A high-order compact method for nonlinear Black–Scholes option pricing equations of American options

Due to transaction costs, illiquid markets, large investors or risks from an unprotected portfolio the assumptions in the classical Black–Scholes model become unrealistic and the model results in nonlinear, possibly degenerate, parabolic diffusion–convection equations. Since in general, a closed-form solution to the nonlinear Black–Scholes equation for American options does not exist (even in the linear case), these problems have to be solved numerically. We present from the literature different compact finite difference schemes to solve nonlinear Black–Scholes equations for American options with a nonlinear volatility function. As compact schemes cannot be directly applied to American type options, we use a fixed domain transformation proposed by ?ev?ovi? and show how the accuracy of the method can be increased to order four in space and time.     

Computation of probabilistic linear programming problems involving normal and log-normal random variables with a joint constraint

A method for solving single- and multi-objective probabilistic linear programming problems with a joint constraint is presented. It is assumed that the parameters in the probabilistic linear programming problems are random variables, and the probabilistic problem is converted to an equivalent deterministic mathematical programming problem. In this paper the parameters are generally considered as normal and log-normal random variables. A non-linear programming method is used to solve the single-objective deterministic problem, and a fuzzy programming method is used to solve the multi-objective deterministic problem. Finally, a numerical example is presented to illustrate the methodology.     

Finite-time stability and stabilisation of singular discrete-time linear positive systems with time-varying delay

ABSTRACT

In this paper, the problem of finite-time stability and stabilisation for positive singular discrete-time linear systems with time-varying delay is investigated. We first present novel delay-dependent sufficient conditions for positivity and finite-time stability of the unforced systems. We then apply the obtained results to solve finite-time stabilisation problem of the considered systems. The sufficient conditions for the positivity and finite-time stabilisable of such systems are formulated in terms of a standard linear programming (LP) problem. Numerical examples are provided to illustrate the effectiveness and advantages of our results.     

Adaptive fuzzy output-feedback fault-tolerant tracking control of a class of uncertain nonlinear switched systems

ABSTRACT

This paper considers the output-feedback fault-tolerant tracking control problem for a class of uncertain nonlinear switched systems with nonlinear faults and strict-feedback form, where the faults which are nonaffine occur on the actuator. As a kind of specialised function approximating tool, fuzzy logic systems (FLSs), are employed to approximate the unknown smooth nonlinear functions. A switched fuzzy observer is designed to address the problem of unmeasurable states, filtered signals are used to address algebraic loop problem and the average dwell time (ADT) method is further utilised to prove the stability of the resulting closed-loop systems under a type of slowly switching signals. Based on the backstepping recursive design technique and Lyapunov function method, an adaptive fuzzy output-feedback control scheme is developed. The developed control method can ensure all the signals are semi-globally uniformly ultimately bounded (SGUUB) and the system output tracks the reference signal tightly even if unknown fault occurs. A simulation carried on an example demonstrates the validity of the obtained control scheme.     

Bringing Scheme programming to the iPhone—Experience

The iPhone SDK provides a powerful platform for the development of applications that make use of iPhone capabilities, such as sensors, GPS, Wi‐Fi, or Bluetooth connectivity. We observe that so far the development of iPhone applications has mostly been restricted to using Objective‐C. However, developing applications in plain Objective‐C on the iPhone OS suffers from limitations, such as the need for explicit memory management and lack of syntactic extension mechanism. Moreover, when developing distributed applications in Objective‐C, programmers have to manually deal with distribution concerns, such as service discovery, remote communication, and failure handling. In this paper, we discuss our experience in porting the Scheme programming language to the iPhone OS and how it can be used together with Objective‐C to develop iPhone applications. To support the interaction between Scheme programs and the underlying iPhone APIs, we have implemented a language symbiosis layer that enables programmers to access the iPhone SDK libraries from Scheme. In addition, we have designed high‐level distribution constructs to ease the development of distributed iPhone applications in an event‐driven style. We validate and discuss these constructs with a series of examples, including an iPod controller, a maps application, and a distributed multiplayer Scrabble‐like game. We discuss the lessons learned from this experience for other programming language ports to mobile platforms. Copyright © 2011 John Wiley & Sons, Ltd.     

Access control in feature-oriented programming

In feature-oriented programming (FOP) a programmer decomposes a program in terms of features. Ideally, features are implemented modularly so that they can be developed in isolation. Access control mechanisms in the form of access or visibility modifiers are an important ingredient to attain feature modularity as they allow programmers to hide and expose internal details of a module’s implementation. But developers of contemporary feature-oriented languages have not considered access control mechanisms so far. The absence of a well-defined access control model for FOP breaks encapsulation of feature code and leads to unexpected program behaviors and inadvertent type errors. We raise awareness of this problem, propose three feature-oriented access modifiers, and present a corresponding access modifier model. We offer an implementation of the model on the basis of a fully-fledged feature-oriented compiler. Finally, by analyzing ten feature-oriented programs, we explore the potential of feature-oriented modifiers in FOP.     

Simultaneous estimation and factor selection in quantile regression via adaptive sup-norm regularization

Some regularization methods, including the group lasso and the adaptive group lasso, have been developed for the automatic selection of grouped variables (factors) in conditional mean regression. In many practical situations, such a problem arises naturally when a set of dummy variables is used to represent a categorical factor and/or when a set of basis functions of a continuous variable is included in the predictor set. Complementary to these earlier works, the simultaneous and automatic factor selection is examined in quantile regression. To incorporate the factor information into regularized model fitting, the adaptive sup-norm regularized quantile regression is proposed, which penalizes the empirical check loss function by the sum of factor-wise adaptive sup-norm penalties. It is shown that the proposed method possesses the oracle property. A simulation study demonstrates that the proposed method is a more appropriate tool for factor selection than the adaptive lasso regularized quantile regression.     

Computing multiple-output regression quantile regions

A procedure relying on linear programming techniques is developed to compute (regression) quantile regions that have been defined recently. In the location case, this procedure allows for computing halfspace depth regions even beyond dimension two. The corresponding algorithm is described in detail, and illustrations are provided both for simulated and real data. The efficiency of a Matlab implementation of the algorithm¹ is also investigated through extensive simulations.     

LP bounds in various constraint programming approaches for orthogonal packing

We consider the 2D orthogonal feasibility problem (OPP-2) and the 2D strip packing problem (SPP-2). Given a set of rectangular items, the OPP-2 is to decide whether all items can be orthogonally packed into the given rectangular container; the SPP-2 is to find a packing of all items occupying the minimal height of the given semi-infinite strip. Both OPP-2 and SPP-2 are considered without items rotation. We investigate the known Constraint Programming (CP) approaches for OPP-2, in particular the dichotomy and disjunctive branching strategies and adapt 1D relaxation bounds based on linear programming (LP) into the constraint propagation process of the CP. Using the dichotomic search procedure the developed methods for OPP-2 are transformed for the case of SPP-2. Numerical results demonstrate the efficiency of the proposed strategies and of the combination of CP and LP-based pruning rules.