A stochastic programming approach to cash management in banking

The treasurer of a bank is responsible for the cash management of several banking activities. In this work, we focus on two of them: cash management in automatic teller machines (ATMs), and in the compensation of credit card transactions. In both cases a decision must be taken according to a future customers demand, which is uncertain. From historical data we can obtain a discrete probability distribution of this demand, which allows the application of stochastic programming techniques. We present stochastic programming models for each problem. Two short-term and one mid-term models are presented for ATMs. The short-term model with fixed costs results in an integer problem which is solved by a fast (i.e. linear running time) algorithm. The short-term model with fixed and staircase costs is solved through its MILP equivalent deterministic formulation. The mid-term model with fixed and staircase costs gives rise to a multi-stage stochastic problem, which is also solved by its MILP deterministic equivalent. The model for compensation of credit card transactions results in a closed form solution. The optimal solutions of those models are the best decisions to be taken by the bank, and provide the basis for a decision support system.     

Developing a Visual Interactive Model for Corporate Cash Management

The operational research/management science journals contain an extensive literature that addresses the corporate cash management problem; yet few, if any, companies make use of any of this published work in their daily cash-management decision making. A review of the literature suggests that the reason for this lack of applications may well be poor problem formulation—the problems that are solved in the literature as ‘cash management’ problems evolve from a ‘hard systems’ view of real-world cash management. However, the problem as perceived by cash managers involves both dynamic and loosely structured components which are difficult to model using classical (i.e. ‘hard systems’) approaches.We therefore decided to approach the cash management problem as an experiment in the use of a novel visual interactive problem solving (VIPS) methodology. The aim of the experiment was to develop an implementable, visual interactive model to support daily cash management decision making. Working closely with a corporate cash manager, we first developed a visual model of his daily decision problem and then agreed on the feasible options and the interactive requirements. At this stage, the problem was sufficiently well defined for a mathematical model to be built and the visual model made ‘smart’.This paper discusses the results of this experiment and suggests that VIPS may have distinct advantages as a problem-solving technique in loosely structured, ‘messy’ problem situations.     

Treasury Management Model with Foreign Exchange Exposure

In this paper we formulate a model for foreign exchange exposure management and (international) cash management taking into consideration random fluctuations of exchange rates. A vector error correction model (VECM) is used to predict the random behaviour of the forward as well as spot rates connecting dollar and sterling. A two-stage stochastic programming (TWOSP) decision model is formulated using these random parameter values. This model computes currency hedging strategies, which provide rolling decisions of how much forward contracts should be bought and how much should be liquidated.The model decisions are investigated through ex post simulation and backtesting in which value at risk (VaR) for alternative decisions are computed. The investigation (a) shows that there is a considerable improvement to “spot only” strategy, (b) provides insight into how these decisions are made and (c) also validates the performance of this model.     

Order Quantities with Temporary Price Reductions

The temporary price-change problem is studied, in which the objective is to minimize discounted cash flows. As pointed out by Goyal in an earlier paper, only the cash transactions at purchase times (i.e. the payments for the goods and the ordering costs) were considered. The cash flows associated with `inventory maintenance' costs which occur more or less continuously over time were neglected, which changes the structure of the model. Examples of these costs include storage, insurance, record-keeping, deterioration and obsolescence costs. In this paper, these continuously generated cash flows are included in the analysis, thereby making the new model more applicable to practical situations. This model is of interest because order-quantity decisions often must be made under conditions of both temporary price reductions and/or imminent price increases. These changes occur frequently in practice.     

Mathematics of currency and exchange: arithmetic at the end of the thirteenth century

In western Europe, a sophisticated banking system for the purposes of international trade had evolved by the end of the thirteenth century. It was based upon the ‘bill of exchange’, which enabled an exporter of goods to receive payment in his own currency, by means of a balancing payment made in the currency of the importer. This paper discusses the arithmetical tools that were available for use in accounting for transactions made in different currencies. It is argued that algorithmic methods based on the Hindu-Arabic numerals were used at the higher levels of banking, in order to prepare tables of foreign exchange such as those collected by the Florentine banker, Francesco Pegolotti. On the other hand, the clerks who were responsible for routine book-keeping would have used a simple abacus and counters, and recorded their transactions in Roman numerals. The paper is based on a talk given to the BSHM at Gresham College on 25 April 2008.     

Cash management cost reduction using data mining to forecast cash demand and LP to optimize resources

This paper presents a two step model aimed at reducing cash management costs in a bank’s branch. First, data mining was used to forecast daily cash demand, comparing an ARMA-ARCH model with a neural network. Secondly, using the prior result, a linear programming model was solved. The optimal allocation of resources, i.e., cash collections and supplies was estimated showing that the model can be a helpful tool to support the determination of collections and supplies at the bank branch.     

Functional data analysis for cash flow and transactions intensity continuous-time prediction using Hilbert-valued autoregressive processes

In this paper we use a functional autoregressive model as a robust predictor of the cash flow and intensity of transactions in a credit card payment systems. Intraday economic time series are treated as random continuous functions projected onto low dimensional subspace. Wavelet bases are considered for data smoothing. We compare two linear wavelet methods for the prediction problem of a continuous-time stochastic process on an entire time interval. Ex poste prediction is used to check the models.     

A sample-path approach to optimal position liquidation

We consider the problem of optimal position liquidation where the expected cash flow stream due to transactions is maximized in the presence of temporary or permanent market impact. A stochastic programming approach is used to construct trading strategies that differentiate decisions with respect to the observed market conditions, and can accommodate various types of trading constraints. As a scenario model, we use a collection of sample paths representing possible future realizations of state variable processes (price, trading volume etc.), and employ a heuristical technique of sample-path grouping, which can be viewed as a generalization of the standard nonanticipativity constraints.     

基于GARCH族模型商业银行外汇风险管理的VaR方法研究

面对日趋加大的汇率波动性,商业银行外汇资产面临的风险也越来越大,风险的计量与预测在管理外汇风险中的作用也越来越重要.引入参数法下的GARCH模型对外汇市场存在的风险进行计量分析,并以此为基础运用VaR方法进一步计算外汇资产的风险补偿金,以达到预测和控制外汇风险目的.     

The American Foreign Exchange Option in Time-Dependent One-Dimensional Diffusion Model for Exchange Rate

The classical Garman-Kohlhagen model for the currency exchange assumes that the domestic and foreign currency risk-free interest rates are constant and the exchange rate follows a log-normal diffusion process. In this paper we consider the general case, when exchange rate evolves according to arbitrary one-dimensional diffusion process with local volatility that is the function of time and the current exchange rate and where the domestic and foreign currency risk-free interest rates may be arbitrary continuous functions of time. First non-trivial problem we encounter in time-dependent case is the continuity in time argument of the value function of the American put option and the regularity properties of the optimal exercise boundary. We establish these properties based on systematic use of the monotonicity in volatility for the value functions of the American as well as European options with convex payoffs together with the Dynamic Programming Principle and we obtain certain type of comparison result for the value functions and corresponding exercise boundaries for the American puts with different strikes, maturities and volatilities. Starting from the latter fact that the optimal exercise boundary curve is left continuous with right-hand limits we give a mathematically rigorous and transparent derivation of the significant early exercise premium representation for the value function of the American foreign exchange put option as the sum of the European put option value function and the early exercise premium. The proof essentially relies on the particular property of the stochastic integral with respect to arbitrary continuous semimartingale over the predictable subsets of its zeros. We derive from the latter the nonlinear integral equation for the optimal exercise boundary which can be studied by numerical methods.     

Defensive financial decisions support for retailers in Greek pharmaceutical industry

In this paper, we present a decisions support solution designed for Greek pharmacies comprising a cash flow management system for early warning of financial distress and a financial advisor based on a neural network. The cash flow monitoring system integrates accounting elements with real time transactions and a predictive linear regression model while the decision support module is developed with the help of a neural network. For any given business unit the system associates accounting entries with information about credit times to reflect the precise instants of cash flows and using inflows/outflows equations monthly, eventually build its liquidity curve and cash flow balance over time. Alongside, a linear regression module is introduced to estimate future cash reserves based on past profitability ratios. Lastly, combining the power of artificial neural networks with expertise in this sector of pharmaceutical business, the financial decision support tool focuses on the retailers that face financial difficulties and suggests alternative solutions for escaping from distress and insolvency. The model has an ambitious and useful purpose, to inform and consult the owners of the business units and other members of the pharmaceutical chain, thus reduce financial risk for the chain.     

A detailed workforce planning model including non-linear dependence of capacity on the size of the staff and cash management

This paper introduces an original planning model which integrates production, human resources and cash management decisions, taking into account the consequences that decisions in one area may have on other areas and allowing all these areas to be coordinated. The most relevant characteristics of the planning problem are: (1) production capacity is a non-linear function of the size of the staff; (2) firing costs may depend on the worker who is fired; (3) working time is managed under a working time account (WTA) scheme, so positive balances must be paid to workers who leave the company; (4) there is a learning period for hired workers; and (5) cash management is included. A mixed integer linear program is designed to solve the problem. Despite the size and complexity of the model, it can be solved in a reasonable time. A numerical example, the main results of a computational experiment and a sensibility analysis illustrate the performance and benefits of the model.     

A fuzzy stochastic single-period model for cash management

The major purpose of this paper is to apply a stochastic single-period inventory management approach to analyze optimal cash management policies with fuzzy cash demand based on fuzzy integral method so that total cost is minimized. We will find that, after defuzzification, the cash-raising amounts and the total costs between the fuzzy case and the crisp case are slightly different when the variation of cash demand is small. As a result, we point out that the fuzzy stochastic single-period model is one extension of the crisp models. In any case, one may conclude that a conscientious analysis in fuzzy mathematics like that presented in this paper provides a financial decision maker with a deeper insight into the more real cash management problem.     

基于汇率回报厚尾性的外汇期权定价模型

主要研究汇率回报呈厚尾分布的外汇期权定价问题。本文利用t-分布能捕获汇率回报序列厚尾特征的优势，推导出基于t-分布外汇期权定价模型的解析表达式，即对外汇期权定价模型——BSGK模型进行了修正，同时应用矩估计法估计出的t-分布的自由度用于该定价模型的计算，最后基于t-分布的外汇期权定价模型和BSGK外汇期权定价模型进行了比较分析。     

A non random walk theory of exchange rate dynamics with applications to option pricing

A two dimensional stochastic process is developed to model exchange rate dynamics. We incorporate the non random walk influence of pur–chasing power parity, to synthesise the theories of international trade and foreign currency options. Our results, which include a closed form expression for the transition density function of the exchange rate and an exact formula to price currency options, offer a theoretical framework for further study of foreign exchange markets     

Valuation and hedging strategy of currency options under regime-switching jump-diffusion model

The main purpose of this thesis is in analyzing and empirically simulating risk minimizing European foreign exchange option pricing and hedging strategy when the spot foreign exchange rate is governed by a Markov-modulated jump-diffusion model. The domestic and foreign money market interest rates, the drift and the volatility of the exchange rate dynamics all depend on a continuous-time hidden Markov chain which can be interpreted as the states of a macro-economy. In this paper, we will provide a practical lognormal diffusion dynamic of the spot foreign exchange rate for market practitioners. We employing the minimal martingale measure to demonstrate a system of coupled partial-differential-integral equations satisfied by the currency option price and attain the corresponding hedging schemes and the residual risk. Numerical simulations of the double exponential jump diffusion regime-switching model are used to illustrate the different effects of the various parameters on currency option prices.     

Generalized bilinear programming: An application in farm management

In this paper we analyze a model which arises from considering manure and fertilizer aspects in the farm management modelling for cattle husbandry. Modelling these aspects may lead to a multi-extremal problem with bilinear terms. Solution techniques for this problem are discussed and a specific branch and bound procedure is outlined. An example illustrates the multi-extremal structure of the problem and the solution methods mentioned.     

面对成比例交易成本的情况下外汇期权的定价

The purpose of present work is to examine the financial problem of finding the universal reservation prices of a European call option written on exchange rate when there is proportional transaction costs of trading foreign currency in the market. An approach is suggested to compute the reservation bid-ask price of foreign currency call option based on maximizing the investor's expected utility. Option prices are determined from the investor's basic portfolio selection problem, without the need to solve a more complex optimization problem involving the insertion of the option payoffs into the terminal value function. Option prices are computed numerically in a Markov chain approximation for the case of exponential utility.Numerical results show that the option price bounds are almost independent of the alternative risk aversion parameter, but the bounds of NT region becomes narrower and the range of values of the initial holding for which the fair price lies within the bid-ask spread is shifted to a lower value when the risk aversion parameter increases.     

FOREIGN CURRENCY OPTION PRICING WITH PROPORTIONAL TRANSACTION COSTS

The purpose of present work is to examine the financial problem of finding the universal reservation prices of a European call option written on exchange rate when there is proportional transaction costs of trading foreign currency in the market. An approach is suggested to compute the reservation bid-ask price of foreign currency call option based on maximizing the investor＇s expected utility. Option prices are determined from the investor＇s basic portfolio selection problem, without the need to solve a more complex optimization problem involving the insertion of the option payoffs into the terminal value function. Option prices are computed numerically in a Markov chain approximation for the case of exponential utility. Numerical results show that the option price bounds are almost independent of the alternative risk aversion parameter, but the bounds of NT region becomes narrower and the range of values of the initial holding for which the fair price lies within the bid-ask spread is shifted to a lower value when the risk aversion parameter increases.     

A Decision Analysis Approach to Cash Flow Management

We propose a method of applying the Hespos and Strassmann "stochastic decision tree" framework, originally intended for investment decisions, to cash flow management. Sequences of uncertain events, such as a strike, affecting forecast cash flows are represented by a probability tree. Forecasts of constituent cash flows such as sales and costs are represented by Beta distributions dependent on paths through the tree. Monte Carlo simulations sample these distributions, and equations provided in the model convert the sampled cash flows to cash balances in each period. Frequencies of cash balances weighted by probabilities along paths through the tree yield a combined relative frequency distribution of cash balances for each period. These and related results may be used by management to plan financing arrangements to meet cash requirements in the future.