Modelling With Jump Processes and Optimal Control

Classical Merton model assumes that an asset is modelled by Brownian motion or geometric Brownian motion. However, these models lack some empirical properties of usual financial series. If jumps are allowed into the model, it becomes much more appropriate. In this note, jump processes are briefly introduced. Subsequently, the impact of jumps on the optimal consumption and portfolio choice is studied. Theoretical formulas of optimal values are presented and finally, a numerical study on real data is performed with rather notable results. Introduction The most frequently used models in the field of finance are Brownian motion and geometric Brownian motion. It was shown that these models reproduce some empirical facts of usual financial series very poorly, e.g. heavy tails. They can be generalized by adding stochastic volatility term. However, financial series also show very large sudden movements which leads to unrealistically high volatilities. Moreover, models with stochastic volatility may become very complicated. A much more natural way is to allow jumps in the model. Modelling by not necessarily normal processes dates back to 1963, when a french mathematician Benòıt B. Mandelbrot noticed that the increments of cotton prices showed heavy tails and as a model proposed α-stable distribution, [Mandelbrot, 1963]. Later, many other heavy tailed distributions were proposed, e.g. Normal Inverse Gaussian, see [Barndorff-Nielsen, 1998], or Merton Jump Diffusion model, see [Merton, 1976]. The common property of processes having these distributions is that their paths are not necessarily continuous. They are known as Lévy processes. A comprehensive monography with overview of existing Lévy processes and applications is the book [Cont and Tankov, 2004]. In this paper, we work with already calibrated models. The used data describes six second shots of Futures price. It is worth mentioning that the fit to the data with jump models significantly outperforms the classical estimate based on normal distribution. The main objective of this note is to numerically assess the influence of jumps on the optimal investing strategy. In the article [Framstad et al., 1998] the optimal consumption and portfolio were derived and compared with the Merton consumption and proportion. We use a slightly modified version of their theorem, however it is not the core part of this paper. A comprehensive overview of optimal control for jump processes can be found in the book [Øksendal and Sulem, 2007]. In the first part of the paper, Lévy processes are briefly introduced. The rest of the paper deals with optimal control. The optimal investment problem is introduced. Theoretical results are presented. These results are used in the following numerical study. It is shown that the influence of jumps on optimal portfolio is very serious. Some interesting relations to the Merton optimal values are also revealed. Lévy Processes We assume a given probability space (Ω,F , (Ft),P), where (Ft) is a filtration satisfying usual conditions of completeness and right continuity. 1 [email protected] 125 WDS'09 Proceedings of Contributed Papers, Part I, 125–130, 2009. ISBN 978-80-7378-101-9 © MATFYZPRESS