A level-2 reformulation–linearization technique bound for the quadratic assignment problem

This paper studies polyhedral methods for the quadratic assignment problem. Bounds on the objective value are obtained using mixed 0–1 linear representations that result from a reformulation–linearization technique (rlt). The rlt provides different “levels” of representations that give increasing strength. Prior studies have shown that even the weakest level-1 form yields very tight bounds, which in turn lead to improved solution methodologies. This paper focuses on implementing level-2. We compare level-2 with level-1 and other bounding mechanisms, in terms of both overall strength and ease of computation. In so doing, we extend earlier work on level-1 by implementing a Lagrangian relaxation that exploits block-diagonal structure present in the constraints. The bounds are embedded within an enumerative algorithm to devise an exact solution strategy. Our computer results are notable, exhibiting a dramatic reduction in nodes examined in the enumerative phase, and allowing for the exact solution of large instances.     

Continuous-time mean–variance portfolio selection with liability and regime switching

A continuous-time mean–variance model for individual investors with stochastic liability in a Markovian regime switching financial market, is investigated as a generalization of the model of Zhou and Yin [Zhou, X.Y., Yin, G., 2003. Markowitz’s mean–variance portfolio selection with regime switching: A continuous-time model, SIAM J. Control Optim. 42 (4), 1466–1482]. We assume that the risky stock’s price is governed by a Markovian regime-switching geometric Brownian motion, and the liability follows a Markovian regime-switching Brownian motion with drift, respectively. The evolution of appreciation rates, volatility rates and the interest rates are modulated by the Markov chain, and the Markov switching diffusion is assumed to be independent of the underlying Brownian motion. The correlation between the risky asset and the liability is considered. The objective is to minimize the risk (measured by variance) of the terminal wealth subject to a given expected terminal wealth level. Using the Lagrange multiplier technique and the linear-quadratic control technique, we get the expressions of the optimal portfolio and the mean–variance efficient frontier in closed forms. Further, the results of our special case without liability is consistent with those results of Zhou and Yin [Zhou, X.Y., Yin, G., 2003. Markowitz’s mean–variance portfolio selection with regime switching: A continuous-time model, SIAM J. Control Optim. 42 (4), 1466–1482].     

Continuous-time portfolio selection with liability: Mean–variance model and stochastic LQ approach

In this paper we formulate a continuous-time mean–variance portfolio selection model with multiple risky assets and one liability in an incomplete market. The risky assets’ prices are governed by geometric Brownian motions while the liability evolves according to a Brownian motion with drift. The correlations between the risky assets and the liability are considered. The objective is to maximize the expected terminal wealth while minimizing the variance of the terminal wealth. We derive explicitly the optimal dynamic strategy and the mean–variance efficient frontier in closed forms by using the general stochastic linear-quadratic (LQ) control technique. Several special cases are discussed and a numerical example is also given.     

Value at risk calculation through ARCH factor methodology: Proposal and comparative analysis

In this paper we put forward a new method to estimate value at risk (VaR), autoregressive conditional heteroskedastic (ARCH) factor, which combines multivariate analysis with ARCH models. Firstly, from a set of correlated portfolio risk factors, we derive a smaller uncorrelated risk factors set, by applying multivariate analysis. Secondly, we use ARCH schemes to model uncorrelated factors historical behaviour. Thirdly, we use the estimated models to predict future values for factors standard deviation. From them, VaR calculation is immediate. In this way, ARCH factor methodology overcomes the multivariate ARCH models drawbacks, which, in practice, make these unworkable for VaR calculation purposes. We apply the proposed methodology over a set of foreign exchange risk exposed portfolios, obtaining better results than those reached when J.P. Morgan’s Riskmetrics is used.     

Dynamic value at risk under optimal and suboptimal portfolio policies

At present, all value at risk (VaR) implementations – i.e., all risk measures of the “maximum loss at a given level of confidence” type – are based on the assumption that the portfolio mix will not change before the VaR horizon. This hypothesis may be unrealistic, especially when the VaR horizon is established by the regulators (BIS). At the opposite, we measure VaR dynamically, i.e., taking into consideration portfolio mix adjustments over time: adjustments do not occur continuously, since they are costly. We allow both optimal rebalancing policies, which entail changing the portfolio mix whenever it is too far from the optimal one, and suboptimal policies, which mean adjusting at pre-fixed dates.We show that in both cases usual VaR measures underestimate portfolio losses, even if the underlying returns are normal. We study the dependence of the misestimate on the VaR horizon, the initial portfolio mix and the risk aversion of the portfolio manager, which in turn determines the frequency of interventions. The bias can be more relevant over one day than over longer horizons and even if the initial portfolio is nearly optimal. We also perform backtesting and estimate a “coherent” risk measure, namely conditional VaR, which confirms the inappropriateness of the usual, static VaR.     

Value-at-Risk via mixture distributions reconsidered

Value-at-Risk (VaR) has evolved as one of the most prominent measures of downside risk in financial markets. Zhang and Cheng [M.-H. Zhang, Q.-S. Cheng, An Approach to VaR for capital markets with Gaussian mixture, Applied Mathematics and Computation 168 (2005) 1079–1085] proposed an approach to VaR for daily returns based on Gaussian mixtures, which have become rather popular in empirical economics and finance since the seminal paper of Hamilton [J.D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica 57 (2) (1989) 357–384]. However, they do not conduct tests to assess the accuracy of the mixture-implied VaR measures. Recently, Guidolin and Timmermann [M. Guidolin, A. Timmermann, Term structure of risk under alternative econometric specifications, Journal of Econometrics, 131 (2006) 285–308] showed that Markov mixture models do well in measuring VaR at a monthly frequency, but the results may not hold for daily returns due to their more pronounced non-Gaussian features. This paper provides an extensive application of various Markov mixture models to VaR for daily returns of major European stock markets, including out-of-sample backtesting. To accommodate the properties of daily returns, we consider both Gaussian and Student’s t mixtures, and we compare the performance of both uni- and multivariate models under different parameter updating schemes. We find that a univariate mixture of two Student’s t distributions performs best overall. However, by the example of the recent turmoil in financial markets, we also highlight a weak point of the approach.     

转点识别在VaR估计中的应用

本文深入分析了VaR估计结果对市场比率运动规律假设的依赖性 .文献 [1 ],[2 ][4 ]都没有考虑市场因素出现结构性的转变对VaR估计的影响 .事实上 ,市场因素受到其它各种因素的影响 ,很可能发生结构性的转变 .故本文在引入转点识别的基础上对VaR估计方法作出改进 ,从而把市场因素结构性转变引入到VaR估计之中 ,且随机模拟实验结果表明引入转点后的预报有更高的可信度 .     

Limiting Connection between Discrete and Continuous Time Forward Interest Rate Curve Models

We study discrete time Heath–Jarrow–Morton (HJM) type of interest rate curve models, where the forward interest rates – in contrast to the classical HJM models – are driven by a random field. Our main aim is to investigate the relationship between the discrete time forward interest rate curve model and its continuous time counterpart. We derive a general result on the convergence of discrete time models and we give special focus on the nearly unit root spatial autoregression model.     

Static and dynamic VaR constrained portfolios with application to delegated portfolio management

We give a closed-form solution to the single-period portfolio selection problem with a Value-at-Risk (VaR) constraint in the presence of a set of risky assets with multivariate normally distributed returns and the risk-less account, without short sales restrictions. The result allows to obtain a very simple, myopic dynamic portfolio policy in the multiple period version of the problem. We also consider mean-variance portfolios under a probabilistic chance (VaR) constraint and give an explicit solution. We use this solution to calculate explicitly the bonus of a portfolio manager to include a VaR constraint in his/her portfolio optimization, which we refer to as the price of a VaR constraint.     

A Generalized Rational Interpolation Problem and the Solution of the Welch–Berlekamp Key Equation

We show that an algorithm designed to solve the Welch–Berlekamp key equation may also be used to solve a more general problem, which can be regarded as a finite analogue of a generalized rational interpolation problem. As a consequence, we show that a single algorithm exists which can solve both Berlekamp's classical key equation (usually solved by the Berlekamp–Massey algorithm) and the Welch–Berlekamp key equation which arise in the decoding of Reed–Solomon codes.     

A note on portfolios with risk-free internal gains

In this paper we show that if a not-necessarily-self-financing portfolio has instantaneously riskless internal gains, then on an infinitesimal time-interval, the increase in the internal gains on the portfolio is the same as the change in the price of that amount of bonds which has the same wealth as the portfolio has. As an application of this result, we derive the Black–Scholes PDE by using the original derivation of Black and Scholes, and we show that it can be made completely rigorous.     

Continuous time mean‐variance optimal portfolio allocation under jump diffusion: An numerical impulse control approach

We present efficient partial differential equation methods for continuous time mean‐variance portfolio allocation problems when the underlying risky asset follows a jump‐diffusion. The standard formulation of mean‐variance optimal portfolio allocation problems, where the total wealth is the underlying stochastic process, gives rise to a one‐dimensional (1D) nonlinear Hamilton–Jacobi–Bellman (HJB) partial integrodifferential equation (PIDE) with the control present in the integrand of the jump term, and thus is difficult to solve efficiently. To preserve the efficient handling of the jump term, we formulate the asset allocation problem as a 2D impulse control problem, 1D for each asset in the portfolio, namely the bond and the stock. We then develop a numerical scheme based on a semi‐Lagrangian timestepping method, which we show to be monotone, consistent, and stable. Hence, assuming a strong comparison property holds, the numerical solution is guaranteed to converge to the unique viscosity solution of the corresponding HJB PIDE. The correctness of the proposed numerical framework is verified by numerical examples. We also discuss the effects on the efficient frontier of realistic financial modeling, such as different borrowing and lending interest rates, transaction costs, and constraints on the portfolio, such as maximum limits on borrowing and solvency. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 664–698, 2014     

Composite generalized Laguerre–Legendre pseudospectral method for Fokker–Planck equation in an infinite channel

In this paper, we propose a composite generalized Laguerre–Legendre pseudospectral method for the Fokker–Planck equation in an infinite channel, which behaves like a parabolic equation in one direction, and behaves like a hyperbolic equation in other direction. We establish some approximation results on the composite generalized Laguerre–Legendre–Gauss–Radau interpolation, with which the convergence of proposed composite scheme follows. An efficient implementation is provided. Numerical results show the spectral accuracy in space of this approach and coincide well with theoretical analysis. The approximation results and techniques developed in this paper are also very appropriate for many other problems on multiple-dimensional unbounded domains, which are not of standard types.     

Optimal Covering of Plane Domains by Circles Via Hyperbolic Smoothing

We consider the problem of optimally covering plane domains by a given number of circles. The mathematical modeling of this problem leads to a min–max–min formulation which, in addition to its intrinsic multi-level nature, has the significant characteristic of being non-differentiable. In order to overcome these difficulties, we have developed a smoothing strategy using a special class C^∞ smoothing function. The final solution is obtained by solving a sequence of differentiable subproblems which gradually approach the original problem. The use of this technique, called Hyperbolic Smoothing, allows the main difficulties presented by the original problem to be overcome. A simplified algorithm containing only the essential of the method is presented. For the purpose of illustrating both the actual working and the potentialities of the method, a set of computational results is presented.     

Asymptotic Gauss–Jacobi quadrature error estimation for Schwarz–Christoffel integrals

Numerical conformal mapping packages based on the Schwarz–Christoffel formula have been in existence for a number of years. Various authors, for good reasons of practical efficiency, have chosen to use composite n-point Gauss–Jacobi rules for the estimation of the Schwarz–Christoffel path integrals. These implementations rely on an ad hoc, but experimentally well-founded, heuristic for selecting the spacing of the integration end-points relative to the position of the nearby integrand singularities. In the present paper we derive an explicitly computable estimate, asymptotic as n→∞, for the relevant Gauss–Jacobi quadrature error. A numerical example illustrates the potential accuracy of the estimate even at low values of n. It is apparent that the error estimate will allow the adaptive construction of composite rules in a manner that is more efficient than has been possible hitherto.     

Real-time assessment of value-at-risk and volatility accuracy

An important question for corporate finance officers is whether risk assessments, such as Value at Risk (VaR), are currently accurate. In contrast to past research on assessing the accuracy of VaR, volatility, and related density estimates, which has focused on backtesting using large samples of fixed size, we develop a class of sequential testing tools for on-line, real-time assessment, based on time windows that vary adaptively with the data.The VaR is determined by a single point of the estimated distribution of the portfolio “gain” and may be positive (profit) or negative (loss). Previous literature has dichotomically tested the sequence of VaR forecasts or the sequence of estimated distributions. A pure test is obtained by converting each observed gain into a binary value indicating whether it was covered by the corresponding VaR forecast or not. A more powerful test results from using the entire distribution, by transforming the observed gain to a random variable that has a known distribution when the forecast is accurate. This, however, also detects errors unrelated to the accuracy of estimating VaR and other measures of risk.We propose an adjustable, continuous compromise between detection power and purity, where “power” refers to quick detection of systematic bias and “purity” refers to insensitivity to errors not relevant to VaR estimation accuracy. Previous approaches focused on either extreme of this continuum. However, we point out that there are few practical situations for which the choice of either extreme would be optimal. Instead, we suggest a compromise that would be much better and very useful in most practical applications.     

Computational adequacy for recursive types in models of intuitionistic set theory

This paper provides a unifying axiomatic account of the interpretation of recursive types that incorporates both domain-theoretic and realizability models as concrete instances. Our approach is to view such models as full subcategories of categorical models of intuitionistic set theory. It is shown that the existence of solutions to recursive domain equations depends upon the strength of the set theory. We observe that the internal set theory of an elementary topos is not strong enough to guarantee their existence. In contrast, as our first main result, we establish that solutions to recursive domain equations do exist when the category of sets is a model of full intuitionistic Zermelo–Fraenkel set theory. We then apply this result to obtain a denotational interpretation of FPC, a recursively typed lambda-calculus with call-by-value operational semantics. By exploiting the intuitionistic logic of the ambient model of intuitionistic set theory, we analyse the relationship between operational and denotational semantics. We first prove an “internal” computational adequacy theorem: the model always believes that the operational and denotational notions of termination agree. This allows us to identify, as our second main result, a necessary and sufficient condition for genuine “external” computational adequacy to hold, i.e. for the operational and denotational notions of termination to coincide in the real world. The condition is formulated as a simple property of the internal logic, related to the logical notion of 1-consistency. We provide useful sufficient conditions for establishing that the logical property holds in practice. Finally, we outline how the methods of the paper may be applied to concrete models of FPC. In doing so, we obtain computational adequacy results for an extensive range of realizability and domain-theoretic models.     

On the Convergence of Bounded J-Fractions on the Resolvent Set of the Corresponding Second Order Difference Operator

We study connections between continued fractions of type J and spectral properties of second order difference operators with complex coefficients. It is known that the convergents of a bounded J-fraction are diagonal Padé approximants of the Weyl function of the corresponding difference operator and that a bounded J-fraction converges uniformly to the Weyl function in some neighborhood of infinity. In this paper we establish convergence in capacity in the unbounded connected component of the resolvent set of the difference operator and specify the rate of convergence. Furthermore, we show that the absence of poles of Padé approximants in some subdomain implies already local uniform convergence. This enables us to verify the Baker–Gammel–Wills conjecture for a subclass of Weyl functions. For establishing these convergence results, we study the ratio and the nth root asymptotic behavior of Padé denominators of bounded J-fractions and give relations with the Green function of the unbounded connected component of the resolvent set. In addition, we show that the number of “spurious” Padé poles in this set may be bounded.     

Boundary Amenability of Groups and Positive Scalar Curvature

We show that if a finitely presented discrete group acts amenably on the boundary of a metric compactification of itself, then the Gromov–Lawson–Rosenberg conjecture holds for . Thus, if M is a compact aspherical Spin manifold with fundamental group , then M does not admit a Riemannian metric with positive scalar curvature. The class of groups having such a property includes hyperbolic groups and amenable groups and is closed under semidirect product.