System Of Semismooth Equations Research Articles

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 L 2-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.

Combinatorial Structures of Max-Type Functions characterizing the Optimal Solution of the Equivalence Problem

Abstract This paper presents a solution to the so called equivalence problem which was introduced in [2]. The equivalence problem consists of a semismooth system of equations which has the following form: This system can be formulated in a more simplified representation by The vector ũ can be interpreted as possible control parameters of a time-discrete system, whereas the vector x can be seen as bargaining solution of a cooperative game. In [2] the bargaining solution is identical to the τ -value, which was introduced in [3]. For the special case of three actors the τ -value lies always in the core, if we assume 1-convexity. Under these assumptions, all properties are expressed by the general formulation of (2) where the functions f g and h are of the form ξ ( u ) = min { u 1,…, u n } = − max {− u 1 , …, − u n }. By exploiting the combinatorial structure of max-type functions we can show that a solution of (2) exists and may be found via Newton type methods which are treated in [1].

A Globally Convergent Sequential Quadratic Programming Algorithm for Mathematical Programs with Linear Complementarity Constraints

This paper presents a sequential quadratic programming algorithm for computing a stationary point of a mathematical program with linear complementarity constraints. The algorithm is based on a reformulation of the complementarity condition as a system of semismooth equations by means of Fischer-Burmeister functional, combined with a classical penalty function method for solving constrained optimization problems. Global convergence of the algorithm is established under appropriate assumptions. Some preliminary computational results are reported.