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A Stochastic Processes Toolkit for Risk Management

by Damiano Brigo of FitchSolutions,
Antonio Dalessandro of FitchSolutions,
Matthias Neugebauer of FitchSolutions, and
Fares Triki of FitchSolutions

November 15, 2007

Abstract: In risk management it is desirable to grasp the essential statistical features of a time series representing a risk factor. This tutorial 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 behaviors. This paper does not aim at being exhaustive, but gives examples and a feeling for practically implementable models allowing for stylised features in the data. The reader may also use these models 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 and mean reversion. We give some orientation on the initial choice of a suitable stochastic process and then explain 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 here we focus on the stochastic processes estimation preceding the simulation of the risk factors Finally, this first survey report focuses on single time series. Correlation or more generally dependence across risk factors, leading to multivariate processes modeling, will be addressed in future work.

JEL Classification: G32, C13, C15, C16.

AMS Classification: 60H10, 60J60, 60J75, 65C05, 65C20, 91B70.

Keywords: Risk Management, Stochastic Processes, Maximum Likelihood Estimation, Fat Tails, Mean Reversion, Monte Carlo Simulation.

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