Portfolio Selection Models Driven by Non Gaussian Price Dynamics

Since the beginning of this century, the normal distribution has played a central role in the mathematical finance literature. However, major drawbacks insight this assumption rely in the absence of closed form expressions for both its cumulative and probability density functions. Additionally, although a great deal of efforts have been spent along the past decades to prove the empirical consistency of this assumption, real data have often offered a contrasting evidence, that is the exhibition of heavy tails and skewness.This paper starts from this point to consider the classical Merton problem of optimal portfolio selection and consumption, when alternatives to standard Brownian Motion are considered to model stock price dynamics. In particular, two directions will be spanned: a)\tthe Benth , Karlsen and Reikvam approach 2 (BKR since now on), who studied the underlying optimisation problem within a viscosity solution framework. The systematic solution provided by the authors relies on the possibility to replace the standard model for stock prices in the Black-Scholes world with one assuming the underlying process to belong to the class of Levy distributions. Hence the optimisation problem is solved via the corresponding Hamilton-Jacobi-Bellman equation, by weakening conditions requested . Since now only theoretical prove for the inverse Gaussian distribution case is given. A primer aim of this paper is, therefore, to test whether or not the above method can be extended to the class of Levy stable processes well known in economics, and hence to test empirically the plausibility of BKR method once a real stock to fit the model is chosen under different assumptions over distributions. b)\tA neural net approach, Ïletting the data to speak for themselvesÓ, which uses the properties of Self Organising Features maps to reconstruct features of high dimensional input into reduced dimensional space. The outline of the paper is as follows: section I introduces Merton problem within both traditional Brownian motion and BKR model assumption; section II describes the numerical solution adopted in this paper to model BKR approach; in section III the neural approach will be discussed. Section IV will focus on simulation results, under different assumptions over risk aversion and interest rates. Finally, section V will give some conclusions and outlooks for future works.