Minimizing Banking Risk in a Lévy Process Setting

Gideon, F.; Mukuddem-Petersen, J.; Petersen, M. A.

doi:https://doi.org/10.1155/2007/32824

Journal of Applied Mathematics

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Research Article | Open Access

Volume 2007 | Article ID 032824 | https://doi.org/10.1155/2007/32824

Minimizing Banking Risk in a Lévy Process Setting

F. Gideon,¹J. Mukuddem-Petersen,¹and M. A. Petersen¹

Academic Editor: Ibrahim Sadek

Received28 Feb 2007

Accepted18 May 2007

Published26 Jun 2007

Abstract

The primary functions of a bank are to obtain funds through deposits from external sources and to use the said funds to issue loans. Moreover, risk management practices related to the withdrawal of these bank deposits have always been of considerable interest. In this spirit, we construct Lévy process-driven models of banking reserves in order to address the problem of hedging deposit withdrawals from such institutions by means of reserves. Here reserves are related to outstanding debt and acts as a proxy for the assets held by the bank. The aforementioned modeling enables us to formulate a stochastic optimal control problem related to the minimization of reserve, depository, and intrinsic risk that are associated with the reserve process, the net cash flows from depository activity, and cumulative costs of the bank's provisioning strategy, respectively. A discussion of the main risk management issues arising from the optimization problem mentioned earlier forms an integral part of our paper. This includes the presentation of a numerical example involving a simulation of the provisions made for deposit withdrawals via treasuries and reserves.

References

H. Föllmer and D. Sondermann, “Hedging of non-redundant contingent claims,” in Contributions to Mathematical Economics, pp. 205–223, North Holland, Amsterdam, The Netherlands, 1986.
View at: Google Scholar | Zentralblatt MATH
F. Black and M. Scholes, “The pricing of options and corporate liabilities,” The Journal of Political Economy, vol. 81, no. 3, pp. 637–654, 1973.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
R. C. Merton, “An analytic derivation of the cost of deposit insurance and loan guarantees: an application of modern option pricing theory,” Journal of Banking & Finance, vol. 1, no. 1, pp. 3–11, 1977.
View at: Publisher Site | Google Scholar
J.-P. Decamps, J.-C. Rochet, and B. Roger, “The three pillars of Basel II: optimizing the mix,” Journal of Financial Intermediation, vol. 13, no. 2, pp. 132–155, 2004.
View at: Publisher Site | Google Scholar
C. H. Fouche, J. Mukuddem-Petersen, and M. A. Petersen, “Continuous-time stochastic modelling of capital adequacy ratios for banks,” Applied Stochastic Models in Business and Industry, vol. 22, no. 1, pp. 41–71, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
H. Leland, “Corporate debt value, bond covenants, and optimal capital structure,” The Journal of Finance, vol. 49, no. 4, pp. 1213–1252, 1994.
View at: Publisher Site | Google Scholar
J.-C. Rochet, “Rebalancing the three pillars of Basel II,” Economic Policy Review, vol. 10, no. 2, pp. 7–25, 2004.
View at: Google Scholar
M. A. Petersen and J. Mukuddem-Petersen, “Stochastic behaviour of risk-weighted bank assets under the Basel II capital accord,” Applied Financial Economics Letters, vol. 1, no. 3, pp. 133–138, 2005.
View at: Publisher Site | Google Scholar
J. Mukuddem-Petersen and M. A. Petersen, “Bank management via stochastic optimal control,” Automatica, vol. 42, no. 8, pp. 1395–1406, 2006.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. A. Petersen, J. Mukuddem-Petersen, I. Schoeman, and A. Tau, “Maximizing banking profit on a random time interval,” Journal of Applied Mathematics, vol. 2007, Article ID 29343, 22 pages, 2007.
View at: Publisher Site | Google Scholar
P. E. Protter, Stochastic Integration and Differential Equations, vol. 21 of Applications of Mathematics (New York), Springer, Berlin, Germany, 2nd edition, 2004.
View at: Zentralblatt MATH | MathSciNet
J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, vol. 288 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 2nd edition, 2003.
View at: Zentralblatt MATH | MathSciNet
J. Bertoin, Lévy Processes, vol. 121 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, UK, 1996.
View at: Zentralblatt MATH | MathSciNet
K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions, vol. 68 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1999.
View at: Zentralblatt MATH | MathSciNet
W. Schoutens, Lévy Processes in Finance: Pricing Financial Derivatives, John Wiley & Sons, Chichester, UK, 2003.
H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, World Scientific, River Edge, NJ, USA, 3rd edition, 2004.
View at: Zentralblatt MATH | MathSciNet
A. E. Kyprianou, W. Schoutens, and P. Wilmott, Eds., Exotic Option Pricing and Advanced Lévy Models, John Wiley & Sons, Chichester, UK, 2005.
F. Delbaen and W. Schachermayer, “A general version of the fundamental theorem of asset pricing,” Mathematische Annalen, vol. 300, no. 1, pp. 463–520, 1994.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H. Kunita and S. Watanabe, “On square integrable martingales,” Nagoya Mathematical Journal, vol. 30, pp. 209–245, 1967.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. N. Berger and G. F. Udell, “Did risk-based capital allocate bank credit and cause a “credit crunch” in the United States?” Journal of Money, Credit and Banking, vol. 26, no. 3, part 2, pp. 585–628, 1994.
View at: Publisher Site | Google Scholar
F. Castiglionesi, “Financial contagion and the role of the central bank,” Journal of Banking and Finance, vol. 31, no. 1, pp. 81–101, 2007.
View at: Publisher Site | Google Scholar
J. A. Clouse and J. P. Dow Jr., “A computational model of banks' optimal reserve management policy,” Journal of Economic Dynamics and Control, vol. 26, no. 11, pp. 1787–1814, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
S. Demiralp and D. Farley, “Declining required reserves, funds rate volatility, and open market operations,” Journal of Banking and Finance, vol. 29, no. 5, pp. 1131–1152, 2005.
View at: Publisher Site | Google Scholar
H. M. Ennis and T. Keister, “Bank runs and investment decisions revisited,” Journal of Monetary Economics, vol. 53, no. 2, pp. 217–232, 2006.
View at: Publisher Site | Google Scholar
E. Fernández, “Distorting taxes and interest on reserves,” Economic Modelling, vol. 22, no. 6, pp. 975–1000, 2005.
View at: Publisher Site | Google Scholar
W. Whitesell, “Interest rate corridors and reserves,” Journal of Monetary Economics, vol. 53, no. 6, pp. 1177–1195, 2006.
View at: Publisher Site | Google Scholar
T. Chan, “Pricing contingent claims on stocks driven by Lévy processes,” The Annals of Applied Probability, vol. 9, no. 2, pp. 504–528, 1999.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
N. Bouleau and D. Lamberton, “Residual risks and hedging strategies in Markovian markets,” Stochastic Processes and Their Applications, vol. 33, no. 1, pp. 131–150, 1989.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2004.
View at: Zentralblatt MATH | MathSciNet
Basel Committee on Banking Supervision, “Second Consultative Paper,” 2001, Bank for International Settlements, http://www.bis.org/bcbs/bcbscp2.htm.
View at: Google Scholar
Basel Committee on Banking Supervision, “International Convergence of Capital Measurement and Capital Standards; A Revised Framework,” 2004, Bank for International Settlements, http://www.bis.org/publ/bcbs107.pdf.
View at: Google Scholar

Copyright

Copyright © 2007 F. Gideon et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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