage if the value function solving the optimisation problem exists and is finite in the limit for an infinite number of stocks. In addition, we present a weak convergence result in terms of market models and look at asymptotic expansions of our optimisation results. Fundamentally, we manage to show that asymptotic arbitrage in our CAPM-style model always occurs, unless Sharpe ratios with respect to the global market of the single securities are close to the same, homogeneous value. The thesis is concluded by a real life investment example which attempts to generate wealth by investing according to our optimal portfolio proportions.