Discrete-time and continuous-time approaches to multi-asset modelling

Abstract

In this dissertation we shall be modelling a portfolio of assets. In particular, we shall be trying to fit an adequate model to the log returns of four assets traded on the Malta Stock Exchange, namely; shares of Bank of Valletta p.l.c., HSBC Bank Malta p.l.c, GO p.l.c, and Malta International Airport p.l.c. After giving some background theory on the Markov processes setting we require for our application, we shall look into the multivariate distribution approach to modelling, focusing on the multivariate normal and the multivariate Student's t distribution. However, multivariate distributions tend to be rigid and that is why we seek an alternative approach. Copulas are multivariate distributions which treat the dependence structure independently from the marginal distributions. They are more flexible and hence tend to be a preferred method for modelling. We shall look into two main types of copulas: implicit and explicit copulas. We shall apply the Gaussian copula and Student's t copula to our dataset under study along with the Clayton, Gumbel and Frank copulas. After applying these multivariate distributions and copulae to our dataset we implement an empirical goodness-of-fit test in order to assess which models are suitable. Finally, we have a look at modelling trade-by-trade data using continuous time stochastic processes. We investigate the basic features of the price changes of the four assets previously mentioned, whilst comparing these features to those of a particular stock traded on the New York Stock Exchange. We then suggest modelling this data using a compound Poisson or compound Cox type process. We attempt to model our data using time-homogenous Markov chains; however, these only depend on what happened in the previous trade. Being aware that the effect may be much more long term we also look into modelling using Action Direction Size (ADS) and Multivariate Action Direction Size (MADS) decomposition.