Volatility skews and extensions of the Libor market model

The paper considers extensions of the Libor market model to markets with volatility skews in observable option prices. The family of forward rate processes is expanded to include diffusions with non-linear forward rate dependence, and efficient techniques for calibration to quoted prices of caps and swaptions are discussed. Special emphasis is put on generalized CEV processes for which closed-form expressions for cap and swaption prices are derived. Modifications of the CEV process which exhibit more appealing growth and boundary characteristics are also discussed. The proposed models are investigated numerically through Crank–Nicholson finite difference schemes and Monte Carlo simulations.     

Numerical methods for a coupled system of differential equations arising from a thermal ignition problem

This article is concerned with monotone iterative methods for numerical solutions of a coupled system of a first‐order partial differential equation and an ordinary differential equation which arises from fast‐igniting catalytic converters in automobile engineering. The monotone iterative scheme yields a straightforward marching process for the corresponding discrete system by the finite‐difference method, and it gives not only a computational algorithm for numerical solutions of the problem but also the existence and uniqueness of a finite‐difference solution. Particular attention is given to the “finite‐time” blow‐up property of the solution. In terms of minimal sequence of the monotone iterations, some necessary and sufficient conditions for the blow‐up solution are obtained. Also given is the convergence of the finite‐difference solution to the continuous solution as the mesh size tends to zero. Numerical results of the problem, including a case where the continuous solution is explicitly known, are presented and are compared with the known solution. Special attention is devoted to the computation of the blow‐up time and the critical value of a physical parameter which determines the global existence and the blow‐up property of the solution. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Polynomial viscosity methods for multispecies kinematic flow models

Multispecies kinematic flow models are defined by systems of strongly coupled, nonlinear first‐order conservation laws. They arise in various applications including sedimentation of polydisperse suspensions and multiclass vehicular traffic. Their numerical approximation is a challenge since the eigenvalues and eigenvectors of the corresponding flux Jacobian matrix have no closed algebraic form. It is demonstrated that a recently introduced class of fast first‐order finite volume solvers, called polynomial viscosity matrix (PVM) methods [M. J. Castro Díaz and E. Fernández‐Nieto, SIAM J Sci Comput 34 (2012), A2173–A2196], can be adapted to multispecies kinematic flows. PVM methods have the advantage that they only need some information about the eigenvalues of the flux Jacobian, and no spectral decomposition of a Roe matrix is needed. In fact, the so‐called interlacing property (of eigenvalues with known velocity functions), which holds for several important multispecies kinematic flow models, provides sufficient information for the implementation of PVM methods. Several variants of PVM methods (differing in polynomial degree and the underlying quadrature formula to approximate the Roe matrix) are compared by numerical experiments. It turns out that PVM methods are competitive in accuracy and efficiency with several existing methods, including the Harten, Lax, and van Leer method and a spectral weighted essentially non‐oscillatory scheme that is based on the same interlacing property. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1265–1288, 2016     

Controlled selection a technique in random sampling

Summary The paper gives a method of constructing a random sampling design corresponding to a given population and a given sample sizen by means of which the probability of drawing a non-preferred combination of units can be reduced up to a certain level. Appropriate methods of analysing the data obtained from samples selected by this procedure are also given. A numerical example is given to illustrate the method developed in the paper.     

Aggregated semi-Markov repairable systems with history-dependent up and down states

Some states in the aggregated semi-Markov repairable system with history-dependent up and down states are changeable in the sense that whether those physical states are up and down depends on the immediately preceding state of the system evolution process. Two reliability indices of the system, the frequency of failures and the time to the first system failure are given. The Laplace–Stieltjes transforms of several time distributions in a cycle, such as the up and down time, the total time the system is in the up, down and changeable states, the length of a single sojourn in the up, down and changeable states are derived. The means of them are also presented. Markov renewal theory, transform and matrix methods are employed for getting these performance measures. A numerical example is given to illustrate the results in the paper.     

Knowledge‐based scenario tree generation methods and application in multiperiod portfolio selection problem

A scenario tree is an efficient way to represent a stochastic data process in decision problems under uncertainty. This paper addresses how to efficiently generate appropriate scenario trees. A knowledge‐based scenario tree generation method is proposed; the new method is further improved by accounting for subjective judgements or expectations about the random future. Compared with existing approaches, complicated mathematical models and time‐consuming estimation, simulation and optimization problem solution are avoided in our knowledge‐based algorithms, and large‐scale scenario trees can be quickly generated. To show the advantages of the new algorithms, a multiperiod portfolio selection problem is considered, and a dynamic risk measure is adopted to control the intermediate risk, which is superior to the single‐period risk measure used in the existing literature. A series of numerical experiments are carried out by using real trading data from the Shanghai stock market. The results show that the scenarios generated by our algorithms can properly represent the underlying distribution; our algorithms have high performance, say, a scenario tree with up to 10,000 scenarios can be generated in less than a half minute. The applications in the multiperiod portfolio management problem demonstrate that our scenario tree generation methods are stable, and the optimal trading strategies obtained with the generated scenario tree are reasonable, efficient and robust. Copyright © 2013 John Wiley & Sons, Ltd.     

Dahlquist's first barrier for multistage multistep formulas

Dahlquist's proof of his barrier for the order of stable linear multistep methods is combined with Reimer's proof of the corresponding barrier for multistep multiderivative methods. This leads to a shortening of Reimer's original proof and gives lower bounds for the error constant. These bounds are then studied for high error order and are used to model the optimal order and stepsize selection in an idealized integration code. Dedicated to Professor Germund Dahlquist on the occasion of his sixtieth birthday.     

Start‐up demonstration tests: models,methods and applications,with some unifications

Start‐up demonstration tests were first discussed in the quality/reliability literature about three decades ago. Since then, many variations of these tests have been introduced, and the corresponding distributional characteristics and inferential methods have also been studied. All these developments, based on independent and identically distributed binary trials, have been further generalized to some other forms of trials such as Markov‐dependent trials, exchangeable trials and multistate trials. In this paper, we provide a comprehensive review of all these results and highlight some unifications of the results. We also describe a general estimation method and then present several numerical examples to illustrate some of the models and methods described here. Finally, a number of open issues in this area of research are pointed out. Copyright © 2014 John Wiley & Sons, Ltd.     

Computation of viscous or nonviscous conservation law by domain decomposition based on asymptotic analysis

An accurate and efficient numerical method has been developed for a nonlinear diffusion convection-dominated problem. The scheme combines asymptotic methods with usual solution techniques for hyperbolic problems. After having localized shock or corner layers and rescaling, first terms of the inner expansion are computed. Using the same concepts gives a method to compute a very accurate solution of the nonlinear conservation law. Because our numerical scheme is based on a uniform approximation throughout the domain, the shock is localized very accurately and there is practically no smearing out. Numerical computations are presented. Another novel feature is the ability to break down the problem according to subdomains of different local behavior, based on asymptotic analysis, which may make it feasible to do computations with different processors.     

Markov interest rate models

A general procedure for creating Markovian interest rate models is presented. The models created by this procedure automatically fit within the HJM framework and fit the initial term structure exactly. Therefore they are arbitrage free. Because the models created by this procedure have only one state variable per factor, twoand even three-factor models can be computed efficiently, without resorting to Monte Carlo techniques. This computational efficiency makes calibration of the new models to market prices straightforward. Extended Hull- White, extended CIR, Black-Karasinski, Jamshidian's Brownian path independent models, and Flesaker and Hughston's rational log normal models are one-state variable models which fit naturally within this theoretical framework. The ‘separable’ n-factor models of Cheyette and Li, Ritchken, and Sankarasubramanian - which require n(n + 3)/2 state variables - are degenerate members of the new class of models with n(n + 3)/2 factors. The procedure is used to create a new class of one-factor models, the ‘β-η models.’ These models can match the implied volatility smiles of swaptions and caplets, and thus enable one to eliminate smile error. The β-η models are also exactly solvable in that their transition densities can be written explicitly. For these models accurate - but not exact - formulas are presented for caplet and swaption prices, and it is indicated how these closed form expressions can be used to efficiently calibrate the models to market prices.     

The Lévy Swap Market Model

Models driven by Lévy processes are attractive since they allow for better statistical fitting than classical diffusion models. The dynamics of the forward swap rate process is derived in a semimartingale setting and a Lévy swap market model is introduced. In order to guarantee positive rates, the swap rates are modelled as ordinary exponentials. The model starts with the most distant rate, which is driven by a non‐homogeneous Lévy process. Via backward induction the remaining swap rates are constructed such that they become martingales under the corresponding forward swap measures. Finally it is shown how swaptions can be priced using bilateral Laplace transforms.     

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     

Numerical valuation of discrete double barrier options

In the present paper we explore the problem for pricing discrete barrier options utilizing the Black-Scholes model for the random movement of the asset price. We postulate the problem as a path integral calculation by choosing approach that is similar to the quadrature method. Thus, the problem is reduced to the estimation of a multi-dimensional integral whose dimension corresponds to the number of the monitoring dates.We propose a fast and accurate numerical algorithm for its valuation. Our results for pricing discretely monitored one and double barrier options are in agreement with those obtained by other numerical and analytical methods in Finance and literature. A desired level of accuracy is very fast achieved for values of the underlying asset close to the strike price or the barriers.The method has a simple computer implementation and it permits observing the entire life of the option.     

Dynamic updates of the barrier parameter in primal-dual methods for nonlinear programming

We introduce a framework in which updating rules for the barrier parameter in primal-dual interior-point methods become dynamic. The original primal-dual system is augmented to incorporate explicitly an updating function. A Newton step for the augmented system gives a primal-dual Newton step and also a step in the barrier parameter. Based on local information and a line search, the decrease of the barrier parameter is automatically adjusted. We analyze local convergence properties, report numerical experiments on a standard collection of nonlinear problems and compare our results to a state-of-the-art interior-point implementation. In many instances, the adaptive algorithm reduces the number of iterations and of function evaluations. Its design guarantees a better fit between the magnitudes of the primal-dual residual and of the barrier parameter along the iterations.     

交叉货币百慕大式互换期权的定价

本文首先研究了涉及两种货币市场的Hull-White随机利率模型.以此为基础,本文给出了交叉货币百慕大式互换期权的定价公式.由于无法得到显式定价公式,我们使用了Least Squared Monte-Carlo(LSM)算法来确定期权的最优执行时刻.最后本文给出了数值计算方面的结果.     

农业银行境外宝汇率挂钩型理财产品的研究

通过It公式建立了农业银行境外宝汇率挂钩型理财产品的数学模型,在一定的挂钩条件下,得到模型的显式表达式.与此同时,考虑了该问题的反问题,确定方程中的障碍比例系数c.最后我们采用两种方法对原问题进行蒙特卡罗模拟,得到了相应的数值结果.     

On the parallel complexity of hierarchical clustering and CC‐complete problems

Complex data sets are often unmanageable unless they can be subdivided and simplified in an intelligent manner. Clustering is a technique that is used in data mining and scientific analysis for partitioning a data set into groups of similar or nearby items. Hierarchical clustering is an important and well‐studied clustering method involving both top‐down and bottom‐up subdivisions of data. In this article we address the parallel complexity of hierarchical clustering. We describe known sequential algorithms for top‐down and bottom‐up hierarchical clustering. The top‐down algorithm can be parallelized, and when there are n points to be clustered, we provide an O(log n)‐time, n²‐processor Crew Pram algorithm that computes the same output as its corresponding sequential algorithm. We define a natural decision problem based on bottom‐up hierarchical clustering, and add this HIERARCHICAL CLUSTERING PROBLEM (HCP) to the slowly growing list of CC‐complete problems, thereby showing that HCP is one of the computationally most difficult problems in the COMPARATOR CIRCUIT VALUE PROBLEM class. This class contains a variety of interesting problems, and now for the first time a problem from data mining as well. By proving that HCP is CC‐complete, we have demonstrated that HCP is very unlikely to have an NC algorithm. This result is in sharp contrast to the NC algorithm which we give for the top‐down sequential approach, and the result surprisingly shows that the parallel complexities of the top‐down and bottom‐up approaches are different, unless CC equals NC. In addition, we provide a compendium of all known CC‐complete problems. © 2008 Wiley Periodicals, Inc. Complexity, 2008     

On a Model for Fires in Tunnel Network Structures

A one dimensional model to describe the dynamics of tunnelfires is used to model tunnel network structures. The corresponding conditions at the nodes are formulated. A numerical methods is presented. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two Exotic Lookback Options

This paper formally analyses two exotic options with lookback features, referred to as extreme spread lookback options and look‐barrier options, first introduced by Bermin. The holder of such options receives partial protection from large price movements in the underlying, but at roughly the cost of a plain vanilla contract. This is achieved by increasing the leverage through either floating the strike price (for the case of extreme spread options) or introducing a partial barrier window (for the case of look‐barrier options). We show how to statically replicate the prices of these hybrid exotic derivatives with more elementary European binary options and their images, using new methods first introduced by Buchen and Konstandatos. These methods allow considerable simplification in the analysis, leading to closed‐form representations in the Black–Scholes framework.     

Extended Lagrangian approach for the defocusing nonlinear Schrödinger equation

We study the defocusing nonlinear Schrödinger (NLS) equation written in hydrodynamic form through the Madelung transform. From the mathematical point of view, the hydrodynamic form can be seen as the Euler–Lagrange equations for a Lagrangian submitted to a differential constraint corresponding to the mass conservation law. The dispersive nature of the NLS equation poses some major numerical challenges. The idea is to introduce a two‐parameter family of extended Lagrangians, depending on a greater number of variables, whose Euler–Lagrange equations are hyperbolic and accurately approximate NLS equation in a certain limit. The corresponding hyperbolic equations are studied and solved numerically using Godunov‐type methods. Comparison of exact and asymptotic solutions to the one‐dimensional cubic NLS equation (“gray” solitons and dispersive shocks) and the corresponding numerical solutions to the extended system was performed. A very good accuracy of such a hyperbolic approximation was observed.