A stochastic processes toolkit for risk management: Geometric Brownian motion, jumps, GARCH and variance gamma models

Abstract

In risk management it is desirable to grasp the essential statistical features of a time series representing a risk factor. This paper aims to introduce a number of different stochastic processes that can help in grasping the essential features of risk factors, describing different asset classes or behaviours. The paper does not aim to be exhaustive, but gives examples and a feeling for practically implementable models, allowing for stylised features in the data. These models can also be used as building blocks to build more complex models, although, for a number of risk management applications, the models developed here suffice for the first step in the quantitative analysis. The broad qualitative features addressed here are fat tails. In the second part of this work to appear in a subsequent paper, mean reversion is addressed with and without fat tails. The paper gives some orientation on the initial choice of a suitable stochastic process and then explains how the process parameters can be estimated based on historical data. Once the process has been calibrated, typically through maximum likelihood estimation, one may simulate the risk factor and build future scenarios for the risky portfolio. On the terminal simulated distribution of the portfolio, one may then single out several risk measures, although the present paper focuses on the stochastic processes estimation preceding the simulation of the risk factors. Finally, this paper focuses on single time series. Correlation or more generally dependence across risk factors, leading to multivariate processes modelling, will be addressed in future work.