Hongxia Yin

No-arbitrage interpolation of the option price function and its reformulation

Several risk management and exotic option pricing models have been proposed in the literature which may price European options correctly. A prerequisite of these models is the interpolation of the market implied volatilities or the European option price function. However, the no-arbitrage principle places shape restrictions on the option price function. In this paper, an interpolation method is developed to preserve the shape of the option price function. The interpolation is optimal in terms of minimizing the distance between the implied risk-neutral density and the prior approximation function in L2-norm, which is important when only a few observations are available. We reformulate the problem into a system of semismooth equations so that it can be solved efficiently.

A Newton Method for Shape-Preserving Spline Interpolation

In 1986, Irvine, Marin, and Smith proposed a Newton-type method for shape-preserving interpolation and, based on numerical experience, conjectured its quadratic convergence. In this paper, we prove local quadratic convergence of their method by viewing it as a semismooth Newton method. We also present a modification of the method which has global quadratic convergence. Numerical examples illustrate the results.

The Convergence of a Levenberg–Marquardt Method for Nonlinear Inequalities

In this paper, we consider the least l 2-norm solution for a possibly inconsistent system of nonlinear inequalities. The objective function of the problem is only first-order continuously differentiable. By introducing a new smoothing function, the problem is approximated by a family of parameterized optimization problems with twice continuously differentiable objective functions. Then a Levenberg–Marquardt algorithm is proposed to solve the parameterized smooth optimization problems. It is proved that the algorithm either terminates finitely at a solution of the original inequality problem or generates an infinite sequence. In the latter case, the infinite sequence converges to a least l 2-norm solution of the inequality problem. The local quadratic convergence of the algorithm was produced under some conditions.

A Strongly Semismooth Integral Function and Its Application

As shown by an example, the integral function f :ℝn → ℝ, defined by f(x) = ∫ab[B(x, t)]+g(t) dt, may not be a strongly semismooth function, even if g(t) ≡ 1 and B is a quadratic polynomial with respect to t and infinitely many times smooth with respect to x. We show that f is a strongly semismooth function if g is continuous and B is affine with respect to t and strongly semismooth with respect to x, i.e., B(x, t) = u(x)t + v(x), where u and v are two strongly semismooth functions in ℝn. We also show that f is not a piecewise smooth function if u and v are two linearly independent linear functions, g is continuous and g ≢ 0 in [a, b], and n ≥ 2. We apply the first result to the edge convex minimum norm network interpolation problem, which is a two-dimensional interpolation problem.

Convergence rate of Newton's method for L 2 spectral estimation

In the paper, we prove the Hölder continuous property of the Jacobian of the function generated from the dual of the power spectrum estimation problem. It follows that the convergence of the Newton method for the problem is at least of order where m is the order of the trigonometric bases. This result theoretically confirms the numerical observation by Potter (1990) and Cole and Goodrich (1993).

Robust optimization model for a class of uncertain linear programs

In the paper, we propose a tractable robust counterpart for solving the uncertain linear optimization problem with correlated uncertainties related to a causal ARMA(p, q) process. This explicit treatment of correlated uncertainties under a time series setting in robust optimization is in contrast to the independent or simple correlated uncertainties assumption in existing literature. under some reasonable assumptions, we establish probabilistic guarantees for the feasibility of the robust solution. Finally, we provide a numerical method for the selection of the parameters which controls the tradeoff among the tractability, the robustness and the optimality of the robust model.

Active set algorithm for mathematical programs with linear complementarity constraints

In the paper, an incomplete active set algorithm is given for mathematical programs with linear complementarity constraints (MPLCC). At each iteration, a finite number of inner-iterations are contained for approximately solving the relaxed nonlinear optimization problem. If the feasible region of the MPLCC is bounded, under the uniform linear independence constraint qualification (LICQ), any cluster point of the sequence generated from the algorithm is a B-stationary point of the MPLCC. Preliminary numerical tests show that the algorithm is promising.

Smooth and semismooth newton methods for constrained approximation and estimation

In the article, we show that the constrained L 2 approximation problem, the positive polynomial interpolation, and the density estimation problems can all be reformulated as a system of smooth or semismooth equations by using Lagrange duality theory. The obtained equations contain integral functions of the same form. The differentiability or (strong) semismoothness of the integral functions and the Hölder continuity of the Jacobian of the integral function were investigated. Then a globalized Newton-type method for solving these problems was introduced. Global convergence and numerical tests for estimating probability density functions with wavelet basis were also given. The research in this article not only strengthened the theoretical results in literatures but also provided a possibility for solving the probability density function estimation problem by Newton-type method.

A smoothing Newton-type method for solving the L2 spectral estimation problem with lower and upper bounds

This paper discusses the L2 spectral estimation problem with lower and upper bounds. To the best of our knowledge, it is unknown if the existing methods for this problem have superlinear convergence property or not. In this paper we propose a nonsmooth equation reformulation for this problem. Then we present a smoothing Newton-type method for solving the resulting system of nonsmooth equations. Global and local superlinear convergence of the proposed method are proved under some mild conditions. Numerical tests show that this method is promising.

Smoothing Newton Method for L1 Soft Margin Data Classification Problem

A smoothing Newton method is given for solving the dual of the l 1 soft margin data classification problem. A new merit function was given to handle the high-dimension variables caused by data mining problems. Preliminary numerical tests show that the algorithm is very promising.