Neoclassical growth model with two fixed delays

“Delay” has been considered as one of destabilizing factors in economic dynamics for a long time. Dynamic macroeconomics is concerned with explaining growth and fluctuations. This paper shows how various dynamics involving cyclic fluctuations can emerge in the standard neoclassical growth model when two distinct delays, a delay in production and a delay in depreciation, are explicitly taken into account. We first confirm that the production delay has a stabilizing effect and the depreciation delay has a destabilizing effect in a one‐delay model. We then determine the stability switching curve analytically in the two delay model. It is shown that cyclic fluctuations emerge via Hopf bifurcation when stability is lost. It is also found that stability loss and gain repeatedly occur. Numerical examples verify the theoretical results when the Cobb‐Douglas production function is adopted.     

Approximating Large Diversified Portfolios

This paper considers a financial market with asset price dynamics modeled by a system of lognormal stochastic differential equations. A one‐dimensional stochastic differential equation for the approximate evolution of a large diversified portfolio formed by these assets is derived. This identifies the asymptotic dynamics of the portfolio as being a lognormal diffusion. Consequentially an efficient way for computing probabilities, derivative prices, and other quantities for the portfolio are obtained. Additionally, the asymptotic strong and weak orders of convergence with respect to the number of assets in the portfolio are determined.     

DEMOGRAPHY IN MACROECONOMIC MODELS: WHEN LABOUR SUPPLY MATTERS FOR ECONOMIC CYCLES

The issue of the onset of Malthusian cycles is investigated by means of a demo‐economic model incorporating the age structure of the population. It is shown that the delayed recruitment into the labour force is a major source of demo‐economic instability, potentially leading to sustained oscillations. We also compare different modelling strategies for age structure, by showing that the results of our general model are borne out by those provided by a simpler model, based on a representation of age structure via time delays, which allows a deeper mathematical analysis. This suggests that simplified delay models may be of great help in understanding the qualitative properties of complex age structure models.     

The valuation of American options in a multidimensional exponential Lévy model

We consider the problem of valuation of American options written on dividend‐paying assets whose price dynamics follow a multidimensional exponential Lévy model. We carefully examine the relation between the option prices, related partial integro‐differential variational inequalities, and reflected backward stochastic differential equations. In particular, we prove regularity results for the value function and obtain the early exercise premium formula for a broad class of payoff functions.     

Perceptions of download delays: relation to actual waits,web site abandoning,and stage of delay

This experiment investigates how slow downloads of shopping web sites are perceived by online consumers, and how download delays relate to web site abandoning and stage of delay. Results show a complex, nonlinear relationship between actual and perceived download waiting, where perceptions level off after a threshold is reached. Furthermore, perceptions of download waiting are found to be more reliable than actual waits in predicting web site abandoning. Finally, delays near the start of the download are perceived as longer than later in the process, and time pressure worsens the effect of download waiting at earlier stages of delay.     

How does awareness evolve when advertising stops? The role of memory

Extant models posit that awareness declines immediately and gradually after the cessation of advertising, whereas anecdotal evidence from managers suggests awareness stays constant for a while and then decays rapidly. This pattern arises because consumers remember advertisements for a finite time before they forget. Hence, we extend advertising models by incorporating the memory for ads. We conceptualize the role of memory as ??delayed forgetting of ads?? and capture it using delay differential equations, which exhibit richer dynamics and expand the class of dynamic models used in marketing. Analytically, we derive the 90% duration of advertising effects under various scenarios. Empirically, we analyze awareness evolution in the absence of advertising for the Peugeot car brand. We not only find strong support for the proposed model, but also estimate the memorability of Peugeot ads to be about 3?weeks. Moreover, if we ignore consumer memory as in the extant models, we would overstate the forgetting rate by 39%. Finally, we discuss managerial implications and identify new avenues for further research.     

基于折射波静校正的未知参数穿墙成像合成延迟时算法

在穿墙雷达实际应用中,墙体一般由两层不同介质构成,且各层墙体参数未知。若对墙后目标直接成像,会产生目标散焦或位置偏移。针对该问题,提出了基于折射波静校正的合成延迟时方法对墙后目标进行聚焦成像。首先读取每根接收天线回波数据确定电磁波的折射回波时刻,通过基本折射方程得到延迟时超定方程组,解此超定方程组可得电磁波在外层墙体内的传播时延,再运用相邻发射天线和相邻接收天线延迟时之差估计内层墙体中电磁波的传播时延,最后各个通道内传播时延依次相加后运用后向投影算法获得目标精确成像。对成像位置精度、图像熵,以及输入输出目标杂波比等性能的分析和比较证明了该算法的有效性。     

BILATERAL COUNTERPARTY RISK UNDER FUNDING CONSTRAINTS—PART II: CVA

The correction in value of an over‐the‐counter derivative contract due to counterparty risk under funding constraints is represented as the value of a dividend‐paying option on the value of the contract clean of counterparty risk and excess funding costs. This representation allows one to analyze the structure of this correction, the so‐called Credit Valuation Adjustment (CVA for short), in terms of replacement cost/benefits, credit cost/benefits, and funding cost/benefits. We develop a reduced‐form backward stochastic differential equations (BSDE) approach to the problem of pricing and hedging the CVA. In the Markov setup, explicit CVA pricing and hedging schemes are formulated in terms of semilinear partial differential equations.     

A PRIMAL–DUAL ALGORITHM FOR BSDES

We generalize the primal–dual methodology, which is popular in the pricing of early‐exercise options, to a backward dynamic programming equation associated with time discretization schemes of (reflected) backward stochastic differential equations (BSDEs). Taking as an input some approximate solution of the backward dynamic program, which was precomputed, e.g., by least‐squares Monte Carlo, this methodology enables us to construct a confidence interval for the unknown true solution of the time‐discretized (reflected) BSDE at time 0. We numerically demonstrate the practical applicability of our method in two 5‐dimensional nonlinear pricing problems where tight price bounds were previously unavailable.     

GREEKS FORMULAS FOR AN ASSET PRICE MODEL WITH GAMMA PROCESSES

Greeks formulas of Delta, Rho, Vega, and Gamma are derived in closed form for asset price dynamics described by gamma processes and Brownian motions time‐changed by a gamma process. The model considered here includes many well‐known models of practical interest, such as the variance gamma model and the Black–Scholes model. Our approach is based upon the Malliavin calculus for jump processes by making full use of a scaling property of gamma processes with respect to the Girsanov transform. The existence of their variance is investigated. Numerical results are provided to illustrate that the derived Greeks formulas have faster rate of convergence relative to the finite difference method.     

Fluctuating Efforts and Interdependencies in Collaborative Work

We present a dynamical model of cooperative efforts comprised of concurrently performed, interrelated tasks. The model contains a stochastic component to account for temporal fluctuations both in task performance and in the effect of a given unit of work on the project as a whole. We show that as the number of concurrent tasks increases, so does the average completion time. Also, for fixed system size, the dynamics of individual project realizations can exhibit large deviations from the average when fluctuations increase past a certain threshold, causing long delays in completion times. These effects are in agreement with empirical observation. We also show that the negative effects of both large groups and long delays caused by fluctuations may be mitigated by arranging projects in a hierarchical or modular structure. Our model is applicable to any arrangement of interdependent tasks, providing an analytical prediction for the average completion time as well as a numerical threshold for the fluctuation strength beyond which long delays are likely. In conjunction with previous modeling techniques, it thus provides managers with a predictive tool to be used in the design of a project’s architecture.     

Designing communication architectures for interorganizational multimedia collaboration

Advances in computer and communication technologies have stimulated the integration of digital video and audio with computing, leading to the development of various computer‐assisted collaborations. In this article, we propose a multilevel conferencing paradigm called super conference for supporting collaborative interactions between geographically separated groups of users, with each group belonging to possibly a different organization. In a super conference, each participant must receive and display the composite media stream obtained by mixing media streams transmitted by all the other participants. Hierarchical communication architectures are naturally suited for media mixing in super conferences. We present algorithms for designing hierarchical mixing architectures that optimize real‐time end‐to‐end delays of media. In order to improve their real‐time performance further, we propose multistage mixing techniques by which mixers can carry out mixing concurrently with communication. Surprisingly, the optimal architectures for multistage mixing are widely different from those of monostage mixing (in which, mixing and media communication sequential as opposed to concurrent). Based on real‐time delay constraints of multimedia, we obtain interesting limits on the sizes of both super conferences and groups within super conferences in optimal hierarchical architectures, which go to show their high scalability in terms of both the maximum number of participants and the geographical separation between them.

At the Multimedia Laboratory at the University of California, San Diego, we have implemented a conferencing system on an environment of Sun SPARCstations equipped with digital multimedia hardware. As an interesting application of the conferencing system, we have developed a telepresenter by which users can remotely attend lectures in progress. We present initial experiences with the system.     

FROM SMILE ASYMPTOTICS TO MARKET RISK MEASURES

The left tail of the implied volatility skew, coming from quotes on out‐of‐the‐money put options, can be thought to reflect the market's assessment of the risk of a huge drop in stock prices. We analyze how this market information can be integrated into the theoretical framework of convex monetary measures of risk. In particular, we make use of indifference pricing by dynamic convex risk measures, which are given as solutions of backward stochastic differential equations, to establish a link between these two approaches to risk measurement. We derive a characterization of the implied volatility in terms of the solution of a nonlinear partial differential equation and provide a small time‐to‐maturity expansion and numerical solutions. This procedure allows to choose convex risk measures in a conveniently parameterized class, distorted entropic dynamic risk measures, which we introduce here, such that the asymptotic volatility skew under indifference pricing can be matched with the market skew. We demonstrate this in a calibration exercise to market implied volatility data.     

Chaos and the exchange rate

Chaotic exchange rate models are structural models built in discrete time (difference equations), and show that with orthodox assumptions (PPP, interest parity, etc) and introducing plausible nonlinearities in the dynamic equations, it is possible to obtain a model capable of giving rise to chaotic motion. However, none of these models is estimated, and the conclusions are based on simulations: the empirical validity of these models is not tested. In this paper, a continuous time (the exchange rate is obviously a continuous variable) exchange rate model is built as a non-linear set of three differential equations and its theoretical properties (steady state, stability, etc,) analysed. The model is then econometrically estimated in continuous time with Italian data and examined for the possible presence of chaotic motion. This paper also shows that the continuous time estimation of economic models built as systems of nonlinear differential equations is a very powerful tool in the hands of the profession.     

An explicitly solvable multi‐scale stochastic volatility model: Option pricing and calibration problems

We introduce an explicitly solvable multiscale stochastic volatility model that generalizes the Heston model. The model describes the dynamics of an asset price and of its two stochastic variances using a system of three Ito stochastic differential equations. The two stochastic variances vary on two distinct time scales and can be regarded as auxiliary variables introduced to model the dynamics of the asset price. Under some assumptions, the transition probability density function of the stochastic process solution of the model is represented as a one‐dimensional integral of an explicitly known integrand. In this sense the model is explicitly solvable. We consider the risk‐neutral measure associated with the proposed multiscale stochastic volatility model and derive formulae to price European vanilla options (call and put) in the multiscale stochastic volatility model considered. We use the thus‐obtained option price formulae to study the calibration problem, that is to study the values of the model parameters, the correlation coefficients of the Wiener processes defining the model, and the initial stochastic variances implied by the “observed” option prices using both synthetic and real data. In the analysis of real data, we use the S&P 500 index and to the prices of the corresponding options in the year 2005. The web site http://www.econ.univpm.it/recchioni/finance/w7 contains some auxiliary material including some animations that helps the understanding of this article. A more general reference to the work of the authors and their coauthors in mathematical finance is the web site http://www.econ.univpm.it/recchioni/finance . © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:862–893, 2009     

Binary funding impacts in derivative valuation

We discuss the binary nature of funding impact in derivative valuation. Under some conditions, funding is either a cost or a benefit, that is, one of the lending/borrowing rates does not play a role in pricing derivatives. When derivatives are priced, considering different lending/borrowing rates leads to semilinear backward stochastic differential equations (BSDEs) and partial differential equation (PDEs), and thus it is necessary to solve the equations numerically. However, once it can be guaranteed that only one of the rates affects pricing, linear equations can be recovered, and analytical formulae can be derived. Moreover, as a by‐product, our results explain how debt value adjustment (DVA) and funding benefits are dissimilar. It is often believed that considering both DVA and funding benefits results in a double‐counting issue but it will be shown that the two components are affected by different mathematical structures of derivative transactions. We find that funding benefit is related to the decreasing property of the payoff function, but this relationship decreases as the funding choices of underlying assets are transferred to repo markets.     

What is going on when nothing is going on? Exploring the role of the consumer in shaping waiting situations

Research on waiting in services focuses mainly on the role of companies in waiting situations. Much of the existing research envisages the consumer as a passive victim of the delays caused by companies. This article redresses the imbalance in research on waiting by exploring the role of consumers in waiting situations. A qualitative methodology is used, involving data collection through in‐depth interviews and personal diaries, to facilitate an in‐depth analysis of consumers’ waiting experiences. Additionally, a holistic approach is employed with a view to examining the ‘whole’ waiting experience rather than individual or isolated aspects of waiting. The main finding of the study is that consumers play an active and deliberate role in waiting situations. This is the first study to empirically identify the behaviours and initiatives undertaken by consumers when forced to wait for services. The results suggest that consumers play an active role in organising and reducing the real or perceived waiting time. They actively seek information on the length and causes of the wait. And, on occasions, the consumer may also be the cause of the wait or may even increase the delay by their actions. In contrast to most of the existing research on the topic of waiting, this study adopt a qualitative, in‐depth approach, with a multicultural sample, and a focus on the consumer whose role in waiting has previously been overlooked.     

Numerical pricing of American options under infinite activity Lévy processes

Under infinite activity Lévy models, American option prices can be obtained by solving a partial integro‐differential equation (PIDE), which has a singular kernel. With increasing degree of singularity, standard time‐stepping techniques may encounter difficulties. This study examines exponential time integration (ETI) for solving this problem and the performance of this scheme is compared with the Crank–Nicolson (CN) method and an implicit–explicit method in conjunction with an extrapolation (IMEX‐Extrap), in terms of computational speed and convergence orders. These findings indicate that ETI is faster and more accurate among PIDE‐based methods for solving the system of ordinary differential equations resulting from spatial discretization of the PIDE. For very singular problems, it is shown that the IMEX‐Extrap scheme becomes unfavorable compared with the other schemes as it is relatively more time consuming and the global convergence deteriorates from quadratic to linear, whereas the ETI scheme yields both point‐wise and global quadratic convergence. For illustration, under the infinite variation process, the IMEX‐Extrap achieves a precision of the order of 10^?4 in 663.016 s, whereas for the same set of parameters, the CN method and the ETI scheme reach an accuracy of the order of 10^?5 in 237.891 s and 22.772 s, respectively. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 31:809–829, 2011     

Markov chains under nonlinear expectation

In this paper, we consider continuous‐time Markov chains with a finite state space under nonlinear expectations. We define so‐called Q‐operators as an extension of Q‐matrices or rate matrices to a nonlinear setup, where the nonlinearity is due to model uncertainty. The main result gives a full characterization of convex Q‐operators in terms of a positive maximum principle, a dual representation by means of Q‐matrices, time‐homogeneous Markov chains under convex expectations, and a class of nonlinear ordinary differential equations. This extends a classical characterization of generators of Markov chains to the case of model uncertainty in the generator. We further derive an explicit primal and dual representation of convex semigroups arising from Markov chains under convex expectations via the Fenchel–Legendre transformation of the generator. We illustrate the results with several numerical examples, where we compute price bounds for European contingent claims under model uncertainty in terms of the rate matrix.