On the theory of option pricing

The objective of this article is to provide an axiomatic framework in order to define the concept of value function for risky operations for which there is no market. There is a market for assets, whose prices are characterized as stochastic processes. The method consists of constructing a portfolio of these assets which will mimic the risks involved in the operation. We follow the terminology of the theory of options although the set-up goes beyond that particular problem.     

The compound binomial model with randomly paying dividends to shareholders and policyholders

Considering surplus of a joint stock insurance company based on compound binomial model, set up thresholds a₁, a₂ for shareholders and policyholders respectively. When surplus is no less than the thresholds, the company randomly pays dividends to shareholders and policyholders with probabilities q₁, q₂ respectively. For this model, we have derived the recursive formulas of both the expected discount penalty function and ruin probability, and the distribution function of the deficit at ruin.     

Variational inequalities and the pricing of American options

This paper is devoted to the derivation of some regularity properties of pricing functions for American options and to the discussion of numerical methods, based on the Bensoussan-Lions methods of variational inequalities. In particular, we provide a complete justification of the so-called Brennan-Schwartz algorithm for the valuation of American put options.Research supported in part by a contract from Banque INDOSUEZ.     

Scheduling rooted forests with communication delays

We show that a greedy algorithm for scheduling unit time jobs on two machines with unit communication delays produces an optimal schedule when the precedence constraints are given by a rooted forest. We also give a min/max relationship for the length of such a schedule. The min/max result (for forests and two machines) shows that the addition of unit communication delays increases the optimal schedule length by at most one.The author thanks Ivan Rival for bringing this problem to his attention. Partially supported by a grant from the Reidler Foundation.     

Ito formula forC 1-functions of semimartingales

Summary We establish an Ito formula forC ¹ functions of processes whose time reversal are semimartingales and forC ¹ functions whose first derivatives are Hölder continuous of any parameter and the process comes out from a stochastic flow of homeomorphism.     

A counterexample in the theory of random orders

We show that the 0–1 law fails in random orders of fixed dimension k, k3. In particular, we give an example of a first-order sentence , in the language of partial orders, which cannot have limiting probability 0 or 1 among random orders of dimension 3.Research supported by ONR grant N00014-85-K-0769     

Connectedness and diameter for random orders of fixed dimension

Let P _k (n) be the (partial) order determined by intersecting k random linear orderings of a set of size n; equivalently, let P _k (n) consist of n points chosen randomly and independently from the unit cube in ^k , with the induced product order. We show for each fixed k>1, that with probability approaching 1 as n, the comparability graph of P _k (n) is connected and has diameter 3.     

One-Dimensional Diffusions and Approximation

Consider the Voronovskaja operator A of a sequence of positive linear operators and let u(t, x) be the solution of the Cauchy problem for A. In the spirit of Altomare’s theory this solution can be studied by using the semigroup (T(t))_{t ≥ 0} generated by A and represented in terms of the operators L_n.One associates to A a stochastic equation; its solution can be also used in order to represent u(t, x).The relations between all these objects are described in the case of the operator A associated with some Meyer-König and Zeller type operators.     

Stochastic flows for SDEs with non-Lipschitz coefficients

We prove the bicontinuity and homeomorphic property of solutions of stochastic differential equations driven by infinite many Brownian motions and with non-Lipschitz coefficients.     

Existence and continuous dependence results for nonlinear differential inclusions with infinite delay

This paper is concerned with nonlinear functional differential inclusions with infinite delay in Banach spaces. Using tools involving the measure of noncompactness and multi-valued fixed point theory, existence and continuous dependence results are obtained, for integral solutions, without the assumption of compactness on the associated nonlinear semigroup.     

A notion of branching rank for semilattices with descending chain condition

Given a maximal subchainC of a semilatticeS, there are some natural leaves ofS attached to it. These are subsemilattices ofS which may have a simpler structure thanS itself. We look atS as build up fromC together with its leaves. Starting with one-point subsemilattices, the (branching) rank ofS is defined to be the least number of steps needed to recoverS. For technical reasons, only semilattices with no infinite descending chains are considered. The main result states that ifR is a subsemilattice ofS and rankS is defined, then rankRrankS. On the other hand, rank does not behave well with respect to epimorphisms. Several examples are presented as well as various results concerning finite semilattices and trees.This work was supported, in part, by NSERC Grant A4044.     

AN OPTION PRICING PROBLEM WITH THEUNDERLYING STOCK PAY1NG DIVIDENDS~

In this paper, a pricing problem of European call options is considered, wbete the underlying stock generates dividends d, at some fixed future dates T, before the expiration date T .without the inappropriate assumption made in that the dlvkdeMs being payed continously.The arbitrage free pricing of the option is determined via a series of partial differential equations.which is derived at the view point of backward s‘tochasric differential ertuation (BBDE). It isshowed how the dividends affect the fair price of the call options. Some simulating results are alsogiven to illust rate the respective in fluence of parameters a.T.r,K.di and F1 on the option pricing.     

A recurrence for linear extensions

The number e(P) of linear extensions of a finite poset P is expressed in terms of e(Q) for certain smaller posets Q. The proof is based on M. Schützengerger's concept of promotions of linear extensions.Partially supported by NSF Grant #DMS-8700995.Partially supported by NSF Grant #DMS-8401376.     

Closed-form pricing formula for exchange option with credit risk

In this paper, we study the valuation of Exchange option with credit risk. Since the over-the-counter (OTC) markets have grown rapidly in size, the counterparty default risk is very important and should be considered for the valuation of options. For modeling of credit risk, we use the structural model of Klein [13]. We derive the closed-form pricing formula for the price of the Exchange option with credit risk via the Mellin transform and provide the experiment results to illustrate the important properties of option with numerical graphs.     

A spectral radius formula for the Fourier transform on compact groups and applications to random walks

Let G be a compact group, not necessarily abelian, let ? be its unitary dual, and for f∈L¹(G), let fn?f∗?∗f denote n-fold convolution of f with itself and f? the Fourier transform of f. In this paper, we derive the following spectral radius formula

     

A Lie-Trotter product formula for Ornstein-Uhlenbeck semigroups in infinite dimensions

We prove a Lie-Trotter product formula for the Ornstein--Uhlenbeck semigroup associated with the stochastic linear Cauchy problem Here A is the generator of a C₀-semigroup on a separable real Banach space E and is an E-valued Brownian motion.     

A spectral radius type formula for approximation numbers of composition operators

For approximation numbers _an(_Cφ)$a_n(C_{\phi })$ of composition operators _Cφ$C_{\phi }$ on weighted analytic Hilbert spaces, including the Hardy, Bergman and Dirichlet cases, with symbol φ   of uniform norm <1, we prove that _limn→∞?[_an(_Cφ)]^1/n=e^{−1/Cap[φ(D)]}${\lim }_{n\to \infty }?{[a_n(C_{\phi })]}^{1/n}={\mathrm{e}}^{-1/\mathrm{Cap}[\phi (\mathbb{D})]}$, where Cap[φ(D)]$\mathrm{Cap}[\phi (\mathbb{D})]$ is the Green capacity of φ(D)$\phi (\mathbb{D})$ in D$\mathbb{D}$. This formula holds also for H^p$H^p$ with 1≤p<∞$1\le p<\infty$.     

Homeomorphism flows for non-Lipschitz stochastic differential equations with jumps

In this paper we study the continuity property as well as the homeomorphism property for the solutions of multidimensional stochastic differential equations with jumps and non-Lipschitz coefficients with respect to the initial values.     

Random orders

Letk andn be positive integers and fix a setS of cardinalityn; letP _k (n) be the (partial) order onS given by the intersection ofk randomly and independently chosen linear orders onS. We begin study of the basic parameters ofP _k (n) (e.g., height, width, number of extremal elements) for fixedk and largen. Our object is to illustrate some techniques for dealing with these random orders and to lay the groundwork for future research, hoping that they will be found to have useful properties not obtainable by known constructions.Supported by NSF grant MCS 84-02054.