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A stochastic processes toolkit for risk management: Geometric Brownian motion, jumps, GARCH and variance gamma models
Journal of Risk Management in Financial Institutions,
2
(4),
365-393
(2009)
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.
Keywords: risk management; stochastic processes; risk factors; fat tails; mean reversion; jump diffusion processes; variance gamma processes
The full article is available to subscribers to the journal.
Author's Biography
Damiano Brigo obtained a PhD in stochastic filtering with differential geometry from the Free University of Amsterdam, following a BSc in mathematics from the University of Padua. Since July 2007, he has been Managing Director and Global Head of the Quantitative Innovation team at Fitch Solutions, London. He is also currently Visiting Professor at the Department of Mathematics at Imperial College, London. He is author of the book ‘Interest Rate Models: Theory and Practice’ (Springer-Verlag), and his academic and practitioner-oriented articles have been published in financial modelling, probability and systems theory journals. Damiano is Managing Editor of the International Journal of Theoretical and Applied Finance.
Antonio Dalessandro holds a PhD in mathematics from Imperial College London and is research associate at the University of Geneva’s Department of Econometrics. From 2001 to 2004 he worked as a researcher at CERN and INFN in Geneva. He has a first-class honours MSc in electrical engineering from the Politecnico di Bari, and his research interests include probability, functional analysis, optimisation and high frequencies trading.
Matthias Neugebauer is Senior Director of Fitch Ratings’ structured credit group. His responsibilities include criteria development and credit market research, including design and development of the portfolio credit model. Prior to this, he worked with the company’s European quantitative research team. He holds a master’s degree in finance and economics from Durham University and an engineering degree from Technical University Cottbus. He also holds a master’s degree in mathematical finance from Oxford University.
Fares Triki obtained an engineering diploma from a French grande école, an MSc in finance from CNAM Paris, and a DEA diploma in economics from Paris Sorbonne University with summa cum laude distinction. After working for multinational consulting firms, he joined the Fitch Ratings Quantitative Financial Research team with the role of risk assessment and valuation of various financial products. Fares is now a macro finance expert at the financial stability directorate of Banque de France and is also a PhD student at the Paris School of Economics. His research areas include risk modelling and international economics and finance.
Citation
Brigo, Damiano, Dalessandro, Antonio, Neugebauer, Matthias and Triki, Fares (2009, September 1). A stochastic processes toolkit for risk management: Geometric Brownian motion, jumps, GARCH and variance gamma models. In the Journal of Risk Management in Financial Institutions, Volume 2, Issue 4. https://doi.org/10.69554/MLKH4177.
© Henry Stewart
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