Minimizing Banking Risk in a Lévy Process Setting

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2007, Article ID 32824, 25 pages doi:10.1155/2007/32824 Research Article Minimizing Banking Risk in a Lévy Process Setting F. Gideon, J. Mukuddem-Petersen, and M. A. Petersen Received 28 February 2007; Accepted 18 May 2007 Recommended by Ibrahim Sadek 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 act 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. 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. 1. Introduction We apply the quadratic hedging approach developed in [1] to a situation related to bank deposit withdrawals. In incomplete markets, this problem arises due to the fact that random obligations cannot be replicated with probability one by trading in available assets. For any hedging strategy, there is some residual risk. More specifically, in the quadratic hedging approach, the variance of the hedging error is minimized. With regard to this, 2 Journal of Applied Mathematics our contribution addresses the problem of determining risk minimizing hedging strategies that may be employed when a bank faces deposit withdrawals with fixed maturities resulting from lump sum deposits. In the recent past, more attention has been given to modeling procedures that deviate from those that rely on the seminal Black-Scholes financial model (see, e.g., [2, 3]). Some of the most popular and tractable of these procedures are related to Lévy processbased models. In this regard, our paper investigates the dynamics of banking items such as loans, reserves, capital, and regulatory ratios that are driven by such processes. An advantage of Lévy-processes is that they are very flexible since for any time increment Δt, any infinitely divisible distribution can be chosen as the increment distribution of periods of time Δt. In addition, they have a simple structure when compared with general semimartingales and are able to take different important stylized features of financial time series into account. A specific motivation for modeling banking items in terms of Lévy processes is that they have an advantage over the more traditional modeling tools such as Brownian motion (see, e.g., [4–7]), since they describe the noncontinuous evolution of the value of economic and financial indicators more accurately. Our contention is that these models lead to analytically and numerically tractable formulas for banking items that are characterized by jumps. Some banking activities that we wish to model dynamically are constituents of the assets and liabilities held by the bank. With regard to the former, it is important to be able to measure the volume of Treasuries and reserves that a bank holds. Treasuries are bonds issued by a national treasury and may be modeled as a risk-free asset (bond) in the usual way. In the modern banking industry, it is appropriate to assign a price to reserves and to model it by means of a Lévy process because of the discontinuity associated with its evolution and because it provides a good fit to real-life data. Banks are interested in establishing the level of Treasuries and reserves on demand deposits that the bank must hold. By setting a bank’s individual level of reserves, roleplayers assist in mitigating the costs of financial distress. For instance, if the minimum level of required reserves exceeds a bank’s optimally determined level of reserves, this may lead to deadweight losses. While the academic literature on pricing bank assets is vast and well developed, little attention is given to pricing bank liabilities. Most bank deposits contain an embedded option which permits the depositor to withdraw funds at will. Demand deposits generally allow costless withdrawal, while time deposits often require payment of an early withdrawal penalty. Managing the risk that depositors will exercise their withdrawal option is an important aspect of our contribution. The main thrust of our paper is the hedging of bank deposit withdrawals. In this spirit, we discuss an optimal risk management problem for commercial banks which use the Treasuries and reserves to cater for such withdrawals. In this regard, the main risks that can be identified are 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. In the sequel, we use the notational convention “subscript t or s” to represent (possibly) random processes, while “bracket t or s” is used to denote deterministic processes. In the ensuing discussion, for the sake of completeness, we firstly provide a general description of a Lévy process and an associated measure and then describe the Lévy F. Gideon et al. 3 decomposition that is appropriate for our analysis. In this regard, we assume that φ(ξ) is the characteristic function of a distribution. If for every positive integer n, φ(ξ) is also the nth power of a characteristic function, we say that the distribution is infinitely divisible. For each infinitely divisible distribution, a stochastic process L = (Lt )0≤t called a Lévy process exists. This process initiates at zero, has independent and stationary increments and has (φ(u))t as a characteristic function for the distribution of an increment over [s,s + t], 0 ≤ s,t, such that Lt+s − Ls . Every Lévy process is a semimartingale and has a cádlág version (right continuous with left-hand limits) which is itself a Lévy process. We will assume that the type of such processes that we work with is always cádlág. As a result, sample paths of L are continuous a.e. from the right and have limits from the left. The jump of Lt at t ≥ 0 is defined by ΔLt = Lt − Lt− . Since L has (...truncated)