Completion of an incomplete market by quadratic variation assets.

Abstract

It is well known that the general geometric L´evy market models are incomplete, except for the geometric Brownian and the geometric Poissonian, but such a market can be completed by enlarging it with power-jump assets as Corcuera and Nualart [12] did on their paper. With the knowledge that an incomplete market due to jumps can be completed, we look at other cases of incompleteness. We will consider incompleteness due to more sources of randomness than tradable assets, transactions costs and stochastic volatility. We will show that such markets are incomplete and propose a way to complete them. By doing this we show that such markets can be completed. In the case of incompleteness due to more randomness than tradable assets, we will enlarge the market using the market’s underlying quadratic variation assets. By doing this we show that the market can be completed. Looking at a market paying transactional costs, which is also an incomplete market model due to indifference between the buyers and sellers price, we will show that a market paying transactional costs as the one given by, Cvitanic and Karatzas [13] can be completed. Empirical findings have shown that the Black and Scholes assumption of constant volatility is inaccurate (see Tompkins [40] for empirical evidence). Volatility is in some sense stochastic, and is divided into two broad classes. The first class being single-factor models, which have only one source of randomness, and are complete markets models. The other class being the multi-factor models in which other random elements are introduced, hence are an incomplete markets models. In this project we look at some commonly used multi-factor models and attempt to complete one of them.